Michel Sima net
Spring r
Universitext EdIoWao.rJ (Ni:dr Amiiiui):
S . .... F.W. Gelving P.R. Halmos
_ ... ..... ....... ......."".a-
Springer lHiolo" i &o r ' 5
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Foreword
Inl ..... U" tbeory hoIdo .. prin .. ~u".. , wbetber In f'W'" matbem&ticl or in vvIouI nelda of applied .....,hema,;.:". It pIt.y1 .. oomral ... in ~ It io the t.Ho of probability tl>oory and ~ an indiopmNble tool In matbo> molkoJ pbysIcI.. III pwUcula. in qua", wn mecbank:o ...d " ..i ......1 mechania. Therdo.." • ........,. tu\hoo1::, ~ to inl.f:&l".tlorJ theory a1rMdy .vaIlabk. The ~m book 1:1,' Michel SimonllH dilfus f""" the previouII1.eXU In ma.ny ~fI'I!'C'ta, and, lor Ih&t fHiOIl , It ill to be particulatly """"""",,1denal pr-ob&l);litio!ol. II &Il10 pro. v'deo _ .pplicatlons 0( inlqntion in ano.l)-.it, !ueb .. II>01ocJ, _ fu",," 'Ional --'1*. and • oertllin degree of 1Dll1bem&11cal lIOpllllil.icalioa, Uule it requiml for proftt.blo ra4i"ll oI.llIis teet. The 10p0I0gIeal b.cqn>und n __ :lo:d II oo.uai. for oxamvIo, In Moduro GCtOCal Ih. Ust. by DO " I w10h to \IlCl!I' 1.I
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Onkfftl
Cmupo
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to
In!.Huble and M.. , 11..,:0,.
(- z;}.., It.u t.u lDfi.=, and thell A...,( - :r,) _ - V,..,z,. Dell nitlon 1.1.2 In 6lI oodw\d Jf"O'Jl) G, ,, e C ill -' 0 ("",p«ti"'lJ',:t < 0). If Gil.." ood",ed """". tho eel. 01 1'- pOOIiti>1l eIocIllC!",- ,..;\1 ..u&!1y "" dmoted by So n (- G'"J .. CO) and C+ :J C+ +
cr.
cr
cr_
1.1 O"I""d G"",p" Pro~ltioD
5
1. 1.2 UI. comml>1Gt.." group G lie gtven, "lUIld P lie • fUbuj
"JO ...:10 tMl p n(- p ) .. (OJ alUl P+ P C P_ The ..."'1;"" %::;
~
if 01Ul mlr
if, - '" E P "/ina on <mJ~ .1"",",", compal,bIe ""th the grnp .lnIdM... "I Moretl~" G ;, tUoIl, M"rhrtd if ,,>III nI, iJ G _ P U (- P).
G.
o
PROOF: Thill is obriouo.
An ","",ed oe\ E io oaid 10 be din!CU!d up"'ard (..... pecti""'y. down ..vd) .1>0.......... to. all % E G. V E G. t~ ""OIlS z E G .w.\sfyin& :t ., z and ~ ::; ~ (..... pectively. % ~ ~ and ~ ~ ~). EvelY onieTo.re """"' .. % E G· lhan Z; _ : t .. ~-(~- :t). Coo' .. ody, if % .. ,, -v and~ .. ... - 1 ......... M. v, .... I _ all j)OIili"", then % S" + to and, S" +.... 0
P .... poo.ltlon 1. 1.4 UI (~,) ... , lie" ......empt, fimte Jamil, ift a di....,u./ gmup O. TIIfId,t",,, u...t z " , .,...1 .. Il101 % V , tzUl. :nw:" z+~ " % " ,+z V, _
PROOF: Su,...,.... that 1: " , ""iota. lbm
:r +, - :r '" _ z + , + (- z) V (-,1 _ z V,. Tbe proof IX 11>0.
Hpl",,),. Con,~y, undo.-r hypoth. ",is (Il) (" "I_li""ly. (b)), eo.cb pili, of ~Iommts % E G"'" , V E C + h.M a IUrw-um ( rwpectl""ly, an Infi mum) in G equeJ to il& 'UpEeit/urn a (_p"o" Ii"",>", it.l infimurn ~) in G"'" . Th" .. abvio:>o..os lOr 0; 1Or~, lei. ~ E G boe alcN-."" bound for % and ,. Thoen l!>ere c:xiwl .. E G"'" such lbat : +" E C + (boa. ..... G .. G"'" - C· ). Nooo 11>f.,..(z+u, V+ uj" greater than ~+" and ea." be writtal ~+ c+ II (c 2: 0). Si .... b +c < % and b +c < V, c _ O. lIenc:e b+ "" Info. (z + "., + u) 2: Z +'" &Dd z :5:~. Nat, lei. " E G. V E G boe arbitrary and U E G+ boe ow:lt IMI z + U 2: 0 &Dd W+" 2: 0; UOOtt hypot'-"_ {al (...... ~i""ly, (b)), z +" and , + u have a ",,,,_mu.m (l" + "1.2-
",., Z,., ",.3.
".1
2. Next, _ .. ,~ that 1Il00 theorem holds lor p < m o.Dd q .. n (_ilb '" :> 2, 0 2: 2) and prove lbal It balds for p .. m and q .. n. By
( L ,,) ••• - L .,' 1'S"-'
19S~
Since.he......:lt boIds lOr p .. 2 and q .. n, _"..." find t ..... """'Iue....... (~j,:r;J"'~, (:f) , ,,,, s o In (;+ ... tlsfyillg
r;
&Dd II, ,. oj + lor all 1 :s; j :s; n, Now. since the t~m 10 ' rue fur p _ m _ 1 &Dd q .. ". d ..re "' • dou~ -....... "" (1ad from lhe I...:. elemems /I and • '" ~ ,~. %,) - ,. 0
(E,
~lI nlt lon
1.1.3 For % E C. tM e~t r v O (_Pft'\ively (- r ) VO,"'" "pea;vely (- :r) V:r) ill called tho pooiti"" part of % (h P"'1i>-dy the nqalive part, ,etpec:tively tM abololute value. or variation, of %) and ill .. rittel! :r+
(,...-th"ely %- , )%Il. NOle that . accordi"ll to t t - definit ....... tM nepl.ive part of:r ilia poIitive element. CJearty, :r- .. (- %). and ) - %1 -14 NOle abo tM Iormulu :r V ~ " :t + U - %). and ~ ~ (/I - %) • . "The liM fol1oon ;mnwlia' rly from 1M Cat:\. that lhe ooduins of G .. preoerved under tranol.t;"n ; the ecoond if a ooneeq_ofPIt)jI,,";tlon I.I,S.
%" .. -
Tbeoo e m 1.1.2 (. ) ., .. " . -.,-
"j
~IIO.tr, if z S " ... then:r· - % ill poIitive. II) " . :S " . - '. and z ill nqalive. n""eb'(, "'" fmd that:r+ " %- .. 0 and . by ,r...wation. t~"'" " .. .. ",.
,,+ -;;
•
{( j ' Let z , WEG beP""'" If,, :!> ~ . tbm snd :z - ~ , - . ConV
",. s: ....
,,+ S w" Md :. - ,.. , - .
r :5~· and
,,+-:r s:
-t :5(-%)+ _ :r- .IIO
~+
- , - .. , ... bene>~r
s:
(d ) : z" v ;r > 1z1, btca. - :r :5 ,,- ""d - :E :z- . NIt. If .. ill an upper bound lor % and - :r , Ibm .. ~ 2: z".
s: 161 + " I/)(a._ s: 1"1and " s: IPI· Similarly, - z - , s: Irl + 11'1 · 1I_ 1z s: Izi + l.t, (f): ~II& z and ,In (e) IJ1 ~ and ~ ve Izl - 1, 1 s: 1%- l'I; (e), :z -+ ,
llimilariy,
...
1, 1- 1"1 S Iv -
:t
% -
r l ..
Iz -
,.
....
III.
C
Noce lhal:r .. z · - r"" .. 0 if 1z1- z ·
+,,- .. 0; t hUl I:r] > 0 lor all" E G,
""""rnpe,
Pl"Opoooit ion 1.1. T Ld {", I'E1 /It. Q j"",il'J .n C .../11 "" ;,vtmM'" " .,." lei • E C lot ...-ti1N". n.. .. 1M ,amllv (: V :t;j'EI 1141 "" mfi'""'" "M ..... '{, v ¥.) .. : V(/\"ol " ,j.
p Rfl()r: We ...-y _ _ that , .. O. Noor 1/+ s: z; £or . U i . If a ~ ,. Iooftr boo .." k>r (r j).( /. 111m a :s :r, + :r, for al l i Ii' 1. Now z, fo"',. ( I, i E J ; tb"" a s: :to + ~- lor I E I • • "d a s: II T ,,- .. ~ • . " "' .... lIo
.. -os I
s: ,-
e_,. be l booom.
0
I~,.: "
and v. _ obta.in • limilar poOp04h lo)n. In p&rbeular. if {1M)' lhal: % iII diajoim from , . If A ill . "'~ jc>int and Inlll a-I A~. It I'oIIowI lhat, If A It a I>(W"oen'p"y .... boet 01 E. lhe diljoint complement of A 011 . _ IUhIo,*",
of E.
Oeflnitlon 1.2.2 E It MId to be Oedekind oom~ ... heo~ _h DOlltmpQ- WM 01 E 11>0.1 .. bouodeoo'e bM a suj:ftmwn. &tuMoI~nUy. E .. Dtdttind oom~ w~ t..:h -.empty ..,beet of E that 101 bo:>w>dood
t t'JOoo I>ao an infimJm Pl'po<ion 1.2.1
E;, DtdU:iM """"'I~ if 4M 0fIIJ 1/""" 0/ u.e I~
""" .......i"".., /otIiJ,: (. J E.....,
~pl,
01:.( ......
(6)
.
Mid A 0/ £+ Uuol if dirtcktI _,... and 100_""""
~ .. m.
ew..,. ........1,01' .......'
A 0/ ~
thGl if d,,~UtI.w--..d Iwu on inji .
~
PItOOr. Clewly. dIll conditionll uel\b ''Y. Con_I!, _ _ t hai Qi')Ddillcill (a) holds. and lee. 8 he • l>Oiiotwpd. 10< S, then c - B .. dlrec:led downw.....!_ Dtt>o\In« b)' m .he inRnwm of c- B, we _ thalC-- m It l be IIlIpremwD of B. 0 E""'Y _ _ w bsl*'" of
otdooorfld _ If*"d. N(IOI' let A be a ... beo:t
Propooltlon 1.:.1.2 Lot D "'" 4
...,....mpl~
• ..u.1 01 E'" .1OCh /JI4I
(. ) D+D c DGnd ~)
:r E D."" 0 So , So:r impj, 11 E D .
Lot M "'" 1M od 01 "'preIIIG ollllo~ ........ mpl~ ~" in D IMt G.. "'-dtd E E· om "'" --;U.... ~+ •. 00\(.£, E Mu u.., "'preIII"'" ...... E E"'" .. diljoi1lt frcrn M .
. . . . . ~ <WF)I:r
.1(. E D: v So "I
P IlOOr. By P1opoo1llon 1.1.9. it ....1Iico.I toloo,.,. that ... l: - ~ ie~ from DOt. tqUi...Jmtly. that Ii .. ; 1.1 .. 0 for all t E D. Now. for all ~ E D Ntil(yinJ .. S " , _ have v So ,. aDeT bo1IJod oj tile J_ !41 1-s,1Jtn" ofoo;k JamiPJ ('1;),0 ;n A. Ltl M, be ~ clGM ol 8lq>l"t:fnO oJu..os.. """"""'PIt ."bod, oj M , !hal ""' 6oundtd . . . . in E . 1'1II:n At. _ B;' . PROOf': Indeed, M. C B . Now. ~ti,,« l.l"' be ,be d illjoint comp&.,rnent of A (... , "'Iui>-alettlb". of B ). and taki,,« D .. At, in Pro~tion 1.2.2. we lee that ......,- (1",,_ of I!:"" ill tho. IIUIII of an cle""'nt of M. and an ~ of l.l"'. Beta..... E ill ilic dlrttt Ill'" of B and l.l"', tbe proposition MUon.. 0
knd gcnenLl'}t r E e'" , ~ "",",pone'" oJ", ill B• .. ..,....: 10 oup"" z' inf{ .. lol, z).
Th«>renI 1.2.2 Ul B. /10
e:.
~
PftC>
,.netUed
If nEE, ~ E E .... diojoinl: . ,Ilea ,"" b6Ild.! A . B by ". b, ",,»«:livoely, "'" dOUoinL Inc\eo!d, • ~ to the d.joinl oompk" ..", it' of
(n} ... B C A'.
1.3 Order Dual of a Riesz Space DclInl.J.on 1.3. 1 A 1i!>eV Iorm Lon.., oro.oed _ ~ti"" If L(r ) 2: 0 (in R ) for all % E E+ .
If L ill JXI'Iili~, the n:btlon P~itlon
% -:: ~
implieo! L(-s)
"pace E Is Mid 10 be
':S L(').
1.3.1 Ld E ... ""...-dtml _,tn" 'J'I'm C into R MlU/)cfiIJ M (", + ~) .. AI(", ) + AI(p) "'" all"" ~ E C AM AI (.u) .. WI"')
f.".""" % E C ami Ctoer)"), > O. Then
t/c.erc: aU,", A ",,_lin...- ~
LoIMIE. P ROOf': EIocb • E E can be .. ritt..., % - ~, .. h:' ) + M (,I). The mappi"ll E:t. _ L( . ) io obriowoIY. linear form on E . 0 1.3.3 Ld E '"'" A lli..",W "",/.or 'P<JCO, "mlltl M .. a....."..." fr=o E· into R + ,1IGh tJwu M (%+~) .. M (. ) + M {II) /.". 011 • . II E £+. Then M ""'" ....iqw PQ4iI;'" I;""'r~ I !hI,,..\" I~·
diJJ~"'fUY;
n.. C>f"der i....t n QI E u" DtIIiebn4 ocmpkk~ .. 'I"'«.
P!tOOP: If L .. U - \I , w~ U. \I ..,.., !;orO po&itj~ linear 10,,,,,, on E , tbeD, lor all ,. E E+. the ~ion - ,. S II S ,. lrnplio.'e -U(,.J S U(r) U(..) and -V( .. ) $ \I (W) \1(.. ). lIeno: ILII,)I S Ulol ) + \I (ol), and"" L is oro. ....
....
""'Conveu.el.r, ....
s.
s
that L Ie ordeo--bounded. Wt need onI)' find .. pOIIiti~ ti~a:r bm N tud1 thu Nlz) ~ L(,.) 10, eed> ,. E e+ b ill lbe diKe .."". of t..-o pOIIiti~ Ii.,.,.... bmo. Put M(.,) .. ""Po;s.rS-' L.(u) fur all ,. E £ +. Then M (,. ) + M (:r') .. IUPoS~ • .o:s:"S" L{r + Y') $ AI(.. +:r') for &l1" , :r! E r . Nert, for ..U < ~ 0 S : S z + r , ..., find 1/, Y' E e+ ,ueh tlu.! 0 S. , S z . o S V S :r! , and : .. (mn;illary I £ . n .. p«M'!I tbe fir3t pArt of tlot Ibtotem and ~ O>Or.~r. that M is the ouP"'mum 01"0 and L In n. [t «'oWIIS to be ))«M"Il that n 1$ OedekiOO ..oN .. (,,) lor all " E
r.
P1tOOl' : For ~ry" E .F;+ , put v(,,),. SU""(HU("'), Tlotn vI""' )" .l.u(z:) lor 0.11 " e £+ and all .I. 2: O. Font... """,,, vI,. + ~) = u(,.) + "(W) lor o.Il "', ~ to e+, tM: subopo.ce of .F(o, R). MOii,OOt r. "'" _urne t41, for all I" I, in 11: + _ 'Ii(O, R )+ and tll 9 in H (o.C) IIUCh thal 191 < I , + h tbIS h . Bod"~ + 9'> - g. n..-n. fix"" & Cli".,.,!Qnn" on 1I:(0 , C) &mo\ml, to fixirq! &1'1. R,..llMar mappirq! r from 'Ii(O. R ) Into C . " and ... """tOO by ,,(I) . ?(~ fJ+iT(I",f) lor tll I e 'Ii(O, C ). For IhlI lEI "". "",..-l not disti"l!uish C li""", on 1I:{O. C) from R,.. Unear "",p~,,&, of 1-1(0 , R ) inlO C . J:le6ne a CI"-r for m " OIl H(O .C) ... reel if ,,( f) ill reoJ ror eva)' I in H (O, R), and pojilive if ,,(I) ill pooiti ... ror f:6Cb I in '11 +. 'The"", of R.li"""", forllll on 'Ii(O. R ) will be 0""'..00 br tbe rel.lIon 1', S 1" ... ~ " , (I) 5 1'.(1) ioo' &II I € H+.
f"",.
Ooflnition 1••• 1 A Cli,...,. form I' On 'Ii{O. C ) .. 6It.id to be of finitfl vario;.. tion wi......:"" L(fl • ""Pri'''''III.C).'''I~ 1 11'(9)1 is finlte roo- every I E '11 +. TMorem 1.4.1 ~II' ''''' C ·/......... '''"'' on 'Ii(rl, C) . 01 finik wNlioft. :n..:n u..: ""'pp11Ig I _ L{/) o~ H+ h4I a ~niqw C ·Ii........ ulennon 11'1 ((If' V,, ) to 1M """"" 0/'l1{0 . C l · M"..,.,... •. 11'1 ~ 1M .... 4Ilut olllll ""';Ii. II ....., lornu " on '/i{0 , C) ... t~fri", 11'(g)1 5 "II, )) lor 411 , e '11 (0 . C). 11'1 it m/kd u..: obol..u oo.Nt, or u..: -"olion, 01 1'. PROOF'. Lt< T be tlot unit~i,.de in the (:1S h . and, - " ... ,.; I hUll 11'(1)1 S 1I'(g')1+ 1,,(9,)1 S Ll / ,) + L(h)· SiT"'" 11'1g)! is .... bitrarily do6e 10 h ), we _ thAt L(!, + h) 5 L(!,j + L{/. ), and
,,+
L{!,'"
.. L(f, + It) - LIft) + LIIl). By Propo6ition 1.3.3, L Ms " H (O.C) . oo 11rm .,." ?f(fl . H ) c:om!Ipondi"f t " . i71rm ; 1nMftII"' . ~(f) .. lU~rE "(n.A).t.Is/ ll'(J)1 (10 IkIIt 11'1 ~ t II'!). P IIOO'" The 6m lUlemenl iI obvious. ,,"ow . for fixed
a ..
. up
• f ..m....I.I.I:Sf
f"
11. +. pUl
1,,(, )1 .
I'or e¥ery 9 e 1( fl.CJ salillfyi", 1, 1 S J. IM~ v:iol.l a cnmplex nurnt.o.r ( witb moduli.. I .""" that ~ (,)I .. ,,«(g). Then 11'(, )1 .. ,, (Re(C,» a. So !l
_"'*
ex.",
:s ..
s
:s ..
,
.. be""", we _ lhat 1" II. Daniril mea8U~ &nd .M(O. R) is. 1Ueoz .... ~ 01 OM eO, R). tiat. Id A be. nor",,"p1y ou'-t or M + ., M (O. R )+. di~oo upward , bootInde ! N-. for " larK" eDOU«h. V(J.I ~ ,/2. 110 ,,(f.) ~ t. lIe"c~" Os" Daniel! ,.-ure. 0
Ln·
Ob.enoe thai 1/11 belouc> to .M{O, R ) for ~i)'" E M {O, C ), b«&1.IIIe !PI < I&1'1+ IIt''l'l· In ]n>pO we IIhalI ex"",i"", !alt ). E L , put .... r, . Show ,bat the family {" )' H if ."mm,b!o
E,.,. aDd ,bat Eu-., ... E ... , z •.
4. Lot (z,),., be a family ia E _ (I . ), ... be & &1Oi", partit .... of I . S u _ ,hal.be family (r .).. , • • O\Lm ..... bIo Jor an ). E L. Sbo-.hal {r, ),.., 110 . "m_Ne I . Lot (z,J.. , _ {.. I .. , be , _ ...... mabIe famitx. in E ';Ib lho _ Iade.~, 1. p..,... 1....1 (r,+1' )"" lIoownmableaDd tIW ,{r, +.. ) ..
E.•
E.. ,,,, + L,tI"· 6.
Show u.a. a family (r ,).. , of ~Iiw ok ,..,.......",m'b!o if aDd oalr if" tho ~ I.., : H E T (/)) • boIondod _ _ _ .......... lb. __ . E,., z , _ ... p"....m .... Lot (.. I... , be. r.... iJJo ... E .ucl> IILot 0 S .. :5 '"
Jor oJl ; E I . Show that .be O.
(V" Z f E I , aDd",,_ tllM ' Iz. ): i E I I .. .....' k.! _ _ ia E. Show , ...., lot: ......,. ; E I, r • • ,ho O>diliono {., (b ) (e)
or'1 C '" So 5 io II.... to ",,<WI_ to tho ""'"
~he«I
{.,
calk
IEJ.
(b) 011'1, and (e)
lor
..n", ,E A,>: "
,E J if &ad
""'J' ifzE J and, E J .
.... 5 bo ,110 elMo '" """"imal ~11erS and, 100- - , . z f' A. put S. _ (J f' 5 : z E JI. p,...., ,bat (5.)' - 5 . , &ad .ha< s. ~. So " So U>d 5.v. '"' s. u So lor 011 :00:. J In A. S __ lha< tho _ So korm 'ho _ '" • H '. h lf~""
-
s.
4.
.... Ai C Abo - " ,Ioat 5 ;. U"Ai 5, lor all ~nl"' • . . - . A. of Ai . DoaooI. by 1 tllo of 1 _ • E A b- .,hldl .bent .,,;0,0 • fig ... tot A. C Ai ...... lhal •
A Boo!
A io
E~ """" .... U wdI ''''''' U _ i"'{O) io clos>tn. If U. V "'" diojooint _!OJ' to of X •• _ ti n P ....
."".a..
Pt-ow. .""" ;r X 10 1y d~ • •Iw:n X 10 ..,.&11,. dioon"",,,ctioo. _ ~i1i .....
n.... M' r _
X o.IW,i"!l ...;~) 2:
/ (' ) ·
......... A
..
'0"
inl ,
":or)"
A into 1t .b.a put ..p I I, ) , ... ...,""
01._
U.........1eot USC fu...:t.io ... I" .... I I,) fr 0.11 ~ E A. ...:I , io o:aIled l ho USC ~oriuIloo
" J. II< I.
""',I a doeo '"'" I>olon« to ~M_'" of A _ (r E X : .,(p) < " j : to, 10 l.5C 11 .... .,...,."';~ .hlo.,.., ... put A .. I + , Slmll&rb-.... deli.,. ~I lot all ~ E R ond all I E c-'(X . C""{ X,ltj . "*"",,ittally .... &.0..:1. io. R.;e,;o _ ~ .""". if H ..........."pI,. .._ of C"'( X, 1£). .....,nd-! 000 ... and if I io 1.. _ _ ."' zI ..... (I(z ) ..... ~ .. ~z ) 100 oil z E X ). ,ben ,be USC 'OS ,)...I _ito&>
_
,
lbol 1..t""N
((I/ n)..) .. II rot...,.,.
~ E
£',
I~ ... E: 10 Ac III< oom_1I< in B,. ih_ S ' that . " 10 lho ...... 1· ht.l~ c( z, 1m l bo _ B, «I B, .
10 Let" > lI_li_ln.OredrI., •• )
boll l S i S n . and 10 O~ - r: ,~,S' ~(eA. - ,up,U ,t.). ond 0 < ~ - ..... 1 'f ~ b~ I 'f j 'f n, PIlt A, '" I' E S:f{t)
""pi '"
LeI z E C•. Pf...-..haI.,. ... 9 io pOOil ;'" if and d M!I: B. be lbe band ito E
~
. , o.
I.
B:.
Foo- 0Wfl' z E pu. ~ . ... laf(....,.,) b all I~"" " ~ O. aI>d .... i~ .... 1.... USC fOC\Ilarisat .... of lboo uppor ... . d " h ._ lbat 9, Iml (z, ODd tbtrfti;n lbao, S z . O""h"", .hal ijz ~ ... C-(S.R:\ &nd • • boo in C-(5. Ii) of lboo h • . 1.1", ,", . . .. h ... ,up.,.c•. O$.o'S' h' .
c:
"'pr
:s
:s .(
,».
1.
~ IhM Ih.er. io • uniq ... 1;""'" ""'I> j from C. into C""(S. Ii) .1lCb Ih&t I (z ) _ fr Iot.a % E B: • ..-d _ 1h4.t ~ ;,.., ..... .,.phism from B. """" all r.!enId _ oubipta of C""' (S. 'ft) (oboenoe IMt if z. ~ E B: _1Ji s: Iz. ~ z).
1_ s:
J.
4.
s: s:
Let rEB: _ g E c'"'(s. R) be V-O. oud>.hat 0 9 9>:. """ ....y i""" from lbo...-de"'" _tor .pooo B . OI>to .... ideilI '" C'" (S, I£) S_.IIM V io ...de< _
oudt. .....
i2
t'"
1(.1- I .. fOr oil. E.A..
""""maI._
Let E be .. Dodddnd ch r ' (-hx» and r ' (-",,) do_ in X . and "H( R ) .... """"" . 'POO!!. and (U,).." • """""'P'Y family '" po>oili .... Ii ....... _ " " E. Aoo""", .......... 'opoloc T . dofi""" bf "'" ......i _ U,()zl), io a H· ,...,.....,
-.
i . S"",", ,hat "'" """"""" r _ )z) is un.ilormly <XKLtin ...... I""" 8 ...10 E ond ~ in "'" ~ T .
,ha, r
... ..
'I'
ate
E
LC:owodod .... ;;, F . Not.e Wot A io dir«tn " on .. -Uri.,. S .. .. _~ if and only if, lor &"f . ' '1'' .. "r d..dod iD _ interYal/. 2,3
2.1
Semirings, Rings, and
01 S,~t.. oJIic.\ .... ~ W ~
dUJoi>tl.
PROOr: lei. R.' be tbe claM oll"'- IUbeets 01 0 ... hicb can he finitely par. tIIO:w:d by s.-u. We fusI. Jbow It...! e¥ery IInloa 01. linl\ let A be an $-eM and B,., U's.'S~ B; be .. ~nJ«o union of S·~ A n lJ' _ n, Sis . (A n 8:'), wheno An n: lw in 1\.' ; bcoot A fl lJ' ill an 'R'-... a
_"'pi,
P ropn 2.1.3 Ld Site m......iring in n . F"., ~nr {int S·.a. B. (k E K , K /iniU) ,lid! tIw .... .4., " •• ~ 0/ ........ o/Ih. ,.11 B. _
en.,.
C n.,
PIlDOP: E~ _ A. ) n Al'l lim in 'R for any DOI'""pty IlUbtet J of I , ... -=II £1 can be linMl y pUtitlono:It &p ... lilt . /00 .. 01 .11 nobtll .In dtid .." .. mUm (\u A., ~ f it fin;~ and".,.,...,..pfJ .... eI n~ A. ... A~ Iiu in C '"' .... i e I . ..mil al /trill, _ A. me. i'htll ... it a _',;"g, ."'" 1M "'If ~k4 !of ... it eucll, ~ ring ~uJ '"
C.
PRO(IPC For.u £ • niSI"., A,.,- aad F - ('''!.IS" d ,,"nJ' lie In 4r . On tbe;,u.,;... hand,
U
U
A~ .. _
IAt,n(
A.,.,- In 'i' , the let E n F
n
A, .. )I.
where tbe A!j .. n tn, soSl - ' .4., .. ) ~ diojoim ; ...
U
E n J'C .
Ie
n
A,,,)nA; ... n(
ISiS., ISiS_,
.. the
fini~
n
A. ,. lI
'S. :5j- I
a
union of disjoint ..-au..
Let C and • be .. In P rpo)Oition 1. 1-4. Denot
V the cl.a88 nl finite
of C - ~ (A,)",I be a li nite _ p t y ,..."ily oIlUboIeU of 11, IPUCh u..t at Ieo.ot _ A; lod"'gII to C, and "",h tlw "'tbH A, « .4~ liM I"C for all IE I. PlOt J ..,(1 E 1:.4, ECI . n..n ia«!~
- (il iA.). Il .0
L
K C/_J
(1- 1,0: )
",' _ J
(-I )I KI . l i n.
, )",,, ..
\I ~. ~
A:)
io a line.r oomhil\t.tion of tbe I" (B E V ). If F Ie An R-_lO< $pace and I a rnappi", from Q UIt(I F 01 tho. !arm I:;EI c, , 1,0. (I finite, ... E F. A; € >t). thea, by tbe ... rnmark. J is . . . 01 tho! form L jH II;· 1,,_ (J finite.
-u'"
"'E F. B, E V ),
Dellnltlon 2.1.3 A Mlbclaloo R of P IO) iI oald 10 be. coun, ·hIe unio(0 ) iI • a -ri", if it .. , ......."'pey. if A n B" 11M in R lOr aU A, 8 E 'R, ..00 if , count&ble union of ~ Is an 'R-1It't. In this eue. , counu.bIe ime, _tiontr.ini", C ..00 contained in a(C); il iI 11.. io(r«l by • CUU/lUlbie union of C_ iI . a-ri",. So every elePPlent in a(C) is contained in • munt&ble union of C-u. DefInition 2 .1.4 A lP'-ayatcm in n ilulul P of oubseuoCO sucb thaI An 8 iI.finite or muntablo union of diljint P_. for all p·oeu A, 8 . Deftn ltlon 2.1.5 A
,.,.~
in 0 iI • clut C of subseu of 0 with t be
roilaorillll poopullM:
(. ) F _ E 11M in C, for all C-eeu E, F sucb that E c F. (b) A countable union of disjinl C-eeu itlJl
Obeerve IIw. if . it ...
"·ri,,,.
~Iem
c...t.
Ie ck H :l1rith rESptCI to finite imcr_ tionl. then
Tbeo..,m :l.l.1 (lP' _ A Theorem) LeI P 600 a lP'-.,.1eno """ C !he A·.,.teno ~ 6r P (~ Df all A•.,slmu l VI'( II;) :S Vp( A ) Ior e....-ry - . _ (A')."2; ' of diojolnt S _ whnoe unloo>" contal...,d in an S....,. A. l~ , ioo" ......:b Inl: I, i E I. - {i.}) .uch thallho .. tieI E. z • E"". _ I;. IIIi1A~)1 di""l"J'III . .. bleb II .. coiltrr.dktion. 0
L:...,._I..
T t-..em 2.2.4 V" .. " ...........e lur P IIOOI': Let (A. ).Z'
be ..
et>efJ'...........,,, "" S.
of disjoint S«u ... bote unloo A 11M ID S. We ..-I only """"" IIw VIll A) < r:. ~. V,,(A;). BUI. if (BlhEJ io ... ",bilrfory finlte pr.rtlti.orl of A IDto S«u, Mod> A, n BI (I;>: I, j E J ) 10 tho diQoint unlool of S-«u Z'.i" (J: E K,.j. K'J 6nite) . 10 ~....e-ntt
~ I,,( BI)I
-
~ ~ . ] ;.. "{Z.J..)t
s LL L
1,, (Z'J .. lI
o Propooltlon 2.2.2 UI" k" ...................., (_,;«If, " ..........e) "" S "",, '" tJ\.e ""~ Gdditi .... ~ 01 " to"R.. 17Kn '" ... qIOUi·,..e . " (-rupt>etiwl), " ..........e) "",, akft4I
V",
V".
,.
.
:s
P ROoF' Denote b,- '" u.. addll j,,, exlmsioo of V I' to R. Tt.en I",(AII .... ( A) for all A E ~ . !It) 1' , ;s • qU8OJi._ure U>d VI" S ..... Mut..,nr, ViiI (A ) 2: VII(A) .. l'l{A) for all A E S, 80 VI' ,(A ) .. " ,{ A) for &II A E S. llen
ba.~reon S and (A.)~al • OOq~of S..- Uy n. 1!.Dd"" V ,,{ B ) .. +«>. 0
S io .. "" , .. ,," if ..nd only if it io.,._ additive. For any 8eq""""" (A')'2;' of S--. (00\ ,w 'rily disjoint) """ for A positi"" function" on .. ..,...;n .,,;
s.-
u,
:s
B Indudcd in A.. In Ihla cue, ,,(B ) E.2;I I'(A,). Iv', : j , any Io!c " , be the unique _reo e>::tet>ding " to R.. and. for every intqer n ~ I. pul B... .. B n Ao n (U,LC' , (foj _
ii;.
PItQQ,; 1'\nI.. IlUI'P'*l that ji hall finik variIIt;c.,. Let (A,)~, be a fini te po,rtltioon of A € S inU) $ ....u.. For e"o'ef)' i € I , t~ .... a oomplex number Co .. ill> modul ... 1 .. tisl"yinc I,,{A,)I ., '" ."(,,, ). Then I _ I ... ta1w;". III $ I.. , "" E.."I1rneasurt and V,,( A):S (foj( l .. ) br 0.11 A E S. Ox..., oely.llUpp\* that" if B quas>-_"', CIe&rly.l [ 1'4 (I. ). !, doc, .. 5 • to 0, ii(l.. ) - I'( A) - E,:s's....( A. ) taIdfI U) 0 .. " _ +00. and bffla! " Is II _u~. F'lnaUy, _un", lhat I' III • 1I>eIIOiill 0; ...... I"{f +g) .. ,, ' (Il +"' (~l ond ,,' (M ) _ ,,·,,' (11). Finall). ."..., i...".."..;,.g ."""en.. {f.)~/: , ino7· lou ill """"r "" ..... 1 in 07" , ond ,,' {Il- "'P.~. ,, ' {f.l·
POWOF-. E •..,.,. I e 07+ is the upper ~Iope of an ind, ... boo .. bel.oop 10 I , of ai! inlervlt.la (0, jJJ such that fJ ~ to I. 5 is oalled the lIIIluraillemiri,,« in I . The ,,·ring ~ by S ill _IT "..., 10 be lhol Borel ".o.!&ebr'" In I.
fJ
Defin!tlon 2.3.1 LH p. be ... complex m" . 8' .... on S. A function F from I inlO C is oalled It.Il indeliwle in!"I!ra1 of I' ... F {Pl - F (o ) - 1'(Ia-, M) &II 0 , {J E I oatWring
dp_O
f:'
br all 0.:.,..., I' is a
LeI , > 0 be «I""". A. F is rlgh\.-a)ntinoous at + 6.tJI· Additiooally, roo-~. n 0 ouch that "P. l S p.«o.,Jl..I) + , /2- ', If .... put Jft - (Oft ,Po + 6,,1n /. N.,... tbe quao,;..~.
exi",
interior J: of J~ rtlMi...:: \(I f COOlainol (n •. A.). Md, since the cburt of J io! indinu..~, J:.lhen:~:cisl H, ~ I , .. "n, that J C J., U· . ·uJ., . From p(J) 5 p(J.,) + ... + p(J~. ) 1oI1m.'l! IIQW p({Q,81) :s:: ~, I'«n • . P-.ll +t . .... . io! arbitralY. tJl) .. L~.:!; , lI{(n • •.o..)). and $I> jOlt. mU9'",. 0
1'«0.
°
Dellnl t lon 2.3.2 The unique meMure I' 00 5 ~uch thn p()o,Pl) _ fJ _ JOoall .. , fJ E I aato.t:yiD« 0 :s:: (J, . nd ouch that p({a)) .. 0 ..""... ·,..,r 4 ""~ to f , is called Le~ ,IIOU"", 011 I . To IlOO that I' ""isla, tal.. F{z) _ z in Proposition 2.3.1.
Eu,clse.f for Chapter f I
I .. R ' , ",O(U0p, n '~' ~''''''''J (G,,,,, lor I :!i ; S t ) io ""lied" 1Iq...... Go io iodtj>£'ode,iO of •. Let S "" .be c_ " ~ ioIinc of """ ompIy Od ot'Id 01.l000000 1 I q _ .. htJooo ...... _ h&ve .... '.,..·1 Show ,Iw. S 10 " oo:n>lri .... """'" 'Ioouch A n 8 """Y """ Ile in S lot ubitr..,. A E S. 8 E S.
"""""'i_
.. be ..... , .. _
l
c.Jl S """"" _iril\ll if it """I ...... ,hot""'Ptr.tOt ....:I io' ~ j ui>ed no """ '" tl>< """ F, .
~""t
\, S - lbal. if" oubpwtiliOli of" - 'ilion {E, ),., 10 a
", ~it .... >
u....
( 50 ).. ... "",p&rtic.ioo>, II (Eo)... """ (Fi),.., _ ,_ ",p&rti' ior>o of E. . - ,10&0 {Eo n F,)(•.r') V" ill defh.ed A p...l1 ... fur>« ...... Mid 10 be ,...nqliciblt If itl uppot in..,01 io ....n. n.io 0I1owo .. to oX6... 5PI' ltIIbIo_ aDd tho _10 o f _ 'y ,,,..--..• .,..' y",!wn" . We Ibm I"'I"N • low imp>nu.' .-.1.. 0U(:h . . Bt:ppo !.-i', , hz .. _ It' ",W 3.1 .1). F'Mou'. he (p..,,,... "ion 3.1.2), .00 tho ru--Fiocbor h iD (Theot . 3. 1.3) "" ~_ of £'11cI.ioonI; &Dd dolno 1'.... ter;rabIo _ and ~ u_ leU. U Tltioo -.ct;'" ioo to ,he deIi.l..,., of "·....,..ur-,,",, ioo. ..c. tqr&1 flO",,"io ... ioo~, ......11 .. boundod m< "ra 3..7 FirS..... deIi .... ho _ and ..,.,.. i"'qr&1 of . poOll"" f...... _ I ioo in.. P"bIo If aDd 01017 if ito _ and Iooow i .............. IWte and ... iUiI. Notioo ,hal il l !o IUlqroblo"" ioo VI: m .b. ''-Y. •here it DO "imp ......... i ...... aI". We tbon _ J........ '. 1-..!11y (Tbooo . ", 3.7.3) .,hid> Ia.., impOrt ..... _ ~h in.1oo of ptObobility and in &aa.l"';O.
a.'
,....,;...bIoo "*'"
or
t-,.
po_
""'_04_ . . . .'-"").
" .... I' Ia . I'. in~ ... whk:tl """ .....m&II... (omaJlo.-and _ _ botb in A _"no wI. _ _ ioo Mid to too dIlr••10 of_ & " · 'UI'O. 0.. .ho otbor _ . & '1'",. Mid to too _Ie: if -=to IiOI1nrs1~bIo a....:r&blo
38
intu it i..oiy. 0lI alOIn lew. _ . ,, ~
n.. ......., 1l1li a
•• 000.ho oemiria,s of MU ouboot.o '" N 10 ... ex&DIJlIe (t! s..ctloao 6.3). If I' 10 and I Ia . l\lllCtion from n i _ • .-rizabIo _ , I ioo ....... ...-.bk If &Dd DIy if It ioo .,.,n' .n, . ... 010 _k atom ... <Mit&i ... an
&tom.
.">mir
( 110"", .... 3.111). A Daniooll • ' ''''' I' ~ ...... _ ..... it on ,ho ,1l1li ~ of inlqr-""""'" alkd 'bo maia prIonpIioo of 1'. Siaoll.&rlJ. I' drIi............... ,.. caJl&d .ho_i&I ",·ok I ...... of 1'. "" II>< rlati ~ '" _i&IIr mtqr&blo -.to. W•• hea $ttod;t .....,... ..I& ......... ic>o bot ......... boooo 3..9
3.1
_u_
Upper Integral of a Positive Function
Let 0 be • 0>0DefI\p(y ~. F (O. C ) I"" ......,. of complu. vaIl>Od funct __ on O. and C ) a ....::tor wbo~ of F {O, C ) web thal rk/, 1m I, and !I! bdoq to H (O,C) for &11 1 E H (n. C). A.wnune ,It&t, for.U I ., I. E H + and Eo< all, E H (fl, C ) OlUi"fyi", 1, 1 :s " + hI"" .... etist , •• 9'l E H (fl. C ) ~ud! tlt&t lg,l S I ., 111>1 S I. , and, _,. + 11>. MQo "" .~r. "'!'P'* IIt&t iufU . I) liM in 1f+ br aliI e 11+ (Stone', Q)Ddition). Now If:< ,:r- be. lei. of functlQ"'I from fl into (0, + ooJ ...·ilh t be foIlowl",
nco,
pt OJ>ttt~:
(a ) 07+ :J 11+ """ 1 - 9 boelongtl to 07+ f(>t &11 I E 07+ and 9 E H + IIU:I:h
.hat, :5 f(b) inf{/.,), ..,P{/, , ). and I + 9
bdor!«
to.1+
br &111, 9 E
r .
(e) 1 5 In :7 + ruo:b that I. S ", &Del V" "(,,, ) S V,,"(/.) + t. If I is tile up. per t melop ,.-" .. inf(g,._I. h~) ~. _ . + II. , it k>I~ .hat ~
~ I~_ I ; lina:
infu.. _I,h. ) +
V,,"(g,, _,) + V" "(h.) - V,," ( ioflg,. _1.11.,.)
V,, ' lg,. ) _
:s
V,,· (g. _I) + V,," (h.) - V,."(f. _,l
< V,,·U. )
+;. +€(I - 2"'-, )
V,,·(I.)+t( I - ~)_
:S
11.00. V,,· c,. l :S V,, · (f. ) + { for IlIl n
P ropooltloo :1.1.1 For ' '''"'11
~
l. ILl ~red.
0
~
(1,). 2:1 ollMI>Ctlono 10, ....1. _ ........ V,."O:::'l l !oJ:S E' ~ I V,,"(f. ).
PROOf', If woe puL '" •
E, S":5"- I.
/rom n
ill/
for ~ "
V,."{g.):S
L
V,,"U. ),
IS· S· V,,'
(L!o ) - . up V,.· (g. ) :S L '~ I
·ll
V,.· (b )·
.~,
o
"'ftC-
P ropo.Jtlon :U ,2 (fiotou'. Lemma) LeI (I. I.l I Ito a _ 0/ tionI /rom n into (0. - 1. ~ .. V,,"(limlof. _ __ I. ) :S IiminlV,,· U.)·
.:j,"''''''''
PIIOO" FQlIlII intet;~n n ~ I . Put 90 - inl.1::'>I•••. The (t.,).l l in
V,.· (limin! I. ) _ fliP v,.· (g,.) oS liminf V,."(I. ).
o n..tI.nltloo ':1.1.2 A flmctlot> I from n into (0.+001 III oald 10 be ,.-Df'dlclble .."",... e, V,,' U) _ G. A..,bitt. E 01 n is MId 10 be ""lIf3litible, or 01 V ..... rr F7"'.1/"" 0, if '" to ,,-nqilclblo:.
E. l,l •. and .. ..,p/. , III ,...""&Ilclble for ~ """l""""" (I.)oll of,... DO&Iiaiblo! fulldiont from n into {G. _I. Evef)' s,,1Mt, of a .....phle !Itt is ,..."..tlpble , and • union of
"""o..lIIy many ,.·nqlillbk acto III ,..",,11&11>1&
o..6.n ltlon 3. 1.3 II pn>peI'ty .. oald 10 be true alll108t ""'"')'w l=e fOf " (abl:rrev\ated 1.. 4 ". or ~.) if &rid only if It illrut outaide .. ".negligible 10)1.. P ropoo.lUon 3.1 .3 A. nn 'I h, ~n
from
n
mto [0, +ooJ. II
P ROOF: I..et E .. 1%EO , 1(%) ~ g(l:j). S;""" in(U .,) '"' 1lUp(j,g) 00 £
r
wbHe 1/ 1.tV,, :s ~/ I ...I · Vp(lf;1) ill finite. I n what foIlf:>,.... , "" fOOl> 116 we o::onsIder \.be t!><pre!I8ion I
f
n.e
r
r
Fj.II').
I",(n •
DefInition 3. 1.4 Do:no::M by C~II') the cklllu", of 1f(n. R ) iii F in the ""m;" .... ,ot ,' !"","Il>"" the ,,-I"~ "'ppi~ lrom 0 ;1110 F. N_ 0>0Slde0- C~v.) as • ....::1.0< ",bel*"" of 'M iJemlnonno:d SpooO! .1JII'). Tboa ,be m
F
imo II Bomadi
qo«G 1mIIi/ I E C ~v.), ~"Uol liu m 4(P) and f (Uol}dp - U(f Id,,). Propwitloo 3.1.1 Ld ({~)~> , /It II ~ ... F}(p.)...d< eMt
L N,(I.)
."
1Jw
" limi.l aioto, =d t...t. I{:r) arl>iI .... riI~ o~nOe (J;),,. . i!$elf WIl'UI'*, Th .. P"J""l$ (al. ( b). (el · rf_. tor """". n in N . pul y~'; I" € n : Ih •• , - /;. 1(" ) 2: 1 /2~ 1. From ( 1/2~)· I y. 1/; •• , - h. l. il IoIIooN Ihat (1/ 2")· V,,· (Y~) N,(/••• , - / •.1. wheooo V,, ' (Y~) 2-~. P~I Z~ _ u...,,~ Y..: By cooostruo;boo.. '~P.en _z. 1/;.. , - /;.1 < IJ2" roo: ..u k 2: n. II:) lhe _ .... t h &""Crallerm f ;•• • - f .. (t 2: I) co"'"'gti6 ~Dlformly Oft Z~ Si nce V ,,' (Z~ ) :5 1 /2~- 1. roo: "'''''- > 0 "'" """ cboooe n !ItICh Ib.u V ,,' (Z~ ) :S c. Then. If N .... """",,IWble >Itt sud> IMt {f;, )>>, o:on'UXCOl lo Ion 0 - N. "'" mo.y ub Z .. Z~ U N I'lId 9 .. If.,1+ L,~ , I/; •., - f .. l. which PO""'" (d). a
:s
:s
n-
T~m
:s
e
3.1.3
( Hie """'''''''''' 10 1 E e '(j00."', III - , 1·.tV" Ie.... to 0 .. h exl mdll.,...,.. the fil\.e"r F : ,**,1)', t hie dt/inltlQn doeo depend on ,be d>oi Olhereexist.lgE 11:+. 9 5 1, sud> l hat v ,,"(!) S V,,{g)+ ~ , Then V,,' (!) '" v,,"Cl - 9) + V ,,(g), .o V,," (f - g) S ~. 0
P ro. -It lo" 3.1.11 lUi'l l ,. h) AM infCl,. h ) Ii
I
Isup(l!. hI - ,upU, .,.)I· dV" $
r
(I" - ,,1 + 110 - 9>1)· dV" 5 (.
3.2 Convergence Theorems Theorem 3.2. 1 (Monotone Convergence Theorem) Ld(/~ )~2:'" ... ir.... . (_p&:tiO(Ir. a ,u"..,."mg) ""VUt""" in .c'CII: It). no.... I - "'P. 2:' I. (_fiO(!~. I - inl.2: ' I. ) u inlt9n -oo). In lIIu _ . (/.,.", """""'l'U to> , in 1M: """,n.
i."
.w"
P ROOF : \V.. "XI"8i(\er the r g . .wl' f« all k :!: n. By the rDOntone (;Dn,el~I>O% theorem, I/o .. lnf.~~ "' .. is i"m~ LlkeYioe, the ""'I""""" (9")" ~ L lDtm , aDd ..., ha~ J gn'dV" :!O Inf'2:. f J. -dV I' 'S c:, wben! c .. ~m 10£"_.,,,/,. -dV". ThUll ,uP.;!: . iI... lim Inl,, _ _ In is ".i.o~, a"d f , -dV" ~ / .,w" 'S
PROO':
W~
J
•
a
Domi .... ted Converxen
J /dj..
f
PlIOOf': It IW!io'I'e lhal If, - 1.1. dV" con~p 10 0 .. p .00 q lend 10 +0:>. Fb:: n ~ I. ADd 0>D0ider the 8equentoM 1.110..,..,"', {g,. .. )v. converg.. a111X111 evICPOI!" tIM!
3.2 Coo,uS""" 11« .. ~, ,s.,st. ~, .•. »< •• lies In C}{P} (Proposit ion 3. \.6). Further,
g,. I) z ..
{
I. (>:} if l/ .(>:}1< 2g{>:} if 11.(z )l 2: 2g{z ).
0
NQt, tIM! ""'I""""" (Po>J.~, I_geo .. ~. to I. and 190 1 !i 2g for ~ .. 2: I . The dominated con~ theorem then.bows ,ha' I • inlqrabk []
3 .2.3 (Contlnui'), wll h Respect 10 a Para mel ... ) wI A ..,,, topoIofior/.,..,.., ...... " potnt i" A, ond tl: _ 1(>: , 0 ) ;, ~)
willi..,., W•.
Far oabo!.ul ail >: E n . 1M ""'Pl"ng 0 - I {z , 0 ) ;,
O>IIIi" ........ ,
00 .
(e) :n,.,." .not. ~U 0/ .... "nd ./o=tiofI ~,..".. n inIo 10,+0: , 0 )",,(>:) ""'" A
i nto F ;, O>IIIin ....... aI 00.
P ROO', Th. follow. im....,.;l;"I,ly from t he corollary to n.wo.", 3.2.2.
[]
At llUI point ..... in'rod""" """" IIOUtlon and make _ dorfinillona. If E , F .,.., .....t (or cornpIn) nornv:d 'P"""", for e:very i~ n 2: I , .... eaIl L" (E , F ) the ...........:I opaa! of n linear conlinlIOUI mappillP from E" in\.o F , _ .... put LO(E , F ) .. F. On the other hand , .... ioolJ(:l.lvely define ..... ....,.;1 "I*"! L{" )(E ,Fj u IoIJo-n: LIO)(E , F) .. F , and L{. )\E.F) .. L(E , Lt" - ' )( E , F)) for n 2: I . Then tben! iI an iIIOmoetry i. of L - )( E , F) onIO L"(E , F). defined by
(.;.h))(11,.···, .. ) .. n..)· "
_I ..... 11, .. (h(.. )· .. - ,) ...J.
"0 '
N_ let X be .... optn JUboJet of E, I " mappi"l: from X Into F, .... )lOin!. of X , _ n > I .... intefl"r. 1 iI Mid to be n limos dilJrrmtiable.t" _ to have V " )1(" ) u deri""'-ive of on:Ier nat", i f aDd ooJy If
,
(a)
I
II ( n - I ) ,i..- dilfM'mtiabLo! at mcb point of an open V 01 .. (,,1th V e X );
o~hood
(b) lhe tn&Ppillll :.: - D(~- '1/("') f":!> II, I D-(/(r "))(4 + (~/" )(% gA(.z) for all:r 01 fl _ N.> Thus
anns:
n /.' (H)·-~ ' U
•
0 (n _ J: -I)! , D
(If." ·lj(a + 8(. - (If>:, ,))(z)1 :S go (.,) Le. In [} - N. Now, ror eodI O:S 1t :S n, dtnou by I)(')/, lbo: m&ppina:' /1)1.")(1(." .J)(.). J. ThUll the it~ funaione. I,..,." 1 Into F for m a mal I : 1 _ F .. SIIId to bc ~la.ted .. bent""" On _ h cmp.)o:t>c.t io k-ln~. L:t I : 1 _ F bc reguJalOi on 1 """ the coroilary to lhe domi .... 'oo cn'-ergen'" .~n" " nd il hM. right, lt&nd deri .... tl..., I _ '[""ly," left-hand deriva\i .....) equal to I {r + ) '" lim.,.., . _./IJ) IIe>ll'«tivdy. to 1(n~nee thec>r-em). Talrl"i I -(0. + O.
PIlOOf": ~ ill .. """"'...,. (J').~I in 'H. + cntl""rging \() I in tbe ",..n. Tht.doot, li,,{(t•. I) - inf(J, 1)1 . 0 be .. rerJ number and pmli", A - 1- '0".+001). ,"" func\ions
r
r
II. _ im (I,,,.[/ - r -in! (?,l)j) ~ int.ograble.
domilW«! by fir, aDd """verge point .."", to I" . Thus I" is
a
I~~
Lot / : n -10, +o "
Propooltion 3.3.3
ftin(;.ti(m.
FM' .....,
in's"' n ~ 0, P-' I• .. L,,S,,s.:r-. , Ilk - IllT')' 1.0.,.' ~ A.... .. 1-' {l(k - Iltr. "lrO JQr 1 S k < " .:r and A•.:I'- . , ,_ - I - I ((n. +001). Tht:n tIw:..........,. (I.Jolt. """"""'" I ,.
PAllOr: Otw......
a
Theorem :t.3.1 LeI C~v If
P ROOF: Given ~ > 0, let, E 11(O, Cj be ouch thal t91 ::;: 1 and gl .. dpl ~ I".w,. -~/2 (Propo.;tion 3.U ). Since SI(R.C) II den .. In 4{p.). there e:r;iN II E SI(i . C) RICh lb., f (g - hi . "y, ... hen! P .. doe projection from C (IIlto tbe cLc:.ed. OOD~ IIeI. 1)(0, I) _ {. E C : 1_1:5 1), ..., .....y auppOee Ihl::; I. Then
J
1/ ill"d,.) 2: I
f
gl"dpl-
f
f
(f - 1I)I Ad,, ) ?:
f
1.. . dVp. - f ;
c
.. hence the result.
T'->rem 3.:1.2 Far net) i~ ..1 A . Vp(A) _ SUI>(A.). ~ , I "( A;}I, IOI\erc (A. ).. , t:>tendr _ !lie cJo.u 0/ all fi";~ ".,rtitioru pI A 'IUD mIL" abt. H • •
P ROOP: G;""n t > 0, Iel " E S I(R , C ) be suoh that 1111:5 1 and I JlIt1 .. dpj2: J IAdV p - t . No... h can be .rnt~ I: ,«eie for~., n ~ I . &nd /
"-'01/1>.
"",,;Shes
""UJiRIal ...
C' ""d , , ..... , F .
c
DeHnit iou 3 •• •2 La n be • """""'pi)' ""' . ( n,. FI»)'~1 be. nonemply funlly 01 0,. The ",.rill(!: F ID 0 ~eratal 1»' {,,'(A,I : ; E I . A, E F;} is ca.l1«I the " . ring g.n,,,.ted by tbe famlly (J. ).~/'
3 .••3 I.. I/o.( ...,/ali"" OIl JHfiniliqn '.-l.t. F ;, 1M orn.dlerl OIl IMM: in fI nidi th4J ...m I, ;, .........,u1e"F;. Lt:I (O' ,.:P) /.Io! .. " ._~rcbl~ ~ h /.Io! .. mapping from 0' """ fl. Then h ;, ~ .:PI F Ii ""J nl, if I, o h iI ........nsbI< .:PIF, lor....:/l l . P~ltJ.on
".ring.'
......
PItOOF: By p~tloo> 3.4.2 , h iI meaourable .:P IF if &nd only if U, 01 11)- '( .... ) lioe In F for """'l' i € 1 and eWlr)' A, € F;. 0
If F , ill the .... rin« .....10& generated by
by a cl-. C. of osuboeu of (A, ) : i € 1. A; € C,).
~t.,d
{I,'
0" then T
ill the
'""'''''''P'Y
Deftnltlon 3.•. 3 t..el. (O.}) be a ,,·_ur.bIe lpooreI " . •~ 0111.· ALN, 101 0 100 u.~ prd Ie; I bg;,' q(II n ;,t I G"d B il6 B«r:1 " .a/gUnl. :nw:.. 0 ' EI S. C B. I" ,..,tic1Jar, if 1 ;, (tJI. ..."II) .......taW. aM ...m Ipo'-'Sim! """'" Iuu .. """"tab/./: 1o.tU, Ih<m B _ 0 "" B,.
n.
n.
P ItC)OI': 5i""" Pi is meuurabIo 8/S. for eo.ri> i. lbe idm!.ily of fI ;, it u.. 11>10 IUI>!oet.s of fI of lbe form p~ ' (V.,) n ... n p;:'(V... ) {i ...... i. € I. V.. e U" for """'Y I :5 .t :5 '" """",iluL.e. beeIs fur the product topoIocy, thlt (countable) built ;, ront "ined in 0 .." 8;. N..,. ewry Open eub&el. of 0 lit _ union ('» I rily oount.&bIe) of di$t1Dct .1.u",u" of thil buII.., il liN in 0 iEi 0
s..
-
"Ibe pal. of a ......"'p'y ~ n and .,,•• is' bra F In n
DefInItion 3. 4.& Let (n ,.F) and I! ite Iloor-.I " .lllgd>ra.. f mo 'J •• bleFI 8'· FT
called._MUtabIe
a " "urable~, 0' a t~!Ipe a ""'wing J. S~ IA4t ""d I. ;, mmJ1" that, for e"")' open 1>1'-'. V or 0'.
r '(V) C limi .. f I;'(V)
c r'(V).
(1)
",here limiDf 1;'IV) - U~~ , n,~ . f ;'(V). Indeed, if z E I-'(V), Ii""", f Ir) _ lim,,-_/p{:t) and V ill open , I,(:t) beIorl.p 100 V 6:>r" large eoougb , and the firn inclusion obuins_ Similarly, if z E lim inf I;;' (V), there ~ta "" inleger n II>Cb that I,(z ) belongs 10 V for all " ~ n .!K> I(z) " 1; ...... _ _ 1.(0:) lM$ in V. Thio gl_ the second inclwd let B. be 11", union of thooc Open boJlo of ..,.diU$ I / ~ (",~ ~ E N ) _oed 0.1 poiots of F. B. ill open , IWd F _ B• .. n. 2:,J~. For. if z E /1., lhen n into F which '"4Ireee Wilh
I
l.aA. II
/J.meuurable.
A rw . " ry and 1Uf!iciem cooditioo that .. fUDCtion I from fl into R: be " . meuu ... bIe II I l>I.t 1-'{l T, +001) lie in M f.,.. aU rW numben r , 100-1, tbe Bo.-el O'·tJrebra of R: ill'" ,, -rilll! ttne ...ted by lbe c1_ mnoi81illl! of R: and tbe!lets
/1", _
I·
Propoeltlon 3.5.1 Each I
E.r ;, ".~.
PlIOOf': Let r > 0 be a fHl numbe-r and \e\ E E i. T het-e uiou 9 E .r" Iud! tlwg 2: 1.11" and < +00. lfn > r . . . . inl'l"'". inf(f, ng) bdonp to.1 · and iI intqn.bIe (The:Htm 3.1.4). Tb... Iot~. r' OT._1) n E ..
!"g ..rv"
E n lDf(f, ng)- ' (lr. +oo1) iI/,,"in~.
[]
DefInition 3.5." Let E be .. oonempty _ and 5 .. "mial&ebn in E (i.e. , .. Itmlring ronuinilll! E). A .... pping I from E into .. eel rr II Aid to be S" o>p!e .....ne-.-er tbe", II a finite putitioo ( E, ).EI of E lnto S·oet.I Iud! tlw I II oonot..,11 on ~ E•.
Propoeillon 3. 5.2 A .....pping I from n ...... a ....lrUoWc ",..... F ;, p m" ,*,c!lle if an.cl """ jf, /.". mdr. imfJ' die nib .., E 0/0, I ;, 1M limit a.t. ift E 01 a ~ ol 1l.I E.ft",pk mapping>. P ROOf": Let d be .. distance on F (compatible . ill! tbe IOp
.....""'JU 10
I
p.t. .... A .
P ROO~:
Fi,..., OUPP 0 ~ &i~. F.,.-"""'Y '" ~ I, there «isu &n 1~1e Z_ of A such that V$I(Z_):;: tlr' and ...m thllll"" ""'I.\IeIICI!! {J..... )~~ , co", er"", uniformly 1.0 1_ (Xl A - Z ... : t hen, puttlD« Z • U...;: ' ha~ V /-IfZ) :;: t , and t&d> 1It
,,,beet
z.. , ""
_"('X" unikinnlJ" 1.0 I,. on A - Z_
No.. "" _ 10 the 0_ ... b ~ 1)_
I ...
(Xl
Put Y• ., U,~ , Y,.... for all J> 2: I. "" that n.~, Y, k nogl\ilbie. By indootioon on p , ...., that {I.. )... » comE' .... 10 I (Xl 04 _ N_ 0 ;""" :r; In A U Y.) :r; bf:Lonp A, for A ouil&ble ; 2: I. and then: «itt3 PO ~; """" iliB,:r; d_ not lie in Y... ThEn, for all p 2: po , "" ha", d(J~)(z). 1,(:.:)) < lIt>. lIentt I (z ) is lbe limil oi lhe !,.• (p){:,:). p - +00, and the proof ill complete. 0
2: 1 Iud>
IN (n.>,
J.
w
ul F bot .. rwJ &ruu:A $J>OO I in .c}v.) ifll'""~bIe; Ihllll ihe coJ>ditlon ill aufficienl. COh'.'sely, ~ I : n _ F he ""","urahle and TOOde-re.~. B)' P,op:ai lion 3oS.!>, I io the limi~ -.e. or a _""""" (J.. ).. ~L in St(R. F). BUI ...at I.. ill l he !imi~ a.e. 01 . _ueru:::e in V ( P n;>pO)!Iitioon 3.1.11), 010 I is the limit Le. 01 . _""""" in V (Pmpo:wilion 35.4). [J
Theorem 3. 5.3 A ....... "''11 and nfficiert! o>roden.te ~
r
r
r
r
If E iI B I U","", 01 0, 1 o!dV" _ 0 m _ ,hAl E is loo:eJly ~~l.iciblo. If / and , aft two func:\ions from r:l into [0. +o[ ...hd> qree I.... e. , tben
rand/·,\.dV" - r g·dVl'- lf I,g. aOO II a~ t h_ functiona from ,. 0 ;". n:al numbe"
r{f
tben
+ g)· dV"
J" 1(, +11) -dV" < +00. "Then tbere '""~. E ,,.. th.al " ::: 11.1 + h). P ut ~ _ . '1.1+ 11) .. I...."'.' , + 11 ,. 0 and" < +"." and put ~ _ +oo .. bcre
....:n
r o{g+h) -dVI' r .. -dV",
,+h _0 or .... +00; tho:n It ~ u{g+hl, to> < M.... w,
r
PROO' : (. ) ....d (t.) 1I.t.~ been I"",ted preylo
Prop05itioll 3.6.2
1'lt(M)l": If , e " .. and f gdVjJ !S
Now Itl F be • Bo n"'h from 11 ioto F cueh that
c
O.
1IpA(Ie. and
1T.U) ..
e""'ll
let ~&o) he tbe
r
IIi!t Ql
r l/I· dV" < +
ep4Ilo.e. 10 G f e P,.&o) (",",,,,,,11,,01,, g e .cj.(j.); '" 111;' f pdoo)- I : 11 - F ;, UK>ItWit, ,, ·iPII"9f"Gble il ""d omI~ if il;, ".~~ IfIJV" < + 00. I ;, " . inl .•• =We i/ .."" <mI, if it ;, " •.....km14: UJ'tn,Iiallp jJ.in~_
a"" r
e"" c
I d" .. f I dl'
for all I € Cj.(I" . tbe _ntW ;IItfll'a1 ] I dp. of ...,. I € ~(I') ill \Wall,. .. rillm f Idp.· For e\W)' ~~~.eI: A, fIlA d;< .. then ",ritWO fA I d".
Sinco! ]
•
Obee....e tW. If .. 1tt A iI _nl;.,)lr in~bIe (I.e.. if 1.. iI). TI.eot.", 3.6.1 oboon th.t.t Ail .. union of.., integrable Itt and .. locaJlr ""&li&lble Itt. Noor. put 1111 - sup",>I(n.CI.IfI";' 111(, )1 ,., ~uPIE>I'J";' VI'U ). 10 that 11'1- UVI'I· Dellt, llIo n :S.6.2 ". is saki to be bounded "hEtIC.e. Propooi1lf 1 /ram fI into. r-...I Bom, 1"
°
PI\OOF: We ~ ooJr J)rO'o'e the fi~ cloirn. Oeow: by F' the dual lpoooe 01 F . Lee ( o. a') '" "'(z) :5 0 (a' E F'. 0 E R ), • relation defini", in F • clMM h.t.lf~ E "hlch """w... D . TMn U{.z),(.z),.') '5 o,(.z) for aJl.z E n, ., (/ 1gd", a') '" /(1" ..') . d" '5 J "t"., ..>d J I gdl'1J gp to E. By the Hahn- g·n ..... theorem. D illhe intu_lion 011"," clMM h.t.lf...... "bid! oonw.. it, t.Dd the proof ill mmjWte. []
Propc.-ition 3.B.5 S~PJlOH " II j:IL>Ati .... a,..l kl I : n - (O, +00) .. " . mtIfl.nL",w.,. PI£/. 9(1) '" 1'"(I - '()I, +001») lor ....., raJ nwmlo!r f > 0, and covider~ , I.e ...,... "'........., en)O. +00(. Thcto jd" '" ~f)dA(I).
1"
PIIOOr. FiM,""ppo8fl th.t.t
or {O,I" ... , I.) (with 0
'*' I n. -=-.bIo:. and D(K>I!~r the ....... function / . . . In P'Q908iIIorl 3.3.3. For ~ t ;> O. the 8eq..en I)).,!:, ;' ''''
.. n _
+CO.
1 to (1 ;> I), SO ,," «(1. ;> I») converges 10
Thl:n:ron,.
j "l ·d" '"
,,' (U ;> I))
... j ' I.-iJ
.r.
3.7 Upp« and!.oooer I....... .
71
Definition :1.7.1 For ~ h E.1. p " (h} .. suP"; I'I(Il.R).. s ~pll} iI ailed the upper int~ 01 h.. O~ t hat . .. t..n h I... in .1 • . p" (h.) is iust the Quant ity
r hdp ..hid>
..... _mum in Scdion 3. 1. I! h. E .1 and g E ?t (O. R ) are .udt that /I < h.. then p"(" ) ..
p" (" - g) + $0(/1 ). Hmoo p" (h , + h. ) .. $O " (h,J + $O " (h. } for all h, and h. In .1, and p OeM ) .. ).. p' (h) for e-.·e'l')" ). E R · and e-.'ft)" h E .1. Simlt.rly, p' (ouph,, ) .. IUP.~ , $O' (h A ) for every inc:noui"ll ""'I"""" ("" )A ~ I in .1.
P ropoooltion 3.1.1 A n, ( ..a~ and ~ """",/>on !hal. h E intqnlWl: U !hal. p ' (h ) < +00 . In !IIu .....,. p " (h) .. I h · 4p.
.1 hoe
p-
Suppoee thai; p" (h) < +00. For ~ ~ > 0, t here exisu /I in ?t{fI, R ). 9 S h , such that p(g);:: p" (h) -t. T h O. then: m.I ,, · inl.,.-obk "'Moho". 9 e - .1 and h e .1 IIOdI tiUII,I S , S hand I (h - 9) ' d" S c . We ""'~ /4U 9 pouiti ... VIm I u pcuitiK. P ROOF: ObriooAly. the modition iII~ . No.- .... ppoee that , ill ,,·inl~ and ot-m: that..., may""'''''' I ill ~ti"". Given r > 0, tllMe uist U E ?t ... . ,..,h t hat II - " I, d" S c/4, and v E .1. such tha\ II - ul S " and dp. S ~/2. 1'0.,.. - v :!> 1 - u :!> ", .. he""" u -":!> I S u + " and (u - v) · :!> I :!> u + v. Th f :!> h" lor oJl n ;:: 1 "nd / .. suP.g" .. lnf. h. """"', ~. PlIOOf' : riM . • u~ thai. / .. pI Jl,. :!> / . g is ".lntqrab!oe (by the lhoomm); and ' A)' d" :!> ,,~) . d,. S
""""""'" '""', .. "eo....
1Ii..o. I(J -
1(.... -
I/ n . ...., hIt.... lldo/J - 1gd.jJ - lim,,_+oo l(l- ",,)d'1' _ O. ",hie!> pnPI"eI that ! and 9 are "'IuaI &I...... """'Y"'bere- The argument;' tbe..",., for (II.M ,. In "",."a! . ..., may app/)' tho ... ta>.lilli argument to ""d 1-: B().ben: are two in'f:rywben::
(d - / - SUI't.; - inf II; Theo>
-
alrnoot e>'f:rywbere. _
II!. + II:. Clearly. tbe IIeCIUI:IlOIOI {g.). 2;'
pL"Oper1\a1.
and []
n_
o..llniUo D 3.7.2 for """ry func tion J : J[, /J'(I) - Inf""J' .... ~J/J· (h) and /J"(f) ,. - /J"( - f) .... caIJcd tbo ul'P'" ioto:gtai and tbe lower ioi"eg.l of
I ~
that /J "(f) Ie the same q\l&l>tlty '"' that in Definition 3.1 .1, _1>w J ill positi""
P ropoo ronsi~ tbe """" ...be .... /J "U,) < +00 and /J "(h ) < +00. If h,. h• ..,.., two in~ funw it, + h. 2 " + h .oJ ......t e....ry ... here, ITO /J "(/ ' + h) S /J{h, +h,) - /J{h,) +/J(h, ). and p" (f, T hl :; /J"(/, ) +/J "{/,).
" gUllion, ~ ~ as in e.7 io intqrable &lid that II~ 2: I~ .
For the
I>"
~
T~rn
3. 1.1 . IUI'I""'inc that
a
Theorem 3 .1.1 M\eIIp' {I} " /i'u/e, !htTo: eNtI a .. iftltgrab/e ",,,,,lion /1 ,lid th4l " 2: / and /l{l,) .. /l" (I) ; 1/ h .. ~ " " ",nd .nttgrabk ",,,,,lion nell tho/. h 2: /"ttd /l(h )" /l" (1). !hen / . ~ h olrrw.t ~ A "_'N'''Y ~ttd ~ o:ond>'ticrn rh.o.t/1Io! ;nt es1dle .. tho/. /l , {I ) on4 /l' (I ) N /iniU
""....,.
P ROOF: Suppooc ,,' (I) io finite. ror ev"tf)'" 2: 1. the ... u:iltII lUI inusrable fuDCtIon I>" sud> that II.. 2: / and /l' (f} ~ ,, (I>,, ) ~ /l"{I) + I / n. o-Iy. II _ infn2:' 1>" - infn2:' inf(h , •. ... 1>,,) is intqrable &nd /l' {I) .. ,, (II ). /'0_ Let / " h be"" in tbe II..... • .1tioort. We 1"0,," that /1 _ h .01",,* """")'W1>o. ...: theft iI no ratooion in .... uminc that /, ~ h- for ev"tf)' i E (I , 2). let be • -'-,-..)ued funet~ on.o, equal to /, at poI~!,1w!re /, and h _ IInM, and . up ..... that /, :s J.. We ha"" ,,(h), or /lli. - ft) .. o . ..
i;
i, .. i. u., and /, -
h
,,(i.> ..
u. F\naIly. Let / be. function from n into R: Iud> tbat /l, (I) , /l ' (I} _finite and equal . "rIIo= u:ioI intqrable functions , . " such that 9 ~ / ~ II and " U) .. ,,(h). Then 9 " h .. / al..-t ......-.ywlw!re, 1>0.""" / io inusrable- 0
",to n:
/0""
P~IUon 3. 7.6 f/" ~nd h ~'" ",,,,,tio ... from n .-..ell 11\.01 " + h ;, th/inm .... .. nd i/ p, (It ) + p" (h ) maku -..e, tA.!n /l, (It + h) :S /l,{I,) + ,,' (h ) ~ ,, ' {I, + h )· 1//, ; n "ttd h : n arfttnJ..,.. then 1" (It + h ) - ,,(/ ,) + ,,' (h ) GtId /l, (It + h) .. 141t) + ", (h )·
n: ;,.nltyro.hk
n: ;,
P ROOF: Tto prove that " , {li + h) :S " , {I, )+"" (h ), ...., .....y IUppo:oee ,," (h ) < +0>. Gi""" lUI integrable fu"",1ou II. 2: h. let h. 1>0. • ...aI·valued fuDCtion equal to ~ Le . If 9 io an lntegrloble f""",ion ouch that , < I , + h , _ ha"" , ~ /, + e., or 9 - it, ~ / , u . II foltoor. that /l(' ) - " (,,,) ~ " . (ft). "Jbenof....... p,(f, + h ) - 1'(11.) :5 p, (f,). or " , (f, + h) :s I',{I, ) + 1'(11,). and finally p . (I, + h ) :5 " , (It ) + ,,' Ch )· Tto ",...blilb the lnequalJt)· I' "(/d + ,,' (h ) ~ ,,"Cit + h), ((jllSidel - It and - h . The Iaot .nation now follows easily. a
h. •.
_ R: IIo! neIIlhatl',U) (rupecti~J,. ,,"U») ,:, foniU. 1/, ("""",'i~IW, II) ;, .... i ..';;,£6I. ",,,,,tiM> from n iIUQ ,11(1\ th4l 9 :5 / .nd pU) .. 1',(1) (""f/«filtd). II 2: / and 1'(11 ) _ ,,' U»). theJo 1"(/ - , ) .. 0 atld I"U - 9) .. 1"(/) - 1', U ) ( rupecfuoti•• 1',111 - I) .. 0 "tid 1" (11 - I) .. I"(f) - ,,"(f»).
Prop.!10 tbe"" exiWI n
..
/ JIA",,,,dp. S ,,( A.
n 80 ).
". (e) :!: ,,(A. n 80) .. ,,(A, ) + ,,( 8, ) - ,,(A, U 8,, ) :5 ,,' (11.) + ,,' {B ) - ,,' (C).
,,,,,,,raj
We """ ~ to lhe cue, &ad ~ A" 8 , be t .... intqrablo ouboeu 01. A , 8 tf!IIp/0 lI< ",.~e to rJ\4I it opIU "," ( i.e. , ,,' C B) _ ,,"CA nB) + ,,' (..1' n B ) for all I1ibKtl B 1>/ 0 ). PROOP, 5uppooe A is I""meatluro.ble. and let B bol an a.tbitrary fUb6eI of {}. We &Irsdy know ,hal ,,-(B) ~ ,,- (An B ) + ",- ( A'n B ). If ,,-(8 ) is finite, .... lei. B, bol an intepable ... oonWninj!: B IIUCh tlul.! ,,-(B) = ,,(8,). Then ,,-( B ) .. ,,(8,) .. ",(A n B, ) + ,,( ..Ie n 8, )
2: ,,- (A n B ) + ,,' (..Ie n 8),
,,-(8 ) "'" ,,-( A n 8 ) + ,,-( A' n B }. eon," ..... ly. IUPII"'" that ","(8) '" ,," (A n 8 ) + ,,"l A' n 8 ) £Or $II integrable 90ts B. Put I = Is and 9 ~ I Ans. 5i""" "U) = ,,-(g) + ,,'U - gl, .... """ that 9 Ie Inte&;rablo! (P ropooilioo 3. 7.8). Henoe A is "·""",,,urM>Io,. 0 go
Dell,,[tion 3.1.4 A fW>C\ion I from {} inu. R is qu"';.integmble Cwith reapect t "I if and only If It is ",·"""",,,,,,ble and ,,"U ) .. ",(f). In thiol ''' lloa/ ... (fl ... ,,(g) ma.tu KfU R. ,...ma."....w. aM I"""""'"'u, mch /hat I.ouI """ 1 the .. umber. ,,-(r) ",..I ,.."(1 -) it {iniu; 11= I ..
P ro»Ollltlon 3.7. 14
0'
qooui.;II~.
F()r"
c",,·
P ROOf: Fif«, SUJ)pOle thai. I ill quui-io~. If pCIl II finite, then I iII~ , p' (r ) ....J "'.(1- ) ..... finlte, ....J ", (I) .. ",(1+) _ p(l - ). [f p, (1l - +00, there erist.s an inuvabk function , < I : iii..... :s , -. ...., uoe ",' (1- ) < +00, whet, p. (r ) 2: 1'.(1) _ +00. If ",' (Il- -00, then ",' (r) Is finite, but 1' , (1-) 2: p,(-Il- - ",' (Il - +00. N_h" I be .. in tbe --.d ' w :,tion. We Pf op(l)-op(,)+ a) if D iIIopenandop COOItInuoue. In tM 11m. cue .4 ill .. c...cl ..,.,vex lei. in F J< R , and in tbe teCOd cue il Is an open """""'" au'-!. of F J< R. B, the Hahn- Ban.ocb IIdritlu, then! F J< R .c~ byperpIane ~ (r, " ) ",hlcb doeI not mM,t, DameIy H .. {(I,I): ..(1)+.1.1 _ oJ, .. t...-e .. ill. tonIi""""" Ii_form 011 F and.\, 0 ..... t..o real numben. In fan, H .. ((I,!): u(1 - II) +),(t_ Il) _ 0) ,
""".In
beca""" it coot.lns ( ~, a)_ Now .I.
t- 0, becalJlie u.,P: In the Iir$t (till, "'" koo>w that I"" ~ .. ih D (Propooilion 3.6.4) . In the fIOODI)nd C88C, "" P E l7'. The", eo:isU ... 0, $0 /d" t-- p. Thio; p"-""'l6 lhat / I"" I.,.
f
/e..
\0 D.
!.em,,,,,
N A, n A, is nqligihk too diMil>CI i. j , $0 V"IA, U ___ u A• ., ) _ L,Im. n.... {p(F ): F i~ .ndIul oJ E) U A"""'''''''' ilLkrNl in R.
e-
PltOOv: Let f:+, be ... in Propooition 3.3.1. O~ that Vp(F) .. p(F ) ( ..... poUi¥tly, Vp(F ) .. - p(F ») for all intqrable oubooeu F of 6+ ('&J*'" tively. of e-). by ~ 3.3.2. It we can J>Rl'I'" that the Proposition boIdri for £+ and £:- , tben ..... conclude that (p(F ): F imqn.bIe, F C £} .. [p(£:- ), p(6+ )J. be. it sum10 consider tbe 0 and p(F ) T. which a>nI..-.dicu the dmllition 01 T. He""" T > fJ/2. Now. if io an illlqrable 5Ut-t of e ouch that p(C) S (J. then p(e) S T. So pC£ - C) > {J - T. and (by tbe p •.,.;.ed!ng argument) tbHe " an in~able subed 0 aI E - e for whid> «(J - T)f2 S 1'(0) S (J - T. from the fact that _ h&voe p(e U OJ S 1. and ,,(C) + fP - 1)/2 S 1. Hence 1, which proves Ih&t 1 .. (J.
n.e..
,,>(:..
e
ot:!
of
P~ltlon
lhat (f").~ , con'''rplIlo I on A-N. For all z € A- N, t he 8eqUffiOl! (f.(:t)).l!:' ., (""),, Illal each alOm contained In £ ie equioalcnt to oat ond only (lDIl Ai . n,., $et £ - U o A.. OOlllai ... "" &lOt """'1 j E I, thmo ;"" nestillible ou!.e\ N. of A. sud! Illal I II oo:JSWot oro At - N,. Now Jet; N boo a ,Itglisible suboolt. of E, oonWning E A" No, a Dd U'';)Eh/ .,,.J A, n A,. elcwly, I 19 the
U.E/
lhE,
limit 01> E - N of • ""'1_ of iiI £ -almp\c mappillf\l, which
II
~"-n.bio.
Now 18
I
be a fUDdion from fI inlll
r
l·dV,.S
"",,-eo
lhat. I
iO. +001· Then
L I'I"I~'
r
11,, ·dVp.
e""Y ..hen: lor O. then: eo:;sU , E:r """" that 9
HI!fII:e
I~
~ f
aDd
~
which P"""'" Illat
r I ' dV,.
".measurable.
r I· dVp +00. Thm! E.1ICIt I . II.
No. "U~ tlLat p.~rau,
[
tel
lP.XiIu a I'.meuurable and loo:aUy dmoet ~here.. So
E'1where. Idp .. ! II.dp .. LoIA)ET.! IIAdp .. EoIA)ET! I IAdp by the domi ...,ed con'dlE"'" 'beorem. C
Then!
3.9
Prolongations of IJ.
on a ouniri", Sinn. and " iI the ~.-c funn I ..... ! IdA Oti SI(S,C). tben we put I . n of ".
r
Tbeorem 3.9.2 I· dV,. ... 1'1' dVl' lor ~ /undW>" I {rom n .nto jO,+CCl). C~(Ii) - 4(1') {tor-emIIl.9.1 &nd 3.9.2.
a
Of particular I n _ is the foliowin,s problem. Oi"", a rneas=e 1'4 on a ... in n , ",!>en ill it true thal,. _ fl. (resp«t;""ly. r. = r.. )1 A panial ill Ii- in P,OpOeitioool 3_9_2 and 3.9_3_
~ &I ........
Pl'OpOOIltlon 3.11.2 u! 41 III! a .oemir'ing of uunl;"U~ ".irlt.,..,blo ..u. and ee I " IDeally jJ+-aIn>O!it en'1wbm! lhe limil Dr (/. ). 2:" il belc>ngs to !h(/J+ ). 0
Propoeillon 3.11.3 U I • /or ~ oem;,;"g 01 p ·ink., W. Kto "nd k l jJ+ /or tJw. -.we E I,, · dIJ "" • . 5, , _ IIi4l SI (• . C ) it..:e.... in v.). n- £bCJ'+ ) c 4:v.); ~, f IdVJJ+ " fl ' dVjJ "nd f I~ " IdIJ /t)f' 0Il1 I E £hCJJ. )· Firn>JI" [J. .. ,.. if and oml. if eocA J' . ~ K' ..
f
£)"
JJ+-Mgl;gibk. PROOF: A""", III ill
P~tion
o
3.9.2.
The next. rerult . ttIOti ....t«l by Propo;::oait;on 3_9.3, deaIoo ... ith "",,,iI'inp 4o fur .,hicb St(. , C) io de_ in £h(jJ).
u,
_t.
,W.
PropGIIltlon 3.11.4 5 , • W 1000 ,tlJI' D/ jJ · ;nlL!§1 Kto, and lei • W W " . 1"ifIg ~ .,. ... 5_K IIIents o f T... lben: ""isl J.. h e Sf~ (.) $t>dI tlw li D, - /.1. dV" ~ t !'liQr \M!ry 1 :.::: i .,: 1, and then
r
r
11OIP( 18, . I.., ) - OUP(/L ./.11· dV p
:s: t .
wbich ~ 1M! B, U B, licoI in T... Finally. Jet ( B. ). ~I be an ;,..:>cni'l\! 1ItqUC>QII in T A, and. put B .. U.~18"; the...eq..eno:e (I";').~ I ron'~rpll.O Is in 4,{p), "" Is beionglI to V . N"", TA job. ".aIg0 _ _ _ ,g........ I~ then stto..'& IIw I s. &I>d benc:e I • • lies In V. FiDa.lty, t hat SI{S, C) c V and. V - 4 {P). D
n
obw;,..,
EurciIu for Chapter 9 I
at " . & ..." intogr&blo &ad tIW 4 . , "" to ...w &ad _p, s.._ f w".
I
ill . ... _
1_ E- c"'l).
_n o
tho .. (f.). ~ • ..,., ... ,..,.
Lot" ..... poeiU,. ""_jell m ' w .... _ H (n . C). Let (""). ~ I boo .. ,,",,""""'" or MI' I .. of n. Su_ thoro 10 .. _""""" ( B.. ).~, of diojoi .... """" tbot A. C B.. lor ~ n?:}. Prtwoo .hat ". ( U.~ l A. ) '"' L.~, ,,· (A. )
,.·_IlBbIo ....
&ad .1oM ". (Un!,
.t,.) '"' E.~, ".C..... ,·
F
.1("'. lor Ch'p''' 3
M
a,
5 I.-" 1>0; . . . . . fl... · ~ "' ''l oa "H(fl, C) ""'" lot 1; bo, ... lntqr.blo"",," SIoow lMI (P( 8 ) : B ....... &bIo. B e E ) io """'poo:t. ( _ ~"'" U .2).
t.;
'V'
ate
4 Lebesgue Measure on R
A. ,bIo pOi ... iI
--
uampkot.
o.:.a. ....
b,o,,, Qe !Ip.. , - 6- ' :!S :t .: 6(p" • • + 1)' 6, n, ..., ha,,, 0 "a 6 - 1. N_ T • .. PI> + '5' S_ ... hence Z .. l1li + L' ;!.I ... /b".
s:
.s
E
'V.
.s
.s
DdInitlon " , 1.1 With III)tatioOD. all -'vM. (po. (10. )'~ 1 1 "' called lbe propo:. ",po."';';" of % (wltb l&\ptd to the bMo 6).
•
( .1 HI ,: bE.""","""" 01 .. Real N"",,*
87
to, I, 2, ... ,~ -
I}. By ..,hat ~" h..,,, just shown, the function ..,: (9o.{ .... )~". CleAtly, ,_ ~ .z ~ ,_ + 1/ ""' , .00 .z + I /~- un_ .... .. 1- I for any" > m. We ooociude lhat r ... '" ' .. 0< r ........ + 1/ 6- . lbe ,""uer equality holding ooly w!oeo " .. .. 6 - I for all i~ " > m. If the oumMr of in~1S n ::!: I .ud> IhaI. tI.. < b - I ill fmite, e.nd if Ir\ io t!oe .mallesl pOsiti ... integer such that ...... b - I for rI > m . t hen :r .. 911 + L ':5 " :5" .... /11' + I /b'" h.oo the form Itl b- (k E Z ). ThUl, if:r ill DOl. 01 1M form k/ ..... (till. ( .... ) .. i!: ,) is .... orily 1M 1"""'" ex.,.nsioo of :r. N_ IUppOot thal z h.oo the form k/II'. TIw: ... an: lwo Pm. In abort • ..,hen" hail the form It/ b" , it ill lhe I",. under.., of at most 1M) .",.",,",.00 In fOCI of eunly 1M) and OUt claim ill true. A real number.z h.oo a ~rminating ""panoion" (ruch that .... _ 0 lor rI Jarp CDOU&b) If and ooIy if II has lhe form It/ .... (n 2: O. It E Z ), and in that " from AN onto (0, I I iou . . f'urtba ......... for " . W IE ,,"" _Wylng" < " , lhe equalily G(,,) .. G (u) boIds if uod 0IlIy if II. " J«, ".pl>Cti~ly. the ",Opel tot.,.".;oo u>d the iJDptOpO:'t e:x.,.".;on of ........ .z eJO, Ifn B , where B .. {It/ II' ,n::!: I. "E Z}.
.let".",,".
«.... )..
Deflnlt ion 4. 1.2 When ~ ... 2 (,... pecti~y. b _ 3. ~ ... 10). the proper ~ 6 e:xpanSion of" iI called iu proper dyadic (I EE Ji
ei..... In'
if 5 n 2: 0 atId 1 $ P $ '2""'. let ("" )'2: ' "" tbe proper lriadic: e>-...... inn ollt(K ...I. N"" let ("")' ~' he a pOint of B atId :r ito image under ~. In o.-der that", IN: In K .... II Io"i« pry and JU/Iiclf:n' lhat """ "r lho: IolIowiu& IIImo DDOd.iIIooo bold:
(al :r. _""
1or~ 1
$ /0 $ ,,+1 .
(b) Q( K~ ... ) '" 0 and ("" )..?I is i;s improper lriadic~.
(e) i1(K ",,) '" I aDd (:r. )o ~, is
il$ proper tri&dio: ""p&It$<m.
In the pArtiot:ula:r calle ,.~ (:nlaiuM in K. C..,,,"( ,acly, let z E K . For ~ n 2: 0, ",rite.:o A lor t he ori&;in or the K .... (:()Otal~ill(l:l'. TWo .:oA'" - a"
.s
has tbe form t: •• 2/30+ 2 for alUiuble l:~~2 in {O. l }. whma! ... deduce that z liM in ~(C). Therefor., K '" V(C) has t he ~&rdin.oJity of t be cooUnllum. Now dmnte t.,o k \be tel. of \be P( KA . . ). If t: lies in k , thero iU impr~ tn.dic expanelon be' 00p 10 C. On tbe d if there eoiltllan intqer n ?: 0.1lCb that l: . '" 2 for all I: > n + 2, thero ¢( l:. ). ) '" P( K... ) fur a.uitable 1 S p S 2·+1; th..., it:. ila pnint of
K - k , thero
ill proper t n.dic ~~po.osion bekmp \ 0 C. F in.oJly, a nee
' '1
and IUfficlenl. condition that Z E K be tbe i""'«f' ~ ~ of two diotit>s .. (", . / 2). 2"'- > + I , thero z belonp 10 I.... and g(l:) - / (11( /", .. whma! ... _ that
».
ThillMt equality il t tut even ... ben z; liel in K . He""" ... ha,'e obcained , be eoplicit ex pc . on of g(l:) for alll: E 1. Ob8crvo! that 9 iI equal to (2p-I)/2"'" 011 NCb f ..... Pro_Ilion 4.2.2 F", ~ n 2: O. plOl K. '" U,s.s,. *' K . .... .. nd IIIriU I K. "" tJw: indio>,.,.. 0/ K • .... ,. Atoo •• o' io. '" (3/2)'+ ' . I K.. tiM let '" be 1M "'n II>~ C~nlM.o"f'l/ conootWrl component of
/ - K. Nw let In ). be an e\o:menl. of C ~nd Z '" \!t( z. ). ). For any ill~ II
o
"'' 'l1lOIit
In slK:rt, , is com.io ....... Inc...... nl':, and has cieri_I", zero elIerywbere, &11Cliono g._ By P"""",ition 12. U , 9 h-. deri .... li'" +00 &I. UOOClUlltably """'Y points
on
4.3
Example of a Nonmeasurable Set
For &oy ... boott E ol R and any mol numl:>er ... .." define E + .. as t be IOtt £ EI . Man "'......Ily, lor AI»'''''-I& E, For R , "'" let E + F be tbe....c ( z + __ ; Z £ E, r £ Fl. Fi....uy, we " 'riU, .I. to.: U:bo1Iiguo _ure on
('" + .. ,"
"P rovo-Ilion 4.S.1 lJ E ;., .. '\'·in~ . 0, IMn D(E ) . (z - ~: z ,1'£ E) u .. ~ ¢fO . P RODP"; For AI»' mol numl:>e< 0 en :r. , . .. ,Zt, theN: are &I. \e&et &11 Ihat "', - z,1is strictly Ieso IhaD 2/ t , and
:s ...
t,...,...,:t,
tben there exisu r E Z for ...·bicb r(I, - I, ) helon&o to J . Thil provts that B ill dtnoe in R. CleM!y, C .. B + I 11 also
-I.n"",philm, ... ith in....... Il-'. Sinoo -I(II{/" .. )) .. ~(I" .. )/2 lor all itLI."Fft" > 0 and I :S p:S 2". ~ _ that -1 (11(/- K)) .. 1/ 2. TI.e. dOLt, -I(II(K J) " 1/ 2, e-.~" thou&h the C..,tor eet K i1-1.nql~bIe. WIth notation loS in Propoeition 4.3.2. E .. h -' (AI n h{ K ») ;. -I."""i«\bIe, ,-""" it iI included In K , but ill ima«e M n h(K ) uno:Ier Ills not ~ measurable. Finally, E ;. not. ~I oet, '-"\lie h{E) ill not _ BlIreIIOt. C
Propoe.lt lon 4.3.4 Ld AI Ioe '" in I'topolilin , .j.t an
p( (M n Ed U (M< n Eo») ~ ~~"(E,) + ~ ~"(Eo) ~U,
ufinu a " ....u.m.e ,...d ..... p "" T ""'" Ih4I P(E) .. -I "(£) Ivr
tdlEe M . P ROOP : Si""" -I ' (M n E, ) .. -I' (Ed and -I' (.W n E,) .. Eo In M. the pr-,tion iI oIwiouf. Thull
.i. ;. not •
~'
(E,J lor all E" C
maximal extellllioo of ~.
1 Lot II be lhe functioD:r .... :r - ~ + 1/21 .... R, wbo<e (:r + 1/21
Le\ ~ > I blope! t , ' b."pt.nOioo>. Show ,tat " .. . a!\oDaI if _ IH1iy if t lltte uiI. ;·"V" ...
D(K)_ 1_1 . 1!. Oom-",.ho «>11 """ 1 ""-th P , opoohloft 4 .3. 1.
01 Propooi.1ooo c.3.2. if G" .•. . G. are poi .... 01 A • • _ .... (U, ~ .""(L +0,) .. 0. 1 _ . An D(U' ~I ,,"(L .,. ... » 0>1>1 ..... 110 ",hoe< t.. ... - 0, (I :S i. j :S 1'). In
~ ~ioII
pol,,,. ....... . I.
Lot { ... ).~ , be an ~""""ion of.ho poiDto in A. I'U - " i ...... p ~ D. I"'t F. " U.~.(L+o. l. Silo-(F')'~ l o p'" POL " 0( ab60. U hao IIw! .. ruct~ of • _ .poooe....d 1 _ N, (J) _ io . oemi-.., ..... " ~ I . V , whld> io I~ """'..... oJ. £' ..uh , _, 10 tho ...... ioo f _ ~
io. 8 8""", _ U> ~ I ). In , hIo ~ .....m ulef>d ...... of t be .....110 01 Soctio '-2 ( ..... ,L . . _ ) to.bceo _ T'be FiodIoed.""" n . . ' . hoof . ... 01 55: P'II'" ,"""'" In that 1\ oJ.,." _ I _ ..n tbr. tooIo of Be" 5 :~ _ _ tlw:<J. jr&.& ,
,.t... ".
V.,.,
U
!D''''' _10m ~ """"" .....oJ ~,nd·""",tall"""l...JI\"" Fb< "''''mpioo. if,., *'"
I! , _ , / . t ,'",. to (0,+00» . ..... if I . 9 1_ funo\ ..... " " f} _ lhal N.{f ) aDd N,CtI) &aito., .bon N,{fg) :5 N.{I)N.(f) (P, ........... t.Ioo> &. 1.2). 'l"borom 1.1 ... ......· oli ..... ioo 01. Minlo:Nol- 0 ... d B '" gd" >- O. From the 1""'IUNlly (f I A)" . ul B'I =:: (of)/ A + (h)1 B , It folkrn that P ROOP: We m&y JUJ)p(l!Ie lhat
'f· r l' d" =:: ~ f· ' f·
AO .B'
fdp.+B
yd;J._o+ fJ _l ,
wbenoole e E (0. + ooJ such that 1/ (" )1 =:: c loct.lly " ....Jrnoot eve.,...hcn:. 111"" """,(f ) =:: M.. (f ) and N,..(f ) _ M .. (l f l) when"" O.
For &II n:&I numl>en! p >-
(I and oJI
P ropoo!ltlon 5.1.2 If p , q, r ""' ill 10. +coJ, ...m IIuu 11r ~ III' + l/ q, a"d il / , I ""' 1_ ",..me", fr=o fl into jO,+oo( . 1IdI !hat N,( fl end N.(g) e,,"
finite, Ihtll N, (f,l) s: N~(f)N, (j). A n«:tua~ An '. &lid
s:
Ilmsup_ _ ,V, !!) S N""It). Henof!. If +00 1:>1:1"9
I.(>
1, J oont.i"" ",bi·
Ifarily ~ nwolle .... which prow!II tl.a.t I is an inte.voJ of 1'(. N(>W .U~ that J is not" pOint , and Let ~ be ill orI&In and , ito endpoint (r P08" lbad' is DOlle"'",),. We I'f'O"" t hat P - f !'dp is inlinlUOly dilf....u iable 10 .r and 111&1 III n'~ den ..... is P - J /". kig" / - dp. P ul A _ (z En: 0 < J{TC) < I) and B _ lz E rI : I(z ) ~ 1) _ For a fixed I> e \c\ r. t be e\e ...""u of J.ucb that r < I> < t, and cIIooo;o. real number 6 > 0 eo t hat r < p _ 6 < p + 6 < •. Thf: fuoctionl! I _ It -Iog~ II from IO. I[ Into R and 1 .... 1- ·· w,g" 1 from [I.+oo[ into R an: bounded by """,,"e>O.Then
'i,"
r,
II .. ·
r ·1og"/1.. 11.. · r-··r .log" I I :5 c · r-··1.. :5 ,,/, 1..
1,, ·
r .log~ 1- I,,· r" -r' -log" 1 :5 c- r+' -1,,:5 " /' la.
eo Ir· k< I I :5 ,,/"1 .. +"I'la for all q satisfying r :5 q - 6 < q + 6 :5 doeIraI a>ndU!lion foI~ bjI tbe ll\andard ...... Ita 01> dilf. ",ntiation i~ sign (Theonlm 3.2.3).
•.
The tbe 0
"00.,
r;. I. ~ 5,,1'_ tMl r ind" ,. I. UII : rI - [0. +en .,,, ""d . /(. ~ an: (llta, "" N,(f'I:5 N. /,(f'I N./C. _,)(1) and N,(!) :5 "'. (f). Thus. if J 10 not.. mpty, Ito infimum 10 0 and tbe function I> _ Np(f) illnc""";ng on J. Put A ,., ('" E n : 0 :5 / (>') < 1) and B,. ('" E rI : I (",)?: I). Foc every u of 10, +00[, (u' - 1)/1> deer : 7' ,," 10 log u as I> goeo to 0 in 10. +00[_11ldeed, fur r , • E +-oo[ oru(, ncr Fio (1/ , ) .
llf(r -
r
r
IocN,(f) -
~ .\otI(! rd;J ) :!O ~(f rd" -
for all" €jO ••,...hlch Ieods 10 \otI(Lim N,(f»
-
~ \otI( N,(f)) s.
1) -
f /';
! " Ie obvlouo for a _ • _ 0; for Q H > 0, tho: a.b<M! ...." be written (0/(0 H))' + (6J{oH)), 2: L .. hich follows from tbe f.-t .hat (../ (.. ""," ~)t 2; o/(a""," b), (6/(0 -+ 6),. 2; b/ (a + b), &Do:! a}(a + Ii) + Ii/Ie. + Ii) ., \. The nw.:.. we h.&.~ U + g)' - "p(ln + gl l },:S "Jrl. (f' + 9"). and"" "',{f + g) Is liniw. SiDoe "',{V + gY-') < +0:1,
Similarly.
N,{g. (I + gr' )
«I
:s N.u )· ( {
V + 9tdll) '{. ,
s:
From \he f_ tbat. N, + g}') (N,(f) + N,(g)) . (r (f + 9}",s. 1_for tlYI::ry " 2: l. s: 2..., ,s>,s.. N,(f_) by MinJo:.......i'.
Then N .C",) N,{E .~ , I.) - Iim,,_+oo N,{g. ) is ..... . han
inaqua!ity. Therefore,
L.i1;1 "',(I. ).
[J
DclInlt lon 5.1. 1 Loet F he .. JUJ 8anach space and P ~ I .. ..,a1 number. We define by ppu.) (~ively, .c~(P» the space ol tbase mappillgll I from rl
into F IUd> tW N,{/) .. (r 1/1'&:4e_
99
Clearly. ?,u.) ADd £'f.u.) .... vector ~pa, I. (s ) ....... ""'VU tWo/toul_ ".e. If.,. F"'/{:z) - L.>". (S) .1 GImM- ..1I poi"" s ~ tIU$ """'"' ""'....-ga, and ta.te I {s ) E 1" arlJitnlril, ~. tht:n I 1Ie/onglID ?,u.l .. nd N. (! - L,so". 10) 5 L.~. +, N,{!.l 1<Jr ail" ~ o.
The,..,
n-.,tm:.
PItOOP:
the.rnu L.2:' I.
A~
"'' ' ' '".91 ,.w. IMJ N,(g) < +oc> "nd 1/.1 :5 /I .. e, I~ e-y n 2: I, TIItn / belong, '" 4(si) 21/ (:0 )[· Tho!> !Io "'longs to SeIR. F) and [g.,1< 21f l. M",oo"", (g., 1. thaI I :5 m :5 n. we haw P ROOr : SIDOO / - "
1:. +(/. - 1.. '1:5 /:
••
~.2
Coo...... .11>000....
101
-.1>eIIot
He!IClt. if euP'2: ' N.(f~J < +00, tllm (f. ).~. iI a c.uduOur tl>oorm3.-- foIlowi i~iately from Tbeoo~m ~.2. 1 .
0
p ...."...ltlon ~ . 2 . 2 lhfine "" orrlt:r OIl t..~(p) .., /IImIIo: j '" 9 if ~nd onIW if I "" ..eo, .nd /(1 II ...t-diruLtd nNel 0/ L~(P), anuUIing 0/ _ iliw~. II Nu .. .....,.......""'~ in L~(P) ~!of - .uPw.doJooo! o/IM 90. ;, 1'lqIUlI • . e. to .n~ "" ",'lforn.: . / b. L~ (Ii) ;, .. DoIdind """p/de Ria: 1JiII«.
be.......
"'''oerpu
_,,"';(1,
PROOF: A..~_ tha, M iI finile. Si""", II iI dirt'C\ed uP""l'fd. there exiI~ an lr...... P"ll (~. ).~, In II .ud! tlw M' - 1/ 2'" '" N,(... )" lOr all .. JIo - 90 )· .. IUp(Io. g,,) - g" , il 101;".0.1"", ...... {(II - g. )* )' '" N, { IUI>(Io, ,,.J) - N,(t-l" '" M' - N,(g,."- 'f 1/2"- !cor all i~ ..
-.1>eIIot
o ~ thaI , in ",,,.. ,0.1. C;'(Il) iI not Dedekind complete. For il>8l.anOe. if _ ....... for p Lebft!gue rneuure on and let E he .. l ube« 01. 10, which
10, II
II
iI no( p-measurlble, lhen the "" II of indicalOr functions of finite JUboeu of E hu 1>0 JUpremum In C;'(P). For every real B!oo ..... spece F . denote by C;:{P) lhespece of p·rneaeursble mappinp from n Into F sud> t.luIt N",,(f ) _ N.. (tlll iI 6nite. and !!quip G'{P ) .,jI b the seminorm / .... N",,(f). Write L f(P ) for lhe quotient .pace of Cf(p ) by the equioalen 9, "'hk:h is t""~fon: ,,· measurahle. Clearly, gill bound..:l.. Finally, N.. (I. - ,) .. N..(g., - gl COD'd gb to 0,.. n _ +Oi>, ..hi.2.5 Ltt" 10:. """,pkt O""i. 1I ~ "" 1t(O.C) . Gm.n a &uu.dN>g ..... tin ...'" Ii..... r fo .... en l..~(p), 1IIAo~ """" ;, N.(g).
P AQOF: We
prO'V: I !O k !O fI , l bere exisU a, E F' such thai. 1",1" .. 1".1' if p > I (ff'!;per;ti.-.ly. I"~ I .. I if p .. I), .... . G~ ill r-e&I, and Go ' ''~ ~ ~ ·1". I·Ia"I; indeed, whon a• .,. O. tbere exilu :' E F' web lhal 1.:'1.. 1, " • . .:' is r-e&I, and a• . .:' ~ 6. 1". 1; _ , il suffice& to t.alol! a.. -la, I' - '.:'. Put g .. L ,,,. ,,~a~I .., . Thtn N. (g) .. 1. On tho ocher hand ,
c:.
f
1"'1' ''
L ..a~ · I' (A, ) ~ 6· L 'S 'S~
f 'gdl' :?: ~ '
Ia. !· !"'!' MA, )),
'S ' S.
L
IS'''·
!"!" !I'(A.)! ~ l_ f.
(l ) wben f be\onp to SI{k , Fl· We.,.,... ~ to tbe cue in ... bidt 1 it &rl ...bit....,. elMneDt ..r £'; (1') l uch IhaI. N~(f) .. I. Gl.- 0 < ! S l. tbere exlsU ,., E SI (k , F) web t b.u N~(f - 11') :5 , '. By ... haI. ..... ha"" j ust...n . tbere exiN ~ E J:f,.. (P) l uch l hal N.c.) .. 1 and :?: N.(II') · (l - t) :?: (l - t)2. Then / fgdp. .. + / (f - 'I' )gdp and (j(f :5 N.(f - \0) . N.(g) < e'. HcllCO 1/ Igd,,1~ ( I - t)' -t' , which ~ (l ). Nex!, au"",*, that p .. +oc and N",,(f) > O. LeI. 0 :5 0 < N",,(f ). Tho_ {:. E fI : 1/(:. )1 > o j CIOnt&lns . I'"intqnt.blc!iCt E which io "'" I'"M&!i41bie. f I E it tbe limit , " . &/nx.! ev..rywbere in E, ..r • ""'luenol! (f.),.~, ..r 'Fl.1E· sinlple ""'pp;np, and lbere • • Ii· ln~ ~""'"' Z..r E sud> t haI VI'(Z ) < VI'(E ) &nd 1M' (f.)~~, ""' " ...... unllormly to fi E "" E - Z. bt e, lor fixaI e > 0, ..... """ find n > I .. thai. If. -fl EI < e/2"" E - Z . mappl", I.. ta.Ir:I!o tbe val .... w" ... ,v, on E. and one oftbe I; ' (v.) n( E - Z), M)' A, io __ I'"nt&lidbie. Clearly, 1/(:.) - v. 1 :5 e/2 fo< E A. Gloer> < ~ < I , tbere nio' , ,, finite partitloo P t A) 01 A into l'.in~ ...... 1UdI thai. 6 . V Ii(A) .. 6 · L 8 tP(A) V ,.( 8 ) < L8EP(") B)I. Choosins: 8 e PI A) .. Wt 6 · VI' (8 ) S 11'(8 )1 and lettl", a be one val"" of / "" B , ..... hA"" tal > 0 and 1/(:.) - "I :5 t for all :. E B . T here .."j ot. a' E F' l uch that 1....1.. 1 and 100'1 > 1-1- t. Now lho mappins: 9 .. 18 ' G' I V ,. (8 ) iI intes:rable
Thill
J)rU"o"eI
/11'9$
1/ 11'9$1
'I')gd,.1
n....
o
.u :.
1,.(
n..
and "''' (1) '"' I. On u., other hIwd , f Igdp .. (tI V j>{B ». f 10' I Bdp. Si""" f 10'1Bd;< .. tJd 1(1 - a)o' IB I :5 dB . ..... _ that
Si""", t and 6 AI1l arbIt,ary, (I) ill true. Ar&bi,,& III .00.,.,. ..... obWn •.,Iad .... (2).
a
If V ill a dcnot _ _ "'~ of C'j...(P), (t J persili,- whm g extends ...... V n 11,.; indeed. tho interior Jr of 8 .. 11,. ill d<nse in 8 . tu>d B" n V i& den80 In Jr. A similar 01 w .... t;on boltle for -.JlM!d fu~ioM 011 o.
5.3
10,11
and H (O. R )
£::CP) if
IUbe~ of i& ~ ~.
I) is IcIOI than
If (!')~i!:' io. ""'Iuentt 6})
s: 6.
1' ( (z E
For""",y £ > 0, lei. V. boo the set ((~,:) E F . F : d(~.:) s: _). Theil, by the .1 . ] ~. ds{/, 9) s: t If and ""ly If (f, ,) belonp to !V(V•. B. t). The mapplna: dB :II. g) - dB(f,,) io .. poeudumctric 011 £( A.I': F). Dell nit ion 5.3. 2 The ullilOnn OIruclur
1" (I", e 8
»
IM",).I.(I'» ;' V )) 8O"'i toO a.t; p and q tend to+OI)liYdy. If And ""ly if 1'. ({ r e 8: (/A (;r), /(;r) t V}) ~ to 0 ... n - +0)_ :
,
106
S.
v
Spocoo
E~
""'I""""" {f~)~~, In £(A, ,,; F). eon,..,rging Io
(»II"""""'"
(»II""'»"
P kOO': Fir........ P!>(IjIe that B "' intt&n-bll!, and Ie\ d bo! .. dillu,,,,,! compUibic wltb ".,. uniSonn ItlUaU.., of F. By inodllCtion an m 2: O....., can defi.o.e • dno.lblol Rquoenee (f_,.)"~"'~1 in £ (04 ,,,; F) wit b the ioOowin« properties:
(a) 10.• " I. lOr all n 2: 1. (b) (f.... ),.! I .. a MIt,. 1""""" of (f...-, .• ).! , Inr all m (e) For eJl m
> O. the lOOt £...•
> O.
of point. r EB for which
d(l..... (r ). I ........ '(rl) > I /r'''' """ _un: ,,(E.... J .. t han If r · ... For ~ m 2: I. put E.. _ LJ.1 ill a Qu>chy ~ in F. fi nally, ""'" that (g..).~ , i!I a IU'-'!umoe ofl/. )"!, . W• ...,.. _ "" tbe ""'"" in which B .. tbe unioD of a R
(a) 9>, . .. /. br all n 2: L (b) (g,., .• ).!. is a IUbMq"",,,," of (g... ... ).
> O.
(e) For eYe .. is uut-jueocc 01. {~"'.).! " and 10 (10,.( ..»a.2:' 1A a Cauchy ""'I""'''''' ill F fo< &II % of 8 - P. .-hen: P .. U..! I p... Tb .. ,,01'" lbe ~rn • nion. Next. . supp.3 Coo·... '" Min MOMW'O
107
F . Let N be. " · r>e«licible ....1xIet 018 such lh.ot (I. , (z ) . ;!: l ill. C,'w=h,y
t
~ In F for e>tiy z E 8 - N . and ... lil
t
PropOsition 5.3.1 S~_~ thGt "'" mn find in A G IGmil, ("' );u IJI mllhoAll, dUjlJint. illl~ "II IIlitA u.~ loIm"f p'.pt"" lor etiC1"J ". mlqnl.b/"lrotecnble- Then L(A. ,,; Fl ill metrUabie. Let (f.). i!:1 be. Cat>Cby IOqlleOOl! in £(A, ,,; F ). By P ropoaition $.3. 1, there ... 'u~uenoe (f., ). ~ I of (f. I-i!: I ",hIdt coo"(>P al_ ltYeI1""bere in A. The limit I of (f., ). i!: I (arbitrariLy ",,\etlded to the wloole of A ) iI I>"meuurabie, and (J•• ).~ I coo\C' .... to I in £{A. ,,; Fl. TI.e. ero.e. the toequmoe {f. ). i!: 1 'IRtf to / in ~
"""""p
Oboerve t bat the condition of P 'oposilion S.3.2 .. satisfied ",hen A ill
a
jJ-
moderate. We IIhall ,",,0"' Later (Seetion 19.2) th.ot it iI ..... atisDed when" ill • Radon mealllI1!. P~ltlon
.tr.o:no..
5. 3.3 Ld F bo G"'" &nadi~, eqWjip,d IIIilII w~_
kfiMil
~
iU norno..
1011
&. L'S poooo
r.JFor tooerr "-_,,,dle,,1 A, ~ topt'/(u of con.....,.n... in rn ....""';,
_I,,", of C{A, 1" F), OM 1M. ..nifarm
w".,..tiWc oo:W\ the t'I'lCtor.paa: "noch,", .", :. l«f ..th /he "nif""" .Inocl_ Df
lo~ooI
con....,."""
(h) For """ry
_I .... , po« ... obl"'n&! .. in .........,_.
~
u.....
~
.....t n"mbo;r p 2: I, IN. !ol: ':n of '~ (I') iI Ii"""
14, .!.n irw/tl.otrl "" '';.(1') I>J Ihe Iopo~ of qO.p;F).
M For 1 :S p < +00, £';. (11) iI .ten« in PROOF : FQ< ~ ... in~bk
St,,-
qo, II: F).
B of A
t""""
and for "''eTY real number
let T(B. 6) be the ocl or f e ' (A.p;F) for wbl 6)/", _ than 6. Lflt d be the <list&nce (~. 0) -II ~ - .1 00 F . o-iy. rlIlU•• :S 6 if and only if f - g beloop to T(B ,6). Thuo, to PlO>e '1000 (.). II "' c"""&h to oho .. tlU>l l he 61tet generated by the TCB.6 ) "' tbe lid of ....N .... boolp(£'. ):S 6. Then ",,- beIOIlp to T {B.6). ",bloc:b Ji- (a). Now 11M 6 > 0 he .. ....J """"-. If f e '';.(1') is.uch lhat f iNd" :S 6"+' and If £' _ (,. E {} : 1/ 1"11 ;> 61 . then 6' . ,,(E) :S f If l"dp :S 6"+'. and p (E) :S 6, ..hIdo ~ (b). Finally, let f he an ."b;\r,.ry element or ' (0,1'; F I and T (B.6) .. ,qhbco-hood of 0 in 1"'- """""" Arguin« .. Boo.., . ..-e _ tbat t~ ";'1>1 a ;t-ln~ ... ~ E Ql B.uch that jlf iI bouOOed on B - E. TI>en tl>eB/lur-e on a a~ H(fl. C). Lflt F be,...,..J Bant.do Ii*'" and p 2: I a real number. Oefl,,;tion 5.4.1 Suppooe II is a ...boet of £';.(1'). II is said to be uniformly iD1~ of ~r p if and only if the follow;", rondillona hold:
(a)
For every t > O. there"09t8 6;> 1) rmcb that rruPf EH f If l' l"d"
:S t for
all I'"im~ lIeU A .."""" ......\SUre ,,(A) is ic!IIIlhan 6.
••
(b) For ew.y ...."/EH
,> 0,
lhere exisU .. p-inlqn.bIe
let
J1/ 1·lo_B d iii....... N,«(f - 9) 1~) S 21:'/ , ; thertfon:,
s ,,'
N,«(I - , ).
/
U"' ,
II-,I'd". l {ll_1I l/ - gI'
"""'"'P
fn_"
......
,.
.
vs_
~.
liD
N.,.. ....III.. lhal coodilion (b) of Definition ~.U if I>I)\ ... tio6li«l. ~ .... ;,,' 1 t ;> 0 JUCh IhM IUP/eHI,,_" l/ r dll > e for all int~ ""'" B. Gi""n n E: N , SUpp ""'" 8, •. . . . and .km"',\1 Io .... ,f.-. of H ouch that 111, 1/,\,dj! > t f...- all I :!: i ~ n - I. There ....m/. E H ouch thal. I " _I II,u...ulI• • ,) I/. I'dll > c, &ru! A lI·ln~ 8Ubooeo; 8. of fI - (8, u . . . u 8 __ ,) for which 111011.I'dll > (. W"!IO $I:rupOIIl l ion 5 ... .3 C<mdition (e) irnptiu .....din"" (G) 1 Dtfiniti(m $. ,./.
The, ""' OJIO~
me.. II .. "',~.
o.
PROOF: FiM, SU~ thM te) boIdt. Fo< fixed ( ;> let t > 0 be • re&I D"mI:>er ru< I/ l"dll + ·II(E) S. !. bonoe cor>dItkln Ca)" ..-lio&ed. Coo. .....ly, ....".. U_ II . dilfuoo and that exonditioa (a ) holds. $uwoee that IUp,(H 11/1/ 1 > f ) doeo noI; SO to 0 as I - +00. There ~ .. > O• (f.).,!:, of OI.ric:tIy j)(IOitJ"" nurnben convera:S"4:\ to +00, and • ~""" "")..~I In ll . ouch thM ,,(i/.1> t.) > a for overy n?: I. >0 be ruch thai II(E) S. fJ lrnplieo IUP'EH 16 1/ I'dp ~ I. For each intoger n ?: I,
,r
.. ""'I'''''''''' _
(S.II
find
Let"
&II
ln~.bIe ",bioi.
I&, 1/. 1""'"
E. of (1/. 1;> I.) such tMIII(S.) ~ inf(a.lJ).
bec._
?: ~. inf(,.,.8). whicb is absurd I. - +00. Tben:1ore, IUP/EH 11(111 > I) 00 0 be!lUeb t bat SUP/€H fA. 1/ 1"dll :::: e for every lI-in1o:grablo! set. A woo..e ..-.Jure II(A) is smaller lloan 6. lor 1 ~~ , 8Up,,,H II U/ I > .) ill omaIlf.r tloan 6. &I>d .., .... hav" sup ,eH II/I">' I/ rdp ~ t. a N_ I
?:
Tboe".
P r-opollit ion ~ ...... ul
II t..
G tub ..,
01 L~(P)
Dtfinili<m '. 4.1. Then (~) /oI 0 be .. re&I number such tilal auP/~ H "II"" In"..... ~ I. AI9o. let B be &II inl-Pgrablo; .... such llw. IUp'~HIlfl"!n_ .dll I . Then
s:
J. Il!"dp s. JI II
UI1 "")nB
II!" dp + " "II(B n (0 < II I ::: al) ::: I + "pII(B)
5.4 Unifonoly
[~Sou
III
s:
Ie
H . "I"Ilu. [ I/ IP"" 2 + "",,,, (B ). aDd H ;. boo'tI(\ed in C~u. ). Con--ty, IU~ \haI. H iI bounded and that (a) bold&. For fiXf!d f > 0, 10.1 ~ > 0 be suet. thai. ruPf~ H [ I/ IP I .. d" is smaller lhan f for """f)' in~ !eI. A. wJw:>.e ,,(A ) iI Ie. than 6. Chooee [ > 0 to thai. !-P(IUP/~HJ l flPdjJ) 6. Then ,,(1/1 > I) ! - P [ 1/\"dp 6, aDd ~ ~/[". I/IPI'M'S thai coOOilion (e) ilauilfied. []
for evny
_=
s:
s:
s:
n.e1E1\::u, wben " iI .rurU!ll!, every uniformly in~ ouboet H cA £~u.) iI bounded. On the com.....,... .. hen ...., tab £Or I' the 011 R defined by the"".. I at the poInl 0 , aDd , for -.oh n ::.: I. Le1. I~ he the function from R iMO R which iI conIWlt, equal t.o n , t hen II~ : II 2: 1) iI uniformly inte&r&bIe of order i , but i l ;. not bounded in £~ (I')'
_=
II " ~ 1I~ ) ~~ 1 """".,.,.,.,. i .. £~CI' ) «> ".. tkrMnl; I ~I 4C1'), tM:n Il .. II. : n ::.: I ) ;, ""ifnnl~ m'.., 6k ~I nkr p.
TbMtem ro .... l
P IIOOF: For fiXf!d f > 0, there mat. 6 > 0 ouch that [lflPl .. dp < t/? for ... u, i~ IH A ",Jw:>.e moasu ... is 1M! than 6. 'I'ben I' ' I'(I/~ I > I) s: [ 1/. jPdp s: "'P.~ , N,(f• ., for aU .. 2: 1 and aliI > O. Thil impbM lhal .... c.n lind 10 > 0 IIUdI that ruP' ~ I 1'(11.1 > I) illlmaU.,. lhan 6 for aI[ I ::.: 10. Now Le1. N 2: I be an in~ such that N,(f. - f) s: "/' /2 for all n ::.: N. 1"bea. for evny I 2: Ia and eve!")' .. 2: N ,
(1
IM 'dp) II.
s: N, (f. _ f) +
(1
III'dl') ,/, s:
11. 1:>'
il. I:>.
~t'/' + ~t'/p
2
2
.. t ' /·.
s: s:
s: ..
aDd [ [I.[>.I/ .IP"1' t. NO'll. for eve!")' 1 < N . [1/. [:>.I /. IPdp.,.. 10 0 .. I _ +iM that (f.).;!!. "'''""'"," to Il.Imoet t\C' 7"'~ For add I:> O. tbm e>;1M.s an lntesn'bIe 9I':t B sudt thal; IlUP. )!. 10_B Ilk l"d" Ie smaller than (f/' )'. For """'Y iolq4> 10 oma!kr l hart (./6'1. T hen. for .011 r • • ~ ",.
r..e.: tie
~ 1 be
I. - I. - 1.·10_/1 - I, ' 10_/1 + (I. - fl · 1/1_ /1 •• +(1 - I.J· 1,, _11.. + 1•. 111 .. - I.' Itt.. ; I .....
N.U. - I.J is smtJler lhan
N.u• • 10_ ") + N.V• . 10_B) + N.{(f. - f) . 1/1 _11.. ) + N.(V - /.) . 1B_II..o) + !V.l.I. - 111..0) + N. U . - 111-.)' .00 80 _ lban • . t' - U' )' ;!!' .. " c.1>Chy ""'I"""" in C~(;o), and It enl.-.-, In .. hlch (I..). "" ton.e.p" to I In troea8Ul"e, io handled hr the ....... ..-swnmt II thai. in P,op<wi,ioa U .S, obow"i"l thai. (f.. )..",. cor>WfI1I'lI to I in ,hoe D-.n of ....o.r p. c
.&eftite.t for Chapll!f" 5
1. Leu, He._ >KIon;" F _ .hal 1-1.. 1101 .. 1. S - tbat 1- " oad I- - "I 2\< ... ly """"" .......
4.
S u _ lhat 1 < p < +«>. and lot f to. lho Dod ..... - . i_uo.Iity (d) 01 po.
N,(I/ I.... "·1- ~9)!i3p ..... V
~ . _t
~~)·
-.J"Pte .., p _
"'. 111+111»)'" (I
100- .111 aDd 9 of c;.",). aDd .bo.. ~ that tboo ~ 1 - 111'-' . 1 f""" .c~u.) ,...., .c).(P);. urubmly """',."".. on - , - bouadod ",1_. 01 .c~u.). ~.
Condodo, lor I < p < +«>, that .100 m'ppHo, 1 - 111'-' . 1 ""'" co;.",) jl>tO
2
.c).(P) io .. _ _ ..........
Let F boo. ,.,.j a.......::t> _ _u", Oll "'(RC).
. 0 < p < I .. JUI .... _
• .....t .... pooit;"" o-;dI
J
II - JI'do.hat.be mappinr; 1 _ 1/1"-' · 1 ;. .. hr b· '.....ptoiom &.;,no 4M""" C).", ) aDd u.... .c~",) if """,pIoI.e. n... C:'(p);'" (»IDp"'~ ~ rioablo V¥*oc .... 1 _ _ ANI SI('R., F ) if ~ iD C'i-",). 3
Let,. be lei: , ... _ .. u",OllI - )O.II. 1'..., O. PU' B, ...
U
E C}(,, ): N.l f) :5 01 """ write D lot ito ct>DYO>< ...., .'D)< If I E C~t,,) _1II>Ch that N.!I) :5 B ' 2"'-', ohow tILot I IO!o in D (in .ho: -..."'" of pw. I. conaido-r f, ... 2/1A and h ,. 2/1 .. ). Coocl...... bN D - C~(.oo)
3. Dod ..... "",., pu1 2 . bat ~ coati,,,,,,,," w..u Iotm "" C~(.oo) 10 \doni ... callt aero, aod tl r do•• ,b&t C~u.) _ "'" loaUy """>U.
4 1M" be ~ " ' to .. ", "" I .. [O. I[ and lot F be • 6) :5 '}.
I. 1M" be • mGtim_ ,0-: Iotm "" C{ / , ,,, f"). Cboooo 6 > o"".bat 1- 0). Dod ...... b.al ,,(ciA) _ 0 lot oJl • E f" aod oJI l>IIocrabIo ....... Cotod ..... 'b.al .. ;" idm'icoJ.ly ......" 2.
t;
Dod..... from pwIl.b.al C(/ ...;F} 10..,. Loc.oJly """--.
L.-,. be. poei' i.. D",,;.u _""'''''. 'pA01-t(f1.C). A a ,, __ _ F • ...... ,;.oJ,!o unilot-m 'PI' b-a>'e . be foIlooo-iq ... _ .'" ~ _ ........ (/').zl in C{ A,,,, F) """-M to I ·1_ .... , at , • In A """", _ aIoo 10 I in ,be ...,......,., T .
_. """"n,_
1 1M" be. pooi1i .... o..iell _U'" "" '''-'' 1-t(f1,C) . aDd lot A be. '" _"-"">Ie _. I.
Su_.h-at - " ", ;n~abIo ..._ of A .. bleb io DOt ,,-nqliciblo contai ... an &I.Ota. Lot F be a IDI!IrizabIe u..niloo-m 'PI'tql'ablo out:.et B ct;o., "" ~
s
'Th. 6>r a111n~ pool .. (~. l) """ lhot h ~ 0 ...... 0 S k '" :t, put / .... .. I •• .• , P ........ th.at (I').~I ""'... ..... to 0 u. £1..( .. ), b ••1w. it doeo _ ..... 'u .. toO 10...I1y olmoIO. ~"j . _ in A. 7 Let .. bot. pool,n. """ioon ... · w.... " " • _ 1I:(n , C ). lei U.) . ~, bot ..... _ of /0"" ,,""abIe fu"",_ f""" fl !n", fi ....... dofj"" / .. lim ... p f., r o< - , . ... in~ ~ 8 and _ 0)' 8 > O• .."...noct • /,.i" "$ able aub~ A of B "'. . d : "~ I'I A) ....... t!.an 8 ...... lot _b>::h t ho 1oIIo>wI.. "' ....... ,' 10 .... 1oIiod: b ...... ~ > O. tbent
I . ~ tl:&! lim IOIp f 2.
\I
I.d" !
f. (lim_lIP I.)J.. (.... E" . d .. 7).
Similarly, obow tI:&t r (~m;ruf.)dp Po.",... ..... 1.2.1 .
s
Uminff/. hebility &pOo R.. _ au_.hat pcs.. _1) .. ("'/ll) . ~ • 6>r 011 ......... 1 ~ O. I. ""boor WOtO&. lho Ia.. of i& ,ho f'o'"y diotribv.I .... .,it/: ~ .. n. f>...a,., ""I Y. " «(5. - n){"r.i)- '"' Of(J - 21" ) &om (0, ,, - 11 inIO R. otuIa ,~
......
L 101 (1 - ~) " { "101(1 - ;)oIz +~q (1_ ';;') +Ro(..), os·s.. wt.or.: Ro(,,) s o. """ """"I\>de IW L 101 (I_ ~) +,.,.~+ I )
""S· s.. 4. Dod""," &om 1*10 , """ 3 Ilia> (Yo: " ~ I)
io
aopt.....
io uDibmly Intqrablt wi,h
& (In tbe "VbI , i': _ ) ....t J ... In..,Jul i....... _ p h .... l _ ~ & A """ if Y io 1100 _,;,.
ortbot!ool~ .....
l.
UO>det.1oo b;ypo:otoImed from P ~ ...... litutiq X~ .". X~ II flqllol \0 0, _.....all lbat P io IIooR d ....ioiblo lot" X. _ X~.
atod"
N_ lei K boo ..
'" tloo
-..", ~ Iat; ... f;dd
.It,,,,,,,. doc ((I /( X. f
~1ooowoI
o.nd • .". - , .
n ~
N . .. rite 4 ( X , •...• X .. )
x•• , )),s .~s.) ." ,ho fotld
K (X , •...• X.. ) ol
&.co .......
~X, .. ..• X,.. ) .. P(X, •... . X... )! n ,s."'s.( X. T X• • ,), o.tOt, P(X ••.... X.. ) It .. pOIy-.i6I 01
lor Choopo of ~(u) ~ by \100 funct ..... I"' (.. E A). Ccx_
u... .
'W
,.
(a J
A t.oo .... aO. ..A 1/ "
.... ftniU.
u.at V dill... from .Q,(u) if ....:I Ii'd. IU>d il> lor oll _;" C _ A"""" ,bat R.e(.) > - 1/ 2.
We iDtead ... _
Lot ...... poin' of 1- 1/2. I.
+ool-A.
wpt1 ft~ .ut.< 01 A aDd If V• • , lie _ ..,btpeM '" Lot",) to ,,,1 t". ,boo j O lor '" E S , t". _ of G,om'* de.albl 0 0) .... m moblo if and 0III1y if ,he ...... L:~ ••.~~.(" + 1/ 2) (ni L liwb'. E.. ....,.. 1/ ,,) io a..;",. Coocloode . bat
1. - 01
Il . + +1
~.
djllion !>.
r.- 0 if ...:l ....1y If <X>l00 if_....:J,. if, b f;'fn)' E E S, •• , ... 10: ..... '" I '" E il .... ti"" , u . '" oI>npIoo "';0;>''''' (P' ...p .... II.,.. '-1.4). 1'\na.II:r.... "'" ._ ",her Im~ ''1 _ """",i. _ 10< I '" boo ",_lItobk U LoI; F be • &n.do ..,..,. with d ...... If I t CJ.v.). ,,- N,(11 .. _ .... 1 1 .. ~ B io tho cloood .ni< booIl 01 S/{S, 1") (TI ", ' . ... $.~.I)_ U TM. -=t ....... -.Ii-. 1M ioIlowi"l ............ Ld 5 boo tho .....ki... 0( fin~ •..-.. 01. N. T ..... £ _ card E . ..... atd E .. tho 0*'d1,..!11y 01 E." a ....... ~ .. ... 5 , ooJlo11 clMI of OF : CItuly. IAdV" :S V ,,'CA )_ To ..cablish the ~...., ;'WI,wily, IIIj>1l(* llIat IAdV/, < +00. For fixed ( > 0 , let I E bf. iud> 1M< r ~ I" and ~V" < IAdV" +~, .nd let (g,.)~l!' he .., ir>c .. "", ""'l'~ in $I· (S) IIdmilti.ng 9 all iU up~r envelope. ChOOfIt 6 eJO. II and put B.. - (% e n : .foe,.) 2: I - 6) £or all " 2: I. ".. B.. Nrm an oeq_ I.b R , and A C u.~, 8... '"
r
r
:r
r
inc:--,,,,
V"O(A):s L V"(B.n ( U
'S,So -!
oll
B", VI'fB..) -
f
B,r) · ·IIPVI'C 8 .1.
1,..,W,, :s I
~6
"l!I
J
g,.dV,, :S I
that V"O( A ) :S 11f" l"tW"l + '1/(1 arbitrwy. we _tude tlW V"OCA ) < l .. dVp.
whenoe .. _
r
~6
I
gdV".
- 6}. Since ( Md 6 ~ 0
P ropOO;ldoD 6.1. . Lei 7 .... ,,·.uUili~ ",ncli<m fro", 5 .. n ;. 011' . '-a,ow Vr." _ Vp", Tbo! proof;. Irit to Ibe
• ie< . p ...."...IIlon 6. 1.3 A ..t A C rill ,,·m..........u. ",;""",,Wo/or~....,. E E S.
, of disjoint ~ luch t hai B c B.. and £"i!: ' Vp(B. ) < Vp{B):;' r . n......rore. Ie!.Iilll p 2:: 0 ~ an
u.i!:'
inl i " mel< thai VII « U"i!:' B,, ) - B ) + L' i!:"" VII(B. ) :5 t, .... _ A _ U',.; .s. B. baa tbe desirfJd P' OfM'lIy.
llou 0
P....,.-Itlon 6. 1." UtI I ~ " mappi"f fn:nn n ;nl ...... ,~"...,.., 1'1Im th~ loillnling ..",.jitiorll B ... """t!Glen!:
F.
(.) I .. P·""FM ... bl .
M
fbrCWO")'Ee S , /IE IIlMli"mil..tm.n1 ~ in E o/ueq,"R«
o/'Rj E·,;mple ....."";, g•.
'VI
ale
(e) F",. 0't'UJ E E S, 1/ £ illhe limir ol..... 01 ... "oui~", in E ", ..
,,' it! E·,;mpk
_ppm".
...,...,na:
PI\OOt': Fix E E S. Let 9 be AD RI E-aimplc rnai>pin& from F; int F and ~" .•. , ~. it.! ,1lI."",. The !leta 8 , .. 9- 1( ~J) I~ in 'R. GiV
S ..
,ho: ,...ill& genen.Uld by S. aDd re that. E' c E C E" .00 C' - E' 10 J>I!Ili&lWe. Ot '.I" i....,;do,ntaDJ that . br t!:o, ~II,I!;, a "'" N 10 ,,·otel~bIe if and 001, If Ibm! ...... E € l uch lI,at E J N.oo v ,,'(£) .. o.
f .
5
T heore m G. 1.a Lei f be a "'apping '""" 11 ;"/.0 a "",,,....,We IJIGOI' F . TIM: loll =;"9 wnJili..... """... . I"",: (aJ
I
i, " •....",,""'We.
M Far""", E E S, 1IIen: ......1I . . . . .gibl< nbHI N 01 E, N
E
5,
,lid!
tJI41. I I E - N .. "'....... nWoI< SI E-N ""II I{E - H) ~ ..,~
P IIOQP: Fir$!, ....""'" that f ill ,,"_Mill~ .00 lei. E E S. By Propooilion U.2, I I E ill t!:o, Umll of a ""'I1lei>Oo (I.). -t l of iii E-tUnpie mappi",p, out.lide • ntsIlcibie JU*t N of E. Now , by the n:rNrU Just .... ta>dl"( the .... En! tl>eon!m, _ ill&)' au~ that N llos La 5 and that Mdt f. io S I£. "",pie. FiJI: % E F and , for ~ n ~ I , let g,. "" the mappinc from B into F "'1\1&1 to I. O. By tlot_ az-(UmW1 as ,bal in Tbooouo ~. 2.~, there ",,"'\.1 9 € SI(S. P ) ouch tlw "'... (9) " 1 &nd 19 1. and by T lIfFor
"'*'"''''
r
"',(f. ) _oup aD
"',(/1 .. )
'SO
b-",p
r
1.9$ !i .up
r
Igd",
r 19$. which ~ that IUp r Igdp .. +co.
lbe ~ of a>n""oxeoot in Ir r ..ure. S u _ tbat " ill &nd lee A be. P'ln!lMur&b!e 8et. If F II • melrizabie unlklrm opve. tbe _ W{V. E n A .~), ,,!>Me E ""teoo.. ~ 'R, form • buio for .be unlJormjly nl -'lA, 1'; F ) (notation of !io 0, 1' ( € En A : 11ft - 1I{;r) :> 6)) coo,~ to O .. n_+. No..-,
~ an prai>e
0
"'*''''''',
" "'1_
I"
6.3
Measures Defined by
LH n be a oonemlltr _ & 5 Jor elle» % E X .M IIlal L.u".\' 1 0.1 ;, fin;u Jor ......,. E E S. TheIl 1M jv1Ictioft 1' : E -
L..,.......,o.
~m.e. oj 1M O. Jor......,.
I'({:'-}) ..
liS;' diff~
an atomic
"'_~. MOLIIOIIe"
c/4uu oj alo"", ..
% .... nge.o
1M {:.-}_
....... X . aM
z eX .
PROO" De6ne " .. \be _un: E _ L.un.\' la.! on S. By F,oc..... l·
L...,.......,
liau 6.1.1, " O( E ) _ 10.1 for all E E 5; in p&nicul&r, Ix} iI ..... llIUIP"J.bIe and f 11.1 . J,; _ 10. 1for eedo. z e X. Clearly. V" < ". Fix E E S. CiVftl ~ > 0, Iet %" ... ,z~ be poinl.OofX n Eeuchthal. L'~2ed 10,,1
,-*,_
.. (E - E n X ) .. v(E)-
L
all
EeS. N.... Iilo. x E X . For.......,. t > 0, tba-e .... jo. . that v( E t { 2. Thea
for
" ({%Il - O
I"n:s
an
E
e S c:onWm",,,
L
la.- I'(l z lll .. I"(£) - I'({"H -
rE!InX •• ~·
:s:
10. - I'({z})1:s: .. (E -
1$0( £ -
euch
0.1
{xHI + L !o.!,
•
(x})
+ v(E) -10.1:5 :W(E - (%}) :5 t.
Hence p(lx)) _ o. for aU % € X . If 8 II. p·in~ ~ sucb l hat V,,(8 ) > O. thea it iI net included in tbe JocaIly "'~l&IbIo ~ n - X . Thil means that 8 oontai .... an % E X , which ollOW"I that" ill ou.om.ic. and tbe proof ill complete. []
Of,flnltlo" 1'1 .3. 1 Let. {o. )o. lha~ O. ~ O. &nd M1pji(1t!f! lhal L.~(nX \0.1 is finite lor evefY E E S. Then " ,E_ Q. il llIt """"""" on S ]If'O"eE tlw I"'l l~ in S. Even tbouch ,,"', S\\reI
t""
"'*
n.
6.4
Prolongat ions of a Measure
'OOO"'tu~ Itt. S .. ..-:miri", in O. &nd "a oornpleo: n " " i'l! on S. I~ _Ith, pttt 10 lhe cxt.cmi •• 01" and i~ with tUp
Lo:4. 0 bola
10 " &rfI kk"lIItal, U ibowD _ .
Pr0p08ltlo" 6 .4 . 1 Ld 4> ... ......iri"9 ttu'l
E - { lsdI' "" • . S"I'ptI«
(., for
n> " .negli&ible """ ......u....:h S E 4> in • ". '''4llpble~. EYio:Ienlly. 1_ remaII>II """ Ior...:ll locaI!y ,...r>egliglble Ht , _hich ttr.emOte • locally " ...,.....ti&ible.
n,..
e ...
Then deli"" 0
n.
SM_
""'
(a) 0IIdI C••., illoth UHntiG1/J " ·int.,.,..",, ."., UHoUiGlt, .... i~;
M f
ill lur all E E C;
"""/ai,,, S .
(~) Ihe ", •..;.., C ,.""rut that l """dI>. For e'm")' "E C, e" ill a ~ and oont.j ... C, 110
PJI()()P: For eoclI "
E n " • _ f 1...,,,
r.....
J._.', s:
~1,
dod""" tluol.
~~'
.. .or:" :SQ, .., + ... +.""
lor all., > 0, . •. , "" > 0 (l_ualiIY of tho _ ",.,). 2
u._,
For IIWIl' , " ' _ 1 )! I. IIWIl' (p" . ... ",,) OS'IO, II' •..:1< 1" + ., . + "" _ I, aDd "-"crt l!otqe< r ~ I,.oall tbe _ " ",,!be Bord O'.~.of. R· , ~D0 "'tUPintJmial t.wwil.b _ ( 1 ' 1 , ... •"" ) &Ad r. Om . bootberhud. 1Or..-.,. '8i ""mile
C.ion w _ ""- ' (G,)l £rom n .111 R Ii>r GVfty I :5 j :5 t, &lid P'" X ~ [X!hs..H' p""", ,hat Let n
2 I••
~
I .... ,_
"".ho
p(x-'(oJ)_
,(")"'. ,, (~ ) "
r'
0 ,1 . .. 0, .
"
"
"'" """" (0 " ... ,", J of (Z+)t .uch ..... 0 , + .. . + 0 , m r . Show , .... , fur """'Y I :5 j :5 t. Pi X , . %) • B (r , n, / n l ({% J) lor ..u Intqero % 2 0 2.
"
eo.......... t- _
.. jnin5n boIliof tdjllenn, Ion.
For"""'Y I S j S t , 1m., ~ n, bolla 01 , be Irtk roIor. Dr ...... b&Uo £rom ,h • .,.,.. ...... ODd .. itk ,.,1 , '. SIooor ,I>et , Ii>r """'l' (0" . . . , 0 . ) E (Z - )' . ,be proboobilj.y 01" dnt.";"C oJ boJII oI".be j. k roIor Ii>r 011 I S j S t II lP"'n 11,- ,be multi_ i .. l lui ... itk ~ (n o/n, . . . ,n . / n ) &Dd • .
'..v
Fr "'""" i " , - .. 2 I. lo< p~ be lito m p, 'UOII on , ..... J.. of 011 ......... of N ...... that P. Cm ) .. 1/ " lor """'l' ISm S n""" P.(m ) _ 0 100- -.y m > ~. For mcb r-t .,.j.,.,J "''''''"''' on fl. wri'" £.(1) for /dP~ For.ntesen m 2 I ...". primoo fl. let 0,(") be .be POW" of" in t .... prime ra.c:toriutiooo 01 m . Aloo, Ie!. ~ ( ", ) be I 00" 0 _ "d;"'idoa m or ...,..
I
I.
1
1
£'(,) _ 0. - 1'. (0, > ~)
ate
~
£or Cbapt.or 6
III
""" ..... .--1 number %:> 0 , deli... 6(%) .. L.~ .Iosp. Pr-ooo thai
2.
£or_binls[n::!:l .
i1,...-. [2il- 2!il .. pcositi~ for - " poi..... p ond ~
3
If n :!: I • ..., to I wben .. < p
s: :In. flO ;,. L Iosp s: £..(Ios' ) -
£ .(Ios' l - Ios(') +0( .. ' )
·.I. 9(" l/n remai .. bouO>ded ... N (In fact, it ~ boo obown .haI 9(n)/ .. CI()ll'"'S to I .. n _ +00). ~ ,~
WIK[e
L .!.Iosp ~ kcHO{z' ). ~. ' 0(%' ) remai ... """tided .. %:!: van... I
Writa 0 m-. kc p/ p at - " primo p, ond let F bo .... funclioft % - L .s'losp/ p
4.
)1,+00( intO R. To oimplify -..oioft, put _. -!osp/p io< - " prirDo p. ond .... '"' 0 £or -=II iDl e,,", n :!: 2 .. h.idI io _ • primo. """ _.--I nutnberz :!: 2. '" [..,that
_
-
' l
F (u)
• ~ ' Ios'(u )
""
..
" - dod""" from part 3 .......
4
,,-
""" '"""Y '" :!: 1. let (1( ...) .. I.
L. 6,("') be l be Dumber of diotinct prime divioon
s_ .1t.r.1 £.(g) .. ""ur.u..r. to 1os(losn) .. n _ ex«0 ",a:«Iilll!
" (H I;])}; Show ,!>at
E. • ... tIIOfI
(H I;]) (•. -! I;]))
III.., )+1/ 1"'11. b
OfId~\bat
il.
• ...wler \ Ioan lE.s.l/p. Dtduoe 11iM, lor -=ry e > O. _ .. -
~" ,, -. , "
d;,o;i..::t ........ ,., ~ oudo that " :S ... q :S " .
p.
+00 (Hudy-~
(16/(10& Ioc .., - II :i!: e) """' .... C
. bon .... ).
to> 0
7 Radon Measures
_w. . .
" ..., _ In &O.aI,yoioo _ dilruen. laI.. r ''1 "'."... 1U'" tinIM< ..... of DuIlriI a "''''''" d - . . .... ' iOOI of ""',.. .). Maay of t ho _
n.o. .
mU..,,,,
d lIIOOIIo,at O,lheroo" a n _ ott U ...:I .........bIo __ .,., ol ...... pooet _ It..! V,. ' (U - 1') S c_
1'."""
t.
1. ' I . Ih .. """"""",,, iDtrod_ l be _ion of L.... n ......urabk ....W _ Int ... iti-.ely , ...... fimo;tio .. 0l'0 "!moo' td of Ioc.uy """'s-t 1I.,o«Iorf( SpA 9. (>:) is ~;ble only if r > I, t: E V.. and t: f V" althoucl> r > I lropliel ii, C V._ Hence /, ~ 9. io< all r , I , and / ~ ,. S u _ f er ) < g(r) !'or lOme r. 11>.,...a: F. S"P~ I ;, ltiPp.orkJ.
ti_..,
n.m,
'"' ...",., ctiono II, from fl 1Il10 10, 11."""
(U.l.
IM1
""* "
Iweupp(h;) C U.. L'5'
-"-
:s
If I, E 1(+ . h E 1(". 11 E 1t(fl .C). and!91 I, + h. lhe function 90 . equal to (,11,)1(1, + h ) at pO!ntl ..·heTe (1, + h )(r) -flo 0, o.nd to 0 ebewben:. is ronlinOOlll on n (wilb i _ 1.1). Idote + h ,.."io'-. MOIw .... 1,,1 1. r.nd 11 - II, + 111· n", 1(fl. C ) has lhe [>tope> lie! required [n Section u . If " • " R.oon ...... u ... OIl 0. then 1,,(1)1 < g,,' 1t{O. K ; C)I . N I l ( wbe .... K _ .upp(f)) £or ~ 1 E·W and for aU 11 E 1(fl.C) Iud> lhat 1111:S 1 .
:s
~Iore, II
has finl~ ...n..IKln_ MootW:' , every ~og 8eq\>CI""l (f~)~ ~. In JI " ",h;eh poinlWI3e to 0 ..., uniformly 10 0 by Dini'.
hao a finite vvialion ill &combination of foor poetll ... 11.-.- tOmw. n."" it ill .. RAdon _ , . , . In 1I>ort. tloe Radon..-.ures on 0 . ... aactly the CH.-r !'or"", of finl" .....tion.,..'tr "H(!l.C), and they "'"" Daniell n ............ Thc,erO'E, ,.., mo.y &ppIy to ~ M","'fOO the 'fOOuI", of SeclIo:m 1 4_ [n l he 1OO!QI>d • ...., ..-iliid. M {I1, C) (resp«tivdy, M (I1, R )) be tbe ,~01 c:omplcx (.....,.,aively, ruI) Radon IMUW'OII, and ,M +(I1) lhe cone of positi"" Radon rneo.ocu,-es on 11. U " e ,M (O. Cl ... e M +(Ol ...., . O. lo:t 4 > 0 "" suc:h that (so( ...)1 S € wd 1" (",,, S € for every I>;n '}{ (O.ruw(g): C) _~~ 1...1 S 6: by Plopoosillon 7.1.3 ...-e """ find &- finite family (h;)"" let H+{o'toJPS>(g)) and &- family «(t,)"" of OOtoplel< .. ~ l uch \hal; !9 - E IE , S 6 and 11, 1- E>E, ....I"'! S 6: th...
s:
0,"'1
!»(g)! :s III ( ~ 0,11,) I+ f :s
.
from which
r:, 10,1. (P(II,)\ + c
:s
r:, 10,1. ..(1);) + •
:s ..(Igl) + 'If,
~he
!lquality 11I(f}1I S .. ([gil folkJ'O-s. n..more, f:VelL \.he!tlSl. condllioa of Section L4 bokIe.
Definltiou 1.2.2 Let I "". nonemply inlen.... i .. R.. and Ie\), he Le'-«uo L' ., Ul'e on the aecnlring of I. The line.r 10m> I _ I 1.1.>. on H(I. C ) Ie .uoo
"",ural
:s
Obten-e that IIIJ...I.I (IJ - "lUI for.....,.,. J E 'H(l. C). if (0 ,,0] ill &lubinwval of I conLal ning supp(1). !II) that t he Ii""", form I .... IIJ...I. 10 o.ctuaily . Radon '" e"lUte. Hencdonb, ,.., shall do!not~ 1"..1. t he /let of lo..e, setnio:mtin....... fuDCIioDS !rom n mto jO, +001, _!>ere 11 Is a giYUL locaiJy oom~ JlaUlldoo-fl" 'J"'C1'.
~
Lemma 7.::.1 E--r
I E :r
if 1M upper ...." " . 01 11K
m
IgE W;g ~Jl.
P ROO ..; for It\'a)I " e O.uch that I (r ) > 0 and ~ry ~ nwnt- a in jO,/{")!. then: ciata a c:om~ nei&hhorbood Vo'" for wh\dl fill) ~ a 011 V . On tbe otllft hand. tbtte .-xist.s a funcboo 9 e H+ .uch tI.." supp(g) C V , g{,,) ... ... .oo m) S .. for all \I e V . T hUll 0 S 9 S I &rid g{,,) ~ ... which provtII the lemma. 0
for tbe remaindrr oIthil oe'J I1IkeI A cl fl . ate""-'.,..... IA -00). tII~ Ihl up"," (rup«tit1(J,. r -) CIL"./"., 01 H U "',"rqraWe, "nd I """""'9U 10 9 ,n 1M _ II &loAf 1M /i1kf" 01 _lioN 01 H .
to...
~
P ItOOr, We ..;1l PI"'J'It the stalemen~ ooly ror lowff 8emio:xltlnllOUll fu nctions, o.od ..., m.t.y ""-VS- thai. 1l haoI a """"'''''' elt~ I~ . The func:t.."..
r
(...."""I....,.y. r), ... twre I ~ _ H . Iorn> .... uporard-di~ "'" 01 loom' oemk:ontil\llOUll fundionl ... boIo: upper e."eIope ill g+ (lUIpectiffiy, • Oo:wnwo.n!-diru:led iJeI. 01 upper IemloominOOU6 functloM "'ho8e \ower enl/& Iopo: is g-) . M(Jttol .... , f r.wl' :!O f IrlVl' + f I; rlVl' for ..u I E II. Th.. ,",.......J only Pf'M' lhe n,o lIatnr>enU of Tlw>mn 7.3.2 ... ben H """';-u of pOIili ... runcti(ln$. If H 10 dil"flCUd UJNW and oomp, 110 lhe ~r _loominOOUOl fW>d;"" 9 ill i!ll~ and f gdVl''' "'P/EH IdVjJ. Since I :!O 9 lor ev that I, :!O II. for each I E H. c:onsIdt< the func:tlon /' ukln« tbe .-.1"", /I{z) - I (z ) at poinu :r ... t.er-.. I (:r) < +00, and the .-.1"" terO at poin'" 'I. ...benl 1(",) .. +00. When I ~ ~ H. the /' forln &II upward-dirulef)' ."..,. Id U . Pr-opoo!IltJon 7 .3.1
PIIOOt': Ewry compKt ..... K ;' lnl"V"bie boa._ I K .. InJ'~ H • . ,~ ,.I. N.... , Ii"'" &II ~ "'" U, let ~ < VI'"(U ) he a real nUInbf,t. n.e..., e· ... ' f E 11: + ouch thai I :5 1/1 and VI'( fl > r. Put S .. . uPP(fl. and K• ... (z E n : f (:r) ~ d fore-y t > 0. Then I S I ll"',
eJI*"
PROOF; If A is inlqJ1o.biI: , l ben, by ~ 7.3.1 , lhen: an Open ~ U :) A ouch that Vp" (U);' &ebil .... ily c~ 10 VjJ(A ). M.,..,.,...... b e....,c > 0. ttwre ....... . pomth..,. upper "'miconlinuouol function I. wlth compKt
,uppon: S, web that / S I .. and /(1 .. - /ldVp < t I l (PfOp(I6ibon 3.7.2). T he .. K _ {z E n : lIz) 2: 6) .. eloooed and contaioed in S for evny 6 :> O. TIm. it isoom ~t and . .. ina! / < I ... K is included;n A. The .et B _ A - K is int"V"bio:, and / S II< +61 ". 1Ienr:e
f
/ dV"
< V,,(K ) +W,,(B ) S V,,(K ) +6V,, (A).
and finally
But. os 6 ..... arbitrary. Vp(A ) S V,,( K)+~ for "su.iubleoompart_ K C A. Come ...!y, if Ior ~· t :> 0 there CXOlUl " """'pod set K C A ouch that V,, "(A - K ) S eo 1.. liN in the closure ("";th ""'r>ee' to 01 the eta.. 01 tIM: I I< ( K arbitrary oompoa wbolot of A), to A io integrable. N.,... t he !>fOOt. of tIM: tim. t ....., slIlemmlli an: complete. Fioally. if A i9 Int~. 0.6". by Induction" ( K~). >. 01 COInpad oubo!oet.o of A M follooo.T K , C A and VIll A - K. ) S I; K~ C (A - U ,,s.s o- ' K. ) and V,,(A - U' S'So K,) S l i n for " :> 1. 'n>en A K, III negligible. [J
4(p»
""'I"""'"
u.;o"
By Propoolitioon 7.3.2, " ou~ A of n ia Ioea!ly nogliglble if and only if An K Ie rqj;pble for all oompe o. th~ erio4. "" '.p,d ,.u.,1 1I 1>J A n.clt. IiIcI VI'CA - If) S r .nd fi ll ..
"""ti""" ....
PROOF": Clearly. Ca) lmp&.. (b). [f K io. com~ sec ..00 ( II.).~, is. ""'Iuen<e 011 in condition (b). lOr t'Vfty f > 0 tbon: ~ P > I such thal. V",( K - u,~ . ~. 11 K ,..q, tIW. V I'{I< _ L ) ~ € 411 putitio .. of rr inlO M_u such th&t / is <XI ; E 1, ,hffi, n~ • !Ieq..en< G mqping Jrcm 0 inlo " .... Iri....w. .,..".. :nw:n I .. Lpn ,,· .........unobk if """ o.sIJ if p if ,......... ~...61e.
Theorem 1 .... '2 Ltl
I
PIIOOI'" If I ill "-'-"'able, tben it .. Lusin ,,"_urab\c by P,opoe;' t~ 7-".3 and 7.4.5. Coo_Iy. if I is LllI!in ,,"measur&bIc, tben i t . II' w EEurabic by Propooihon 1.4.4.
0
7.4.2. a Lusin ,,"~bIc m..ppinc from n inl.O a ~ opoce ",Ill _ simply be caJlod .. ",w EE' ....bIe mapplDf[;. Now 'Theorem 5.2.5 may be refined .. 1OIt.:.>w.: In view
of 'Theoo~w
Tbeo.em 1 .... 3 Ltl F .. G BGnadI sp SI(* . F) and N,(/ ) .. L Glvet> € > 0, Iberuxisu 9 E St(R . P ) ouch thai 1 , 1S 1 and 1/ / gIl"j 2: 1 -t (by the same arxummt .. \bat at Tbeooun E>. Hi). T here cxieU a finl~ number ol mutually d~m, COIiIj*O/. ~ X , luc:h lhat 9 tabs the COllIIt&lOI. ..JUIl": "" ~ X,. U X .. the uorion 01 the K ,. tben / 1/ ll l(odV" S t. Let U, he an open ~hood 01 Ki ouch 11w!.he U, ..... mutually dlljoint, and lei. It., M" Q)IltinU(IUI fuoctiorl from n inl.O 10, II .,qual to I on K i • ",bolooe IUppott iI OOIIIpoc\ and contelned In Uj . If we ~t " .. E, .. tben II(:r) .. g(:r) on X aDd Ihf:rll S Ion n . H~ / I/lIlll« dV" S t .Dd 1/ IIIdJctO;Q 11( • ....be .... X ""tendo O¥ft" .1" . .. d[.. rcd
V,,(K ,n X. ) '"' Vji(K,)+ V,,(K , ) - V,,( K , u X,) -
VIl l A).
Write P lor lhe inum :lion r:A all elcmo::nte of .1". Then P lie! in.1" by Tb&.....,." 7.1.2. Now. If { n it coJled .he Radon ........ un: dc~ned by tho: point"""""'" 0 (") (>= in any .el X ouch that 0 ..ani!Iho!oo "n X ") . P l"OpOIIIitloo 7'. ~ . 2 Ld a aM " boo ... in Dl:jinili"", 7'.5. 1. Thot,. ;., alOrr>i O. ~ ""..!.II. finite ~bed; Y of n "",h that E~n_ Y 10 (,,)1/ (" ) ::; e . We can lind disjoint nei,hborhoods V. of lbo! poin!.ll ~ E Y , and, for L
o (,M, ) -
..,..
Ii -
L !o (I/l! IM - t ~ L ""..
VI'Ul
10(,,)1 1 (,,) - 2£ .
'EU
L 10 (" )1/ (" ) ~o
VI'U) ""
L 10 (:r)1/ (:r)
.,0
1'1_, VI" (h) ::; E.tu 10(:<Jlh(: that 'Y. ::; I, . upp(",.) c V., and "...(, ) _ I . n,.,., 9 - E ..,y G. ,>,. I.... in H +- . 9 ::; It, and
V I" (h ) 2. V 1' (9) 2. E .... .. ~ 10 (1I) 1 · Jbr BUitabie numbers a.,. ~ .. a., 10 w)1 can be mad
Not« thal lOO (:I:) (:< E ft"""" that 0 (,.) '" 0) ....., rep<e>ientali_ of the dilf= t d ' " , 3 of · ' 0"'"'.
Conveneb'. t".
PJOptCtion 7.5.1 and Theorem 3.$.1, any atomic Radon _ .. O!:/ined by poi'" m E f'OIr~' ,t.omlc Radon "",pm",,, on O. all mappings from 0 into' topo. . , ) IJPf":"! an:
7.6
,,-_urtbIt.
T he Riemann Integral'
Let n bo! , locally tX>mpIId. HAUlidoT/I ~. " a Radon meuure on 0, &nd 'H lhe """ of (»IItinoous functiono from {} int1."" (:ODIpocteupport. '111m I"(f, + h )1fV1' :5 I" l ,dVI' + I" loIfIfl' for all h . " Denoc~
C. t'o/dV" - of" 11fV" for
,,,,d r
all I € 0 and Q > 0, 11fV/,- S /" l,w 1'. /.10""0''''', by Tboll.lf:Ction from 0 ill\(J R wil h comp8Cl 'lUpport. If I is R"m&DD in"","abie. for eve"l. 10 Riemann Inl~ if &lid only if I" IdY" - III IdV". and in 1m. CMe 111 IdV" .. I IdV".
Ii...... £ ;"arbilrary.......... tW 1",/ ..w"
Thflo."m 7.6. 1 U I I k a bo~1I4td' mapping, orith """"pacl IUpport. from 0 ;..t(I a .,.,., B<mad. _po>~n 1/ (0:11.
!hal l E,s;s~ 9,,,< 1s
PIIOOF: First, '''Pi> that ((lndit;on (_) ia $&l.isfied. I ill ,...inlopt>ble. 110. b &II € :> O. there aN elem.enta a, . .. .• "~ of F and eJemoma 11 •. _.• g~ of 1t(0, R ) such that :$. , / 4. if k .. 1/ - g,a, _ . . . - g~a" I . Let 9 be equal In !of/lg,a, + _.+g".a,,1 nn the let K .. (o: E n : [g,a, +. ,' + g,.a"I(o:) ::.: !off and In 1 on K . Then 9 ia continuous and
I "me"". A " A .. included in V . In whal. follows, n '-;11 he .. oompa« Ht.UBdorif II""'" and " a Radoo measure on n. Let P he .. aet oI 6nite coven"" of n by p·lntegrable seta, oudt that A., n A.. .. II""",liglbIe for all ( A. ). n of P and all diotil>C\ k,.1:, of L. SUppoR that !or """" entour~ V of n there e.:IsU an elernent (A.l. u o f P I ud> tlw. all tbe A • .... lmall of o m.-. v.
V.
P ropOoliUon 7.6.1 Ld I b< II RiM1l4nn jt.in""",w. ""'PJ>ing /rom 0 into II Banadl .~. Then. f 0 and lee. A Ix tM...t of poinl8 of fI at ..'hich 1M ooclilalion uc: ed. t. S~ A ill c:om~ and jt.ne&!i&ll.ole, there ""isbl an Op(z ) - /
~
lSI
mo- ;'"
PtIVp. S ! , 2
t€
0, an ".., ,lid> thal 0 S AI(",) - m(",) S t for all ... E P t:OD$IItiD3 of IV, Coo~ly, .... h&"" tbe b1~11III ptOji06ilioa,
oro..
n-
Propooi t ;QlI 1.6,3 Lei I : R k ~ndal. II, I~ """'"" t > 0 , tAcn: trim.n", to P n.clIlIoal M (", )- m(",,):5.~, IIw:n 1 is RicmGn~ ;,,~. P ROOF:
Clearly,
/111/- on L
LA . ' in!
. CA.
j(z )ldVP
S~.
a
Theorem 1.8.2 LeI 1 ... Gil inun..! 1 R GM" .. JlG40n ..........., "" 1. FWdI~, III I ... a Riemann ~ , inI ..., .We ""'PJPI"ffwm 1 into. &m.a q.:we, n .. I .....iMa ovlftdc .. a>mJl'Cld...mtcrwl J _ (... !oj 11. F".. tw 0, t/wo! trim a > 0 ""'" mol, I"" o. let ...... . , o1indtJd luflCltlm I from I into R
PItOO" AU funct;"'" t;' E SI(S, R )
l
it
aM
...w. «>"'I"'CI $uppni.
Illimann in«Wlblt.
~ ••,U.(S.Rl·,U inf
Id).. < r -
~rt.
I""
, ~ I , Id 1.. ,61 be .. (Ompo.ct rubin\.01"Yal of / (OnI.Iini"l! IUpp(g). For evffy t > O. there "".. Ill an Inl.tf;tr n ~ I ~ tlw 1r(:I')- g{p)I :S ' l Ib-a) ",".JI.to , It in ja,61_wriD« I,. - )'j < (b- a)fn. Wriu; Co for the oupren''''D of, on [.. + (t - 1)(6 - a)/ ,. ... + t{6 - a)/n). for e.-..y I :S I; < n , and Id '" be the fun'- of St{S. F) which vanllh outside .. 1Utd complId. ..... bI ... ",,",111 of J. T'IruB R, II the closu«; of SI(S, F) in G,. 1"he<ek>re, RIemom>
7.7 ..........
Con.e'....
1:103
inte,rablo: funaiono (with ....pect '" .10) -r be defined thl'OUlh SI(S. F). Ele...etlUry I_ t _ pooc : . J in d ill ""'Y. The followina; eump!elllh.lIt1 a>n"".,u limn ,,(I) E C . ~ " ' / .... ,,(I) to a R.>.In ..... '''"', .nd Uo" ).~, ""''''''J'U ...,..." I " .
I",
PlIOOf': If K is. """'~ .ubtie\ ofO, {""f1([0 , K : C ), n ?: II is pOInt.... bro,Med. By the Banri-Steinhalll theor .....""'J'U I f /dj. oil19 B. ~ltlon
I~
1. Radon J.k __ w
"··· r
P ROOF: By ~ 7.6.1, 100- - " ! > o. t~ exist a,. .. ...... E F, ,f. E' 11(0, R). and II E' If+, IUCh that If ,o, + ... + "..... 1 s nIl.
I - (g,o, + .. . +", ...11 S II. and /1w4< S t. We WI §nd A e B,ur::b.hAl /1kW < /1w!1' +t and I/{g,o, + ... +,. ... )ch- - /(g,o, + .. _+,..... )dJl I S ~ for all "E A. Then 1/ I1 fram 0 LIltQ C . C de hcd with the nom! 1_ II I - ",p.~fl l/ ( ..lt. We Ny \h.u a Ion 0/0 h4A a r:»Lpo::.it;o" 7.7.3). 0
""'I""""'" U that n"l u.w to DI'Udong 8. LeI/be I 6otondtd /rom fl mil> .. n-[-D, and b auobl , ,-1( [1, +000" . l>- t. lf"" 2: I if"" intqer&Dd K Ael, Ihttl ..... mUllt DOt ha", V ",, (11 - K ) t lor all " 2: no; tilt • .,;.." taki"l L. compe.ct 1O\...ch that K e Land V,.,, (O - L) < t lor every 1 ~ .. < no , ..... have v",, (I1 _ L ) :s: , lor all " 2: I , WIIt.-.dictiDa; tbe cboOoe '" t . n.."""f, for f!IdL int.qfI" no 2: I and eo.ch coml*=! ""'" K, there aM an " > .... ouch thai. V",, (fl - K ) > t .
:s:
."""'ptd
1!oS
1. _ M
UN
Propositkm 7.8.1 Ld C/I~ )~2:' k" _ _ ;~ M' (O, C) ~ to a lUukm ",eo , ... Il, "nd ._IMIIIl~ ' ~ 2: I) if fight- "",,, Il if and (.u,.). ~, CO""'''iiU .... ~ , Il·
..,...1,..,.,ndtd
:> D, t1..,ru xisu " comJMC' oe\ K 0Y0Cb thai V ",,(0 - K ) S ! I<x all .. 2: \. If I E 1(+ ill such l hat I S 10_ ,.. and 5UPP(1) C 0 - K, thea If Jd;>1 - I 6"", _ _ gd;>,.1 S • !of each g E 1i(O, C) aati:dyi", 191 S I ; thcrcfO«!, f IdYll S ~_ Till" ........ ttw. VII"(O- K ) S e . Now fu 0 I!O th.o.t, fo< MmpN't "'" K , u.."". io an n 2: 1 fo< ... hN:h V,..,.(O - K l :> t. IA:t ( V.l .~ I be an inc,""""",,, """,ucnce of open lieU with COlI'po.ct cloou,..,. such that o _ 0 ' 2:1 V, . In "'"' 01 lbe dioc,..i '" immediately followi", Definition 7.8.1, ~ ean ot ruo:t; ... l/.ridly i~",.eq_ (n. ). ~, " f int~ 2: 1 iIUd> that VIl ... (O - V. ) :>. lot all t , """'Ug&I "",,,,,•.-Iy. &od ~ h.o."", \ t (which to ~bIt by lhe remark IOlIowin« n..tinition 7.8.1 ). n...n the (OI$f1,lCtion can CODti""" by iDductioo. TIlt _ L~... , - L; are diojoinl . H...,... the IUPP(p;) are diljoiDl. , and 1, 1 S I !:l!: ,g;. MOl"" ..... , ' '"' 9, + "' +" _1 on L; , I'ore&eht ~ I. 009 to conlinllOUl 011 0 '"' q. "",(9) con....... to 0 .. t - +00-
if, _
U'O!. '
"flwo,do.t.
Bo'
and
11'-. (!:.,.. g; )I S VI'-, 10 - L,.. , ) S t 13. ..,bt.
111en Re~.(g) > ~/3 if t is e\..... , and Re,.., (g) ~ - e/ 3 if k is odd . But""'" tho ~uenQO (,.,.. {g»),> , (."not bt .. c..llChy 1OlQ""""" and ...., ... ri~ &1 & contradicUon • .-bid! PlO w.. Ihal /". : ,. ~ I) is I~'t &nd IMt (,....)..2:' (IX1~'lIeo1 nanowly \.(> O. 0
I
1.d n be a "'p,,,,,,M'.t _ • ...a I : n _ II: a fOndion. \\-.. defi..... ho: _mo.ion of I ... pOlo ... En .. tho .... mber in II:
wi": I ) .. Ii .. "''' /(6 ) -
......
.-.
b", ",f 1(:£),
fiChl.-hand
side io dofinod (i.~. wben ~m.up . ..... / (6 ) &hd lim h.t._ 1(:£) OM _!>eo botb foqU.O! to + • ...,. bot~ «Iw.! to -ce) .
...!>eo.. . .. the
,.
I . SMor th .. tho:function r _ w(,.: 1l io u"""' ..... i«>rumuo..on i .. domalR 1.
If I . fini'" ""
n. ' - u-..t II _ ...(0. 1) _
0 ud .hat, "" ..-err" E O .
!i",..,p I {z ) - 1M. ( ...)-t..· )
,ho.
(I..«, < wlo ;1) be a ................ _ r :S; ....., • •• lE V' V / (z ) _ 1 (, ) b - , . ..,;p,Ixw1wood V of G . c.o",utel). Itt. :> ",(" :/ ): _ ,hat '""l>i . .. ).u • ., / (z ) - I t.) ~ ,for .....Ilable ~ U of ..). 3. !.d I : {} _ It be .. _ oocaicool i...., ... fu .... \on. Show .hat. if w(",/l .. 6..rte .. _ .. E fl. tlloa !iminf ...... w{z; /l _ 0. For I~io, """" by """,,,,ditto:... oIoowi"l lhat, ..-.. tho: 00DIlU)" IIypotbesio, .1>«0. "" .... poi~" z ..-bitnorily _ \4" ouch.1Iat I (z ) io"larp .. d -' od.
4.
2
r... ..-err ....a..J n"mW,"p/ q I,, "' • . : if . .. 0) . pu. fir) .. , . Show.haI; 1 • _ ,11M ..,(4 : n - 1"00 ..,. an " € Q .
[.g {}
be ... IoO. .ntI'I _ ! .....""",\1..""",.". Q. 1> ....... of n, COftOt."", .. -.q ......... (U~)~~ , of.,.,... wboot. "'- V ~ U~ Iuof """'1*1 _ _ C
lei
Loot 11 boo tho of 1M plano R > .. _ ~ .. , ow .... lbe ~ of t ho line 0 _ {OJ" R ........ be poiD" (l / .., t / n'). *~" t.ho ... N or otn'& or --'I....
..-.- """'"
'--
""" nery polDI (0, wl in 0 ODd <Wry ;ntqn" n > 0, .... T. {r) be the tot of ."'- poin.. ( ... ~) E 11 • ..cb 1M' U:S I/n ....... Iv-":S u. l'Uo, .. __ of tho ~lt.er of....;at.bonI.IIim..., ( . ,~). &Dd _ , ..... hio _iooo ""I....tty H• ....JorI( ';>kv T 001 n. MOt"""" , _ lhal , i. Ihio T ,_ 01 t.ho nbtoto T. M io """poet ....... metri...bleZ. Dod""".!>at 11. e .cJ with .bol lopoIo:>c T , if ItlocoJly ...... poet. P ...... ll>aI .bol .... "'..,. ind"""" by T OIl io ~, S. Let A \>e l boo tot (OJ ~ Q, c_ br, Il. .. _
.··'."'.III. Thtl". ...., l ,,,,tM. R- Q - lJ..J,, O•. aodll .b.
;hat at _ ~.
doli_.
n,''''' lOp''''..,.
"... D. ho an im .... I"'i....
Coot.cIoodo IbM _
neiKbborbood of A lal1 ...... U .
• . Dod"", rf'Olll p&tI ) l /\at t_ """" ""btoto of n __ 'iDi", .bol d . . . . clooctioOll
0 - 0 _
(if",t/n' ) _ II'" Ir.-
IO,-H:Io{.
""" ~ E R. . - ."'" L n (, ) O("" io ·mal .... l han E "",,(1n + I )/N', Oood .... th.... b.-,. ~ ... K,.be"""" E....,.,o(.., io hite 2. Lg" be the Radon _....., "" n dOd. by ,be point no, Cw In 11 - OJ. Let U be &n " " , " , , " ' _ of 11 OnI..aitti", D. R-u ;"'" ~ can find an In.... ,.
-lQ.II.
lbe
Ie:!
7. Ro.cioG M:, ..
(1rIttpoJ of II m..M 10 1M "r ·ure" "'" (0. I ....... , doIlned tor \be _ (:ls"{1- z)--' .. _ b poiD:. pl. For ~ ... _ A of (O, I. ... . n ) . ... PII'
!'t A) .. J I .. oj.. I
11 , . ;be fwr>r" > O. put A ., (o !S p !S .. : tIIP)1 > .. ). a:od'- t/o.al I'{A) !S (1/,,') J g....... Coac1_ ...... ,.(A ) S I I(~""·I· 3. i.e< / : 1 _ C bo OOAIIrIoo;>ua. For evay t > O. tbnt ""... 0 > 0 _ IW lI(z) - / (soll < c b ... z , ~ E I NlioIyi ... )z - ,I!S " . Oed""" I..,.,.. ,.... 2 t/o.al. Iot.-,. ~ .. IMp. ''"-9.... """" 1/ (" ) - a..,(" )I!S
(I + 11111« "" 011 z E J. " -
, ' I"" ' (8..,4," ..'&"
lin,. "'l'.."
1/ ''')1. ,.. "''''''" .......... WlII"j.
"~,. < _ b """"' , I.
S-.hat.ho roodi • .,.,
'"
" Y (~that, lor oil .. 2: (I....t o.ll bubo...JlI< of '''''' ,ho poIrw-ia1 .... (1 -"la- Oj. io...
o $ , $ n. to ~-'"" "'... 2. COO .....I;. , ',_ tho rooditioo i. ... ioIiod. u" V be dw _ ou"poooo of C(f, C) _ ..... br ,be I. ( ~ 2: 0). aM let ~ be ,be " - ' lot", "" V whldl """" , . UI Co lot _'Y k 2: (I. U"'" part 5 of Exa ..... e, _ thai, lor OYer)' k 2: 0 , ..(B • •/.) """verp 10 c . .. " _ +0, 3. 1'0_, lor _ n 2: I , ...... be .... Ii...... fotfl>
"" C( t , C ). S""- ,bot u. ill comim,o... aDd ,bat, lor ~.t 2: 0, ,be _......., (.... {f. l) . ~l ron,ugtJI '" co ' 0calIy """'--' HauodorfI'~. E. _ oubol*'O of c (n , R ). aDd P. _ _ """" in q o, R ) ( i . ~.• • ouboot of C(O, R ) . Ch I"" , aDd .J> beIone: to P b oJI J. 9 , h E P aDd all ..mil' pooi. j... .-l """,bon . ). Su _ ' baI, "" NCb h E 11(0 . R l, th u..n.
n.. roaditiott io drariy '...
f
' 'Y. A-.tmo
,be,.ro.-e. in ... bat folk ... tba. it •
...ioIItd.
I. Le. E be'M _ _ ...bo~ ol ClR.R) _~ by.ho runc\«- ~ (i ~ OJ,. 00« d 1 hao .. (_ &;_8)_ 2.
4,
and
o:mo,"""
~
b!' . ranoIinile h> 0, lot ~:> 0 1>I pool>. Ifm (Ie ," 2 tI~'_ w!.eo> A 10 """ ~ 1 _ r.c.t;d.; BHIi g"ij.1'o 'iIi" ..... " .. ' ..... £nmpOolU.p. «lf)_
,,0./
t
10
"C, ••
I . 1.0< E boo ,1>0 ,uboonditJo;o· )_Stielljeo·
2 .....
1"• ., ' .... )
Fi., . -.I ....... be. & :> 0 aIOd • IUUtc Iomily U,l .. , .. x (n , C ). ..... 10< h ..... ..,...1_ f.....:i"* fIom 0 _
(0. I I. .. IUd! ""'I..aI 10 I .... K •
U.., oupp(f. ) ..... 11M """_, '''ppOn, P ...... 111M .......... 511lte 1am11r (o,l,.J of d*1nd po;nlo of K ...... family "',l,.. ito X ·(O) ..... ,hat
2,
I..(/.) -
:t" J /'Uo,)J.(a, )1 S ~ 1ot.oJi
E f.
_,p>I.
o..d _ _ put l ..... , .. boto M {O.C) ""'Iuippod with ... """. If beIontIlO /O" ""'(I) - "U)I ~ '" 0 .. n _ +00. Let
..... ""..Ibal .UI>,."II'.(f) - .. (Ill do.. '"'" .......... '" 0 .. .. _ +00. Tbett, lor ....i..... I > O.... ""'" find • otrictly i" " -;" 1 " " _ (,.,).~, ill N _ ..... _ (/.)'21 ill H ouch lhat If'.,., I/.) - ,.(/.)I :> 0 lor all ; 2: I. Put
,.,., _,,_ I.
Eq"ip C(O. C) ' ;t h t iloo topk>o of unilomt ..... . .... ' 0 " " compooct teII_ l'IMm . .. 10 ....tl-"-'tI. q O. C ) iI _,,;" ... Oed""" ftoon 1<JcoIi'. thee>..... ,hat thmo uboeq""""", (I.. ), ~I of (1' )' 21 .. bIdI ..,., ' UgEZ ;"
u'" ••
C(O. C) '" • fwtoet;"..
2. B, P",,,,,oil;"" 7.8.2.
I
E CO(O. C ),
(~,
_ " : ; 2: l) io
tbon _ ' " '"' mirillf "" .....ring ()J p·i~ IU.I. SM_ 1h4t. 1M" "'""'" """'Pver. 4u..) c !.hv.} and J IdY ... - J IdY" for all f E l h(".}. by Propooilion 3.U ."., ~ri Eo, - E, is ~·l>f3liglble, heOCO= K Is ~.In"","abIe and V ... (K } _ V,, (K ). N....., to.- each E E . , thom! il lLII i~ ""'l""OCO= (K~"~, oloompocl. . ul»ctt of E such that E- U.i!:1 K. ;" ".pq!!dble. Then
V ...( E - K~)
_ _ _
V ... (E )- V". (K n ) VIll E ) - V,,(K. ) V,, ( E - K~)
a>&lWflt"" to 0 .. n - 00, .. hlo:h implitol tht.t E -
Next. V~( E)
u..;!; , K• .. jl+.""!tli«lble.
_
... pV~ {K.)
-
aupW". (K. n V) + VJI+ {K. n cr')1
."
" l! I
_
..,p V~(K. n V) +IIUpV~ (K. n lr) '~ I
oi!:' V~ I E n V) + V". (E n U') .
-
lor Mo. ... 6ote io ". __ ura!>le. FUrd"'f~.
V.pl''''
V". 01 +-ta,
V,,' IV) _ ,ul'(V,,(K): K c V.KQ)mpACt} oupVJI+(K ) :S V". (U)
_
V". (U ) -
•
oup(V".(B ) :BCU, B jl+, inlegrahlo} "'pV,,(B) :S V,,' (UI,
•
V,,' (Uj - V~(Ul ·
eo.
_
luontly, each ,,'-eJl~""" illlouIly ~-~bk EadlloaIly ~""Cli&ible "'" A rneoets eYer)' E E • In • ,,-''''CUr;l''''", ADd 1>0."", .... ""Cli&ible. lie\. An E_ So A Is loWly 1'+""""C~r;lble. Coo~. let A L
168
8.
~!'Y
be aloeally I'.-'''-'&li«il* ""'" A ,,-. e'O«IigiI»e, aDd fi. . ... p by P roposition 3.9.3. 0
"'-eeu.
So. gi"",, • Radoo .......,,., I> on R .., """ that"""", metl8uml ~ on lIttnlrinp S&tilfy ;. _ ft+ (01' fl ., jj. ). Now. tbe £UlXwlng f'I'OI'O'itlob 108 theoU......y. PropOSition 8 .1.1 Let .. Wc • ......i.-ing in 0 mn. ~ ~ wml"'cI .d K.
(e) V" . (U ) _ sup{V"..{K) : K C U, K v. l1>eD
s:
/
INdv .
inf
1V1f f dJ" f E H+'/:5 1,, 1 :5 V,.. (U). Thus V". (U) ~ v· (Ul. It IoIIowf that V". (X j :5 ,, "(X ) for a1llu'-to X of fl. Now V,..( £ j _ lDf(V".(U): U::> E, U opo::n J- inf" v "(U) .. v" (E), fr -tI E E ... 10 v' (X ) :5 V". (X ) for each ouh8et X of f1, by defi nition of V". (X ). Finally, v· (XI- V" . ( X I . The ,,!&in ptOloolplions of v . " d V~ "'" t b.. (~ idomtk;al. In pIlrtk:uw, ~ E E: • ;. .,.int. pabLe. Sio.), and.., f fdj;. .. f fd~ for all / E el::u.). 0
J
/d,..
Lhu.),
1'hI: roilowi"ll; "",-,It ill putleularly
J J
irn~.
•
£n., "
for CI>a;u< 8
169
Theorem 8. 1.2 s."""", tlwt mel! open , ,,",,1 01 n u m ~ ..,04>1 01 ~ Kto. Ld " ... oem;,;"g In n ndI Uw 4> ;, th.~ Ben! .. ··19'1' ... 17Icn e.....,. .......... ~~. "" .. ,lOCh Ih4t V~. ( K) < +o(),."..u a>mpacl ..,u K ".".;a!w ''cn M;"'(. ,C ) may be Identified ...-lth the ...-Ide ol M' (n. e ) (11".0" '/0 &.1.1 ), and the study 0( -U coo·.co ........ oimplillcd.
I
!..oK n boo • lof!d by poi'" m' a . (z E X ). p""", . hat ~ io """La.- if and only 'fE .. z n« I,)io lin;" lei< <Wrf comPK' ''' K aDd.hat. bo Ihio __ •• to. RodDoo ' ure ...... (""" " io dofino5l io _ 1.
4
'1** "..". .hI: Iioold Q of .... iooaaI
Ref",-
to
~
2 aood . -
• polnt ~ - (t - . )rr" -.I .. ae;,.. "" -.It '" tho In! " , . . I(l.+, ..). ~ , It - 0){2··'1 and !b - (t - d)/2"+' ..8(f.~, ..II. Leo Z M . poW. of E - (I) _ io lite ""iCiu 01 "" ""-'tJ """'it:....... 10 E. S~ tlLal ,be riIJt1-lwmd .mi>at ;'" 011 at z .. o.
E>wt[, [ 3.
Let z be .. poiDI. 01 E _ {OJ ..hid> io ,be .~
E. Show
,~
b~8
171
ri&bI """pol", 01110 I.. tenal con-
I
hao DO left·bud don...i .. (!a.I.. (It" iDfiAne) at z. 51""" E 10 _rizablr, llOIlDpact, totally d .......... eeI . ud h.u "" ito "[eel poiDI., it io Itomoc>n~ to ,be Cantor ... b)< •• ' "'i e. , ..... .... ical _Ill •. TL".1ore.1 h.u DO Irft-h&nd don ...;.., at "[' . ,.tobly """'1 poi .... 0110.1). 4.
to
1'...... ,be ............,., 01 .. function 9 from 1 illtO R ouch , hat Itl :5 3. , 11M .. ~ ,'iciblo ... of pol".. 01 diooool.lnuily. Md I (z ) ril )cll _ f g . 1",.•,400 b evay z E I (.... ,be dottUtWec! _ ,.. Il, diotribuWld _ulo 1. 1.0,- ,. "" T...:I _ _ to In! ... t e with t'OSpe« 10 ,h.io ind-'- " "m' ~ 8.\
('n :1 ' $.1 .1). 11.2 Let ,.' . I'~ M . ... _
...... _ .....1Ii"'" 5". S" (with 0' , cr _ .bW g ' . b . 6 ,. : A t ".4 ~ _ ,, '( A' '''~( A~) _ ,100 5 I 5 _ ( A' ~..t ", A' £ 9' . .4" E ~ I (.. _ .~_ io n ., 0' >
n "
" '' ' ' . . . . . . . . _
r
.,,_
n of R '
( .... /t t
rnplIoc _ u n OIl S. We ohaII.,. that 1'(1' .....;,.. boo, a nouemp(y tn,
Y .. not>o:mpt)'
8U~
" •
tal Y!!I
~""'lI3U",bk
MId the T -ae15 are " .illl
~
r
!d("l.
g.1 M_un 1"""""dooop to U, ",1g of .. su.l>inler.....1 J of I , .tooD /J f T """,- and ., ~ nw:o.eure on T .
9.2
Fubini's Theorem
be ,wo '''''''''npl.y ""'" and S , S" I"", ""mitiDgllin rr, rr, •.......,. ~the (t..s _ ( A'led in1O" finite number of S.oete.. Tlierrlon:, S" .. """'""I, callod the prOduct cI. S' ... d 5", ~ by S'" 5".
T boorem 11.2.1 Lei ,,' 4JIlI p" lit ~ "'''''nI'U on S' aM 5", rupoclilldr, _Mid" lit 1M ",ftditm Ufi"N on S bu A _ A' x A~ ... I"(A') · P"(A~) . nom" if • m........ on 5, mtkfl l" MOI"""
S"~ """ptCti""ly,
( L
a"(I'(A'J
1"'8'11) (
L
B " EI'(A~)
L
1,"(8")1) ~
1 (I"s,, ~)(B' >< 8")1:5 V(P' ® p")( A' >< AN),
IY(PI A ') n (('(P\A )
D
DeAnitlon 9,1.1 The II ' U""'" ,,' and 1"',
_II' s
" n is e&l1ed tbe prod\l Sf(S, Fj, obsen", t .... t o ls' , ,) lief! in SI(S", F ) for all r' E X ' and lhal s' _ J o (s'. ')dpU I;'" io SI(S', F) M~, J djJ Jo (z' , ')dll" - J od" , Heocoeb\h, uru- ot","""ix su.ud, ....un'" lluLt ,,' and II" are pOait ive and put " _ p,' ® ,,", Denote br,J" (.1' '', .1"+, reopecti""ly) the """ of tboooe fuDC\iooo from fl (fl' , fl", ~i""ly) inti> 10, +001 whicb Are upper .... vdoopeo of ioore&ling 00Cj1lft>lW iD Sf"CS) (St~ (5'), St+ (5"), "",poai,,,lyj ,
r
r r
Propn 9.2. 1,
r r
(I' ® I"}d" n 0>«+00) " 0 ;mpl~thal.'Sg" ~ toj" ,lO thac. r g'®g"d" .. (r g'd,,')(r g"djJ'"), o.nd lbe derilw:t laequalilJr IOlIoows. a
.r+
r
r
Propnodtlo" 9.1.3 A' " A" it ,..negligible for <WrY ,,' .....,;bk HI A' .M ""'Y j/' . ...-...u H I A"". Mo"",~" , A' >< 0"" it ~ " . JUgligibk for ~
/ocal/r ,...~
s' ·,...~ od
A'.
yr
ate
P ROOf'", To ~ the first 7 7Iion , .... m.y 7181iUme Aff _ult;' then a ~"'""'"' (If P lOp(I6ibon 9.2 ,~.
;I'.inl~rable:
the 0
Theorem 9.2.2 Lei J' 60:" I"· m...... "'w" ""'","9 from a' ;nlO G m~Irimh'" spa« F . TheN /: (:r', ... ) .... l'(rJ ;. f'.m.,.",rnw".
ring gmt'ratal by S' ($'" and S, l"eIIptrtiveJy). Let A • A' )( A" he .. oonempty S·$Ct. If 0' if; an 'R' IA'· 3implc mappinl from A' into F, then 0 , (:r'. z" ) _ o'er) .. 'RIA.... i.mple. Thm: aiw .. 1"'''''Ili!ible",,~ N' of A' loeb that I'IA' ;. the limil, Ob A' - N', of .. 1Ieq_ (O~)~~ I of 'R' IA' .• imp!/: ""Ppinp (Proposition 6.1.4), 'rbo:a fI.-t "' tM limit, on A - ( N' " A"). of the ""'I""""'" (0.).,., of"RIA...fIirnp!e mappi"., wlLldl pr<M!OF lhat f "' Jl-meuurable. 0
PIIOOf":
Denote~·
'R' ('R- and
~pocti'~ly ) the
In partlk....wn ,," .. ",,,", , ,iltlt.
T beonm 11.2.3
LJ.."
PROOP: Suw- tW I W- the "",,"'ani ,-.Jue ~ outalde A;. when! (A; };", io . finite or ...... wlt. famUy of d~nl nonempCy S.-eu. For..wry i E I. the"' ....u . I'-""Ililibie oubeet. 11', of A; such that I IA; io the limit , 0
E f'l'. and dV" " (%', .)d\'#" is 6n;~ A•• oon.q\ICIICC, Hz'..)4JtId ' 55 'lion rollows.
f
f
o
TMon:m 11.1.5 kl I .... p .........1<J"ObI.o: ""'cCi"" frr>m n .. p.mkrole. ~ /!me""""
i,,'" [0, +]. II I
z' ... t/(tI, -)dVp~ ...... z" ... j ' /( ..z")dVP'
r r -r r
...., ~ and moleraJc lor I"
j'ldVP
-
...... 1"', .-..po/II:1..",j~; "'~,
dVl"(;Z' }
dVpN(;.-")
1 (z' , _)dVpN
l(-. z~)dVp'.
,
••
9.2 FbbW'. n.w.ew
II j/' ;, modmtIe, t.'Ieft!
c:om"'r)e~lee.
dVjl'(:z')
dV,.'
1(¥ ,:tdV,.N,
f{r" 'JdVjl~
o
tbt rcoof·
,.''''
r
J'
PIW()f': Th. is • diroc\ _I' Propaoldon 9.2.11
.uenc:e of 11leooem 9. 2.5.
r) into r
(~tiwdJ.
o
ut F' , F", ~n.OI F" ~ Ihroo BaIl4dL IJ'OI
I'
r
J'
.....
ppi..,,..,,,,, F' " F" iJOI F . ul
N.n UKJIliallr,.' (~""I., j/'J ·in' ,iLW. mill'" ,;"',.."", rr ( rup
SO_t
.. £ JS,
FiI S E S. and let (A~ II A: )~~ l bot • lIOqu..Oe of disjoint S-atU wboooe union oont~ ... S. Put E.. _ s n ( A~ ,. A~) M.
DefInit ion 11.3.1 Let n be . IIOOmlpty opea out.:t '" R ·. and let S be tM d-. _ 1*;"3 cI. tile ~mpty !/Ct and the rect.",p. A ..I>«< cloo~ A II ..,.,.·;ned in fl. S Is called lhe natural aemiri", in n.
PrDp08ltlon 11.:5.1 hl tile ...,/4.;"" 01 Ih/iroiti,m 9.'.1, rl u .....nt.NIc ....;.... "11Ii.rjoinI. S·.-..:/4"fI'U, mch ~I .' i ct ""')" ... _"""'" t<J "'" of .... e 1- n ,~;~o lp; tr' ,(p.; + I) (m 2: O. p; E Z).
f2"'J
Foe _ \,.. W ' m 2: 0, let q ... be the c .... """,i-U"3 01 all «
::r. .,
s: , s:
A. !1eI In P" .. hXb oon\.radlcu \.be floCt tbat,. dol!! "'" I:>cIoot to U.. €p A. ",.,.,""', A.:;" n fr< ;. 8 fo,. .-.:b m ~ 0, and ,.". ean ooeoo .. pOint ",- in ;r,;; n fr; I>Wnl>o< ...""""
tho.!" _ r I .{i." ""1IO ._ r itllepab!o. .. bt;n Rq,, ) .. ,ttktl1_. ati ... bu• ..,. ........ " .. ~ 18 ........ ~Ml)" uil.. Ho ''''' , '" U" Iou
p",...
I: e"J,/ib .. , _ 0 "'10,+00/
bu .. Iimi • • _ by f:"" ,.., ,/ib . aad. _ +
".
">I (
.
Lot 9 be . hoo fu ...""
._J..• ./i
..!.. ~" G -·) Kt ... boot Ii of
t.bont.,..... = ..... CK... &l>d>
jO. 2"b ball (::::. , )
I~
/(
~
10. 11.
(DCI.a5io1O of part I). Oeduce drat
4.
s :z
_
,ho.. 5 io iIIRniteb- dirr.". ,I&bIo.
Lot:::: E jO,2'o'[. ~.hat
IIh- (D (:r., )I:5 CK.~
,>I" 1.N_' . \ - ...-'
S. coo,,", ... unifomlly on.be """'~
. u..... of jO . 2.-j 10 .be fuDCIioa ~
,bat.
"""7"
Jr - l'~P'''' . -;:;rdr
• U _ I .'" i.:r •
011 IPIU$1n t ~ I. and dod""".bat. ,ji . .ui.:r_ 1 • 5(:r) -
": ., G) (.a- «3m - L'll &..
•
1
"7. dr .
>I'
I . ,. :5 ::itio
.p"""" .. p 10 5 · l p. ..., 151 ,iIo .,.... ...... .,. cl R ' .,:.b _ , to 1110 ~"" .hroqll ( - I, -I) and ( I, I). &ad ....ilariy let R.a ... boo 1110 of R ' witb rdp11, .....
to 0
·1
1. ID'ho __ ."""of_lt_ 5. b - . in....... 01 {D. 2, 4. ' , ' . _
~
O. b -n ... (i, ..... ;. )
b -n intqrr I S k S 3, 1>\1'
K •.• _ ,. .. (u. «2k -
1.2kJ) ).
So> K ••o • (~" .., o· . '0 ~"' ... ) . / ••• , (12k - I. lIrJ) .. hoeno it _ _ .h.or. j, _ i,f'J •. ... j. _ ;./2. p ...,..,.h.or..be .... K . .. (10< 011 .. i! 0 _ 0II • .and k) ..., dlojoi
fo< """'" d>Iqt< n
f,,,.,,;...,,
3.
E.~ , {4/7'-') be. puint of 10.1) (0 S /, S 6 for 011 . _,1>0., (" .).~. l'*Obeo.cl.'
.... in.,....~. By \\'h\tney·. ~n 'Ir.oo "~" the . . . . . . . COO'II in_1y d ifl'....... iIohIe f""",100 I from R ' into R wbich .,.,..... F _ 10 _ h , bat 01(.) _ 0 lor all. '" , ([0. 71). 'Then
all poOrwo of ,(10,11) .... tlItICri 5 , "'"MaI ~ ... M~ (S) , allrl_K" ;, Itl , ..""""..m'" M ~(S).
from
n iroto [0. +001.
77>( ..
,,' V) -
$UP-fOlr if) ;. ;. v-mfal.nohI.: /r all .. e H
Propoa!~1ou
PIIOO': Oeoote by C ,he dose. of tho!oe S·f!Ily if it "' ,.,.." .. ,,!urable !or All a 10. A; a mawl", / [rom n into a """ B&nId> Ip&O! ill allally p;~ If and ODI)' If it "' tdia1ly ,....im"l'"abie!or aU n E A and tbe lim! L..~ .. r il l d,.,. ill finit~ . in wh;.:h P'II g from n Into ~ """ Banach SpfoOO "' II&id to be 1oc61ly " . illlq... bIe if ,· l s", " .inlqrabie for IlII E E 5 ; iD thill~" 10 j>"meMIIl&bIe. P rop
E""
:$
Fl.rst, ouwoee Ihat 11 _ C< . 1 ... , ",!>ere / is finite, the "" "'" o>ml'l"" " ..mbe.... and the A, aN: di6>int S--. Ci ..... E E $ , 10. ~ i E / t~ ""... a Iinlte p&Riti< All j E J;. Id. ptB,.;) ",n through the claBo offlnite peni.1ono of. B;J into
s.....u.
s_ _
(1, I· V"j(E) _ ~>. I· VII(E n A. ) ;(,'
,.
- D" -L N, "EI"("' L ..1W)I '101 iV . 1'( ...... )
.. L L SUp L Igl'(f') 1 «;1 ;EJ, "'EP(II."I .. L L V{gIi)(8 ,.J) :s V (g,,)(£). pt8, ... )
«;.1
,E',
le,>ef;&! 0, \tw:re ni:!!tII a ill SI(S, C ) ...m S __lhal. I.. - jI . 1. l otVI' ,/2. For .u fini te ~itlo.,. (E.)"" of E Into I~
In
J
:s
L J ' I~dl' .f./
:s L lf~'IK,dl' ~I
1a,4V" +E/ Io 9 1' iii.
J
:s VUI')(F:) + 10 - rI' i.dV". So
"""'(lit,. V,,)(£ ) ..
f j,1-
l.tiV"
:s
f lol l. eN,. +f la - f l·
l"dVI'
:s 11 (,,)(£) + f/'l:s V {gp)(E) + f. and. ojncel .... arbitnzy. "'" condOOe thu (191' V,, )(£1 :s V{g,,)( E). 0 ~CIo) oflocally ".!~ funo;tioro:o from 11 b>lo C if_ UInlpie>: motor fIll'Ol'. and the JIl.I.pplft,c 9 - 91' from ~M Into M (S, C ) if Unear.
The let
"" .. 1121' If and 00.1,. it lf' - 91 1- V" ... O. Eq,uivakntJ,y, "" " rnl' If and only if " .. Ih kIaoJly '"'to j wbt:n:. When I' is pOIIl ",,",
".0.1_
Inl{g, /, .9\lI') ... int{g,,. h I ' j.nIr if E n X
(II) .4 m.o.pl'in9! frgrn
<mI, if /tx
n
;, IoaJI,
in./o a "",,,,,,,w. II"'« ;, 9p·m""~",w.
;, p .......... n:.bJ.o: on X .
if.n
(c) A mq p;,og! ".",. fI ,...... Banado 11"'«" gp .............w.: if.M """ if J, ;, 1""'= , ....101.; it ;, .......,.,...,Ir 11"""4' .w. i/ an
J
J
P IWOF: Lo,t E be. locally p .J>ef:liJihle .uboH of n. Fix A E S . G;""D r > 0, t,,"",cxiot8~ > 0 0IICb dlDl J" I, l dVp S r forever)' ,,·in~ ... beet 8 of A .... too. ""'''u''' Vp( 8 ) it Ie8I tliM ~ (Section U ). Lo,t ( A. ). 2:1 be a Iotq\IeDCe of dioijoill1 S·~ ;,..,luded ;n A Ai5dllhat E n A c U.2:1 A. and E..~, V"(Il,, l ~ 6. Then
."
- L. j ll l·I A. N"
." .. j 191· !UA. N" S.
0,
and bo,><e V(/,,)"(E n A ) S. r. ThlI "" ...... thai. E • locally I#,,,.,pi&ible. Now . If ! if. p.me&8urable .....1'I'in« from (} into. _ri"hIe BJ'IIO". II • 1!,-lMUIIrabIe by Pros-imn 6.1.4. NeJcI . let E be. locally 9#'nrslilible Il0)l.. We abow tha,
• !'-U gliglblo! for all A E S and all 0 > O. For """" intqer m ~ I/o, let (.4.....). >. be a - t _ of S·oeu contained in A.uch that
E- -
.,.
(U .4...... ) n {,, : liJ{zll ~ r} :J E n A n {z: Ig(zli?: r }
,.
.
V,,(E-)
con~ w 11"""',.(0)..(1) outside a II'"~bIoe aM 8. n..,., (""')"~1 CO.aga! to l(.U,,.(.)oo(I) ., _ 0 O1J\.Side U>t aet 8 n (% , g{%) -! 0). wlUd> Is j.I.oq!i,sibl/:. Bec"_ J I"., - o.II,w" - J I". - ".l dU, IV,,) br aU p ~ I and q 2: I, (O"I ),,;:' II a ~""by teql>tnl:tI1igibie, and therd"on: that , - '(0) Is locally gl'"~~ So • ~ E ~ locally ,,,·neglWbIoe E n X II loooJIy I'" nqli,sible. Nooi Iel I be a m.t.ppinl from fl loto • metrizahOe Space. Fim, ""PP""" tbat I", IIl'"meMurabie. For every B E $, ...., (An find • IU*, A E S of B n X tIC) lhat B n X - A Is I'"ne&lisiblo: . T l>ert exis\$. I'"ncg!igiblo: ruhle\ N E Sal A """" thai IJ. _,. .. 5,._,. and I (A - H ) io ~ (Theor-cm 11.1.2). !>iDoI! N U (B - A) .. 'I'"nqligiblo, ...., ..... IW I II g".
,.""never
_ .able
l1>te8l1Tt\bIoe.
c...,u&tly, oupj)o8= En: Ig{r. ) 01_ Thill I, .. 111 JocaILy j>-";""""~ tTCiYWbcre. and / .. / locally 9"''';''''' nc.-Iog .... .,.,.,.,.of1t-.u......:b tb&t / .-I8beo ouWOt and, '" all n ;:: I , put F• .. (>= E E" : ]/ {r.)] S n). ~ tho I ·1 , . U"II ,,..inlqnbio, t.bo _!'Pilip / ·I,.· g ~ ,..in~.
+
u.2:' e...
Morea ..." ,
j l/ l· l,. 4(!,!Vs.o) ... j IJI,I,• . !, ldV,.
s j !JlldVs.o ,
r
and ...... I! II,. loCi 5 to III .., n leDds to +00. _ _ lhat 11I4(lilVs.o ) .. ftllhe. Tlw:.do.... i io ,j>-lmq:rablc, and I io .....,iall)' g!>-Ln~abIe. C
Tlreonm 10.2. 2 LlI .: n _ C i.e Ioooilt/ "';nllf'
r
/no from fl into +on]1IUc:h tloat " 2: I g. Then " . Ix - /I,pg Is I'.i~. "" is ... intqr&ble and 1 ~"" - I"· Ix dp.. But I -Ix S ~, ... hen O. In PIOpoiltion 3,3.\, ~"by v - I" - (" and ....Ul"" the F... are S-au. We """"lude tlt&t lhere • .IJl $..eI B C £ ouch tlt&t v{ B ) - (f,, )(B ) - .,,(B ) _
IlUp (,,(A) - (f,,)( .4) Ad ... ~ "
.,,(A») > G.
v{A ) - U,,)( A) - . ,,(04 ) 2: 0 for all S....t.:u A of B . Put ~ _
J.",eorettJ • J>f'O""oe '1'''
...
0
Notioo that tM modlt .... 01 TbeoiEm 10.3.1 AlE ""t¥>ed if fl bck>np to 5 or...... $hall _later, If I' io rqu\o.r. Theorem 10.3.2 Ld,. k
....... " S.
"""Iitibol ..........~ on S. " ... Id" k" """ mOl·
no.. foJ./t)lIJi~ ~
0",
qooi""'~
(a) F.,.. coery UKIIliAllr ... i .. l¥"ble jwodion f fnnn fl mID {O, -too{ elld.! S f. TIn.. ~ition (_j imp/iN OODdilion (b j . ..hlch La tlll1l imp/leo 0) &lid write 1/1 (or the function t~t it equal 10 IlIt:z) at
n..o.-
N_ . for 8Wf)' l"tp~, II 2: I, put A. _ (>= Iii 11: k(>=) 2: II}. Since k · I .... is 1l--..abIo, I .... is .t.lHDNMIf'&bIe and!lO A~;s .,.."..aew:ablt. The funclklnil / . LA. dec .... 10 0 M II _ +00, and they ~m.o.in bounded by / : roo: fW:d , > 0, 'MS"'" t~~lo<e lind an inlq;er N 2: I IIUCh thai. f / ' lA,. tW S e/'l. Then, if h : fI _ I NJ.io&B O!S Io !S / and hd b, I' and b, u. The n:tull IOIIowtc on IoCCOI!nl QfTheorm> 10.3.2. 0 P ItOOV: W$ rn.y
~upp\lu , ~. crnd~. ~ .. ~niqwlg del~ed ~ tit~..,
--
U.
U I I' <mel
By the RirR d«oinpOllilio)n t~ Re~ '" ..;, + .,here u;. ~ M (S. R), V. ~ M (S. R ). V. c: I' and V. J.p. Similarly. 1m ~ _ ~ +..:'. Then " .. u. + .... _here~... + iv.' it ahooIuu:ly conti ......... ';111 , ~ 10 I' and ... ~ + ill,' It disjoint rmm " . N"", lei. " ~ I . +8. b$ a"odoer <wwtpo:$llon '" " ..·ilb reptoCllo 1'. n.en", - " '" 6. _ ". II abeoIul tly _lmtOUtl ";111 ,et ..... t :.0 " and 10 " . - 11._ T hUll ". - II. .. 0 by P' pOfiIIon 10.3.2, wheoce I ... ". and So .. ". , 0 PIlOOI": ,,_
v. .
thaI I' It
pClSili~_
v.
v.
A _ MY and ouffidenl condit ion thai 1_ I'" · 7'u ..... " and v Oil disjoint;' lbat they be 0CIDd OIl
11>erdo. " lnlll' ." ) '" o.
poali_ For ~ A E: S. I,, · 1..1(".v ) '" En F, ..'hlc:b PfOI-"" WI I" ,1n1U., 1') '" o. (J
v...).""
Pr-opwitlon 10.3.4 Ld k c ""I........ 0/ """"".... "" S .1Itdt thai ~ "n.:! "" .... m... -.l1J "npIa~ lor oj! diltirKl p E N .. "J q ~ N . T1toL, /0dno- Nikodym lheor-em, the", "";111 ".intecr&l)h, fuoctloom I, and h from n InlO (0. +0:[ IUd> thai 1"·,,, '"' h" for alli S; S 2,
+
We .....y.uW* that /, .."ilihel
Oil
0_ A &nd tbal h / A II" : w,rsblI,
S,,, (Tbawew 6.1.2). N_
inf(J"h) _ 0 p-&Imost eoerywbert, to 1,, · II , ..
avv-nlrated on
...... 1" . II. iI a",cerura,ed on
Ft -
{>: e n : It (>:) .. 0, 12(>: ):> OJ .
f or eto:h inU(ef j:> 1. ,bert exis1; disjoint S_ E:[. Fl, incloo.d In A. auch that 1" ' 11 , isOlnied by Ef and 1" ' 11; by Ff. Then 1,, · II ," carried by E:, - nJi:' E{ MId tbe 1,,", ...... 2.
P. ox ee jl", ... ilh the COII$Iruct\orI step-by-Jtep, .." obtain the .......""" ( £,. ).~ " which hu tbe deo:i.l"fJd ~y. 0
In I*rticuw, if fI beIonp to twed on disjoilll.
lOA
S...u.
_ u .... "" Ibtn_l_ ............. ~n
S, tbe
Combination of Operations on h'leasures
The foIlowI", !..:II ...... Immediau:, MId require linlo: comment. Let X be a _ ,ply 1Ie't . S a -run", in X, Y a ,,,,,.."'pty ... t.t of X , and T a .....iri'" in Y . If " ' and II, ...... t _ me&II\lJ"ej on S such t hn and " ' IT exist . tben (p, + II, ) IT alto exilw and iI equal to (PL , . ) + ""' IT)'
""t
P ropo:.lt ion 10 •• , 1 Ld H .. Gn
...... t 0/ M ~(S), ~ ~, """ Ittll .. U. ~"'" in M (S,R ). Then. ·b, . owI n/!icKrol IMt " 'T e:riIl ;, IMt " I ......t J.... 01111 E H aM tA4.t IUP"' H u( B ) .. /in;u. for""'" B e T . In tIIU ...., II, . - sUP~H " Ir' ~"....rcI-dimc:ttd
.......ntion
p ROOf': .u.ume that "" exiIu for 0.11 " e H and that ouP"" H v{ B ) II Snit.! b eto:h BeT. TIle .. Y • II, nu",abIe. the T _ ...... lI""im"l"bIe, and II( B ) .. "'P~ H u(B) lor &II B e T . F""LX A E S. for every .. 2: I, tbere t::ri8I. ". e H . uch t hat II(A n Y ) - II.( A n Y ) S l I n, and B. E t ouch that B,.
CAn Y
and A n Y _ B.
"( An y - U 8.) -
"".-~bIe.
1I.(A n y -
. ~,
U B.) S(P- ". )( A n Y )S IIp, .~,
10 II( An Y - U.i:L8. ) S IIp, for &II p 2: it p' ''''Il!&lble and that II" exio\.t.
N."..
I. ThilL
ptO'IflL
t hAt A n Y - u. ~, B. 0
,.
.
Propoooltion 10.4.2 Ld H Le" nommpl~ ,ublet oj M+ (S), Ioo>unded a.Iooor, And kl # Le it. ' """""IIm in M {S, R ) . 1/ # 17 en..t., Ihf:n I'h ~ OUP.., H " h '
P ROOF: For all ... " .., In 1l , !u(>("" ..,) _ (v, + '" + l"l - ",1)/ 2, """""
,11(>(""
"")/:r ., 9UP(V'/T' "'/T I·
It IollowI tlw (aup~J ")/7 ., auP.€AvI T) for ~ry nonempty HniU sub6et J of H. Sirxe # iI the auprmlUm of ""P.." "' , ",he.., J extODemply finiU IJUboeta of 11 , the prOOf i. oompleu. 0 Nert, ~
(j>,),u be" ! "bUMbie family of po8ilive M""""" 00 S ",ji b sum f'. Then f' /. exloU if and (lilly if ~fT exU;1Il fu< a U i E 1. and lhe family ()..-( B)"" .. aummabio! for Nd> BET. In tbil caoe, f' h - E"" "'/,. N"", let {l be" nooempty lIIlt and S a !lemiling in {l. If I" and I', ~ two 'rp·n .... on S and if g : {l _ C is Ioc&Ily inu"n.bIc. lor botb I" and #1. then g 10 Ioc&Ily W, + ",,}-;otO'gnl.bIe and + f'1) - 9f', +
gw,
g"".
P ropoooltloD 10.4.3 Lei H Le A RCnemply ~t 01 M +(S) , IIuruled a"'-. ondleC # boo it. n prem""'. Theng : {l _ (0. +co( ;, """'"'ill f" i"~" (and ""'" if) it ;, """'"'I, v.intoy, d k '''' 411 I' E 11 and (g ... ; " E H) is IIundrem 10.... 2 Ld 0' , 0" boo tW>o> """"""t~ ,d, ,,"" " , I'~ I.... _ r e i .... H"IIIiringl S' OM S" in 0' Galli ra,..,li",,/). if g' , 0' _ C ;, 1«o11r /,' ·inl" ' M V' : 0" _ C ;, 1«o11~ sI' ·inl~. IMfI g' 0 ,.. ;, 1«011, /" 0 /'~ . inttgJdbl. olld (g' 0 g") . 0 P") .. (g' ") 0 (i' I'~ ).
rr.
..w.
u.'
P ROOf : 0bri0uI.
10.5
Q
Duality of V Spaces
Let n be .. nonernpty ett, S .. toerniriotl in n. and I' .. comllla .......ure on S . Let p E [I. +00] be Ii....., and IH q be it. -I ill smaller lhan 1711. N,(I).
In ,*""",Iar. for all t > 0, "'" have "'I(A) ~ ~ lor all A E R. """" that Ipl(A):S (t/ ITIl' , which POO<el lhat " is.oh6olmely cont;nUOUllOl'i\.h rUlpo:ct (Tho"H~n
10.3.2). NOOI let F I:wl tile cl_ of ,..."., I'O'litive fUnelionoi 9 E !hv.) lor .. hid! 911'1",FN.(9), aDd I1· ~ .... IItradictlon; "'" QOOClude that Ivi ., 9 ·1 1'1. FOl" tbill, . "'" can find .. ~ ( E.. ). > , of disjoint that 9 van ..ht:a on n - U'~ I E". Then fl - u.~ I Eft is locally v-' >("gl;g;bl
L ' :$fS·
TIl ..,)1s: ITI · N, ( 1.. -
L I S ' :$~
1... )
10.:; o.../itJ' 01 V'
S-
209
coo>e.&&, to 0 "" R _ +CO, and """ is a -additi''e. Mon!O'I'ei", since I,,(AII < JTI' !PI(A) for all A E S, " is . "",asun:, Siuce 1"1 :S ITI' !PI, eKb -..tially p·inttJn-bie !Itt is .utl.ally ... !~. By the Rodnn- Nikodym theomn, .!>ere a locally p·iLlle&f&bIe fUDoCT.lon, . ud:llhat .. ... ,p. We have "I nTi locally ,...1""", eWi1*'he~. /uthe Ii...,.... formoITand ' ,. DLL Lb(P) .... contin\KlUlland qrel!oo Sf(S, C ), they .... idemical. c
.,xi.n..
:s
Now auPI>OM t Mt p '" +CO , Write M for the elMs of
,.._utabIe
leU.
Lemma 10.U Sf(M , C ) .. .w- in ':; (p ).
PROOI': Let f E £j)(P ) &ad N ,. {J: E (} : If (J:)I > N",(f l}. Given an j~, .. 2: 0, 1M I he Lbe _ oflhooe i '"' (p,q) E Z " Z sucb that the ... U&M R, ... J"/:r' , (p+ 1)/2" J " (q + l )n~J i n _ U 1(0. - 1'0'). For eKb ; E I , PIIt E, '" r '( R. ) and chooooe any Ci E Ro. l1>eo 0 '" r:~ II£, · Co beIonp to St(M , C) and N",,(f - o j :S 2-· . J2, "'Mdt prooes the ~
J,/2" ,
o
Propooltlon 10.5.1 Ld T lit .. II ..... ' /orm"" St(M , C ). In nkr tJW tMrt. ...... , E 4;(p ) I1Idl UwII T (h ) ~ / ,hdp I", allh E St(M,C), iI .. ~ ,, " .... ~ w.t w.1l1ot.J;"!I cond.tiMu MId:
r.) T (I ,,) ... 0 I'" """'1' I'.Mgligo~ S· .., E. (~) T (I "" ) """wYJU 10 0 10 w. ....Ptr ..t.
I'" a.rt)' - " " " ( E.. ). ~ ,
In
S roNdo Uti
(c) II E E M andT(I,. ) .. 0/", all S -..II F .ncilldtd in E ,
'Y'
w.n T(I ,,) '" O.
P ItOO+': First, I UPf>OI'II' that T ( I,,) .. / /I ' I"dp for all E E M , for. l uit.t>Iy chooen II E 'b(P). COI>dition (b) holds by the dominated (On.e' KtI>Ce lheorem. On lhe od",. haOO, let E E M be l ud:I that T ( I,) .. 0 for all S· lIN F ineluded io E; if F is .. BoN!I IUt.ed of E n {" : lit.,) '" 0) sucb that E n I" : lit,,) ,. 0) n F' is p·""SIi&ible, lhen g]" '" g], p-almoot e.",,,.1w:re; thUII T ( I,,) ... / /l1"dp. '" J gl,dp '" T ( I, ) '" O. 'y, .... ume that conditions (a ), (b), and (e) .'" "",i5Iled. l1>eo T ( I,.) '" 0 lor every locaI]y p·necJi&ible _ N. Tbe function ~ ; E _ T{ I ,,) from S into Cis .. ",hW;h is bounded ( P ropooition 2.2.3 ). If" ill it. rMriction to S, tben ~ ... ~ ( P tOJl(llSitlon 6.U ). Since ~ It bounded , tbe", n\aU, E 4;(p ) 1ud:L lhat ~ .. 91' . For all E E S, 7(1,,) '"' ii( E ) .. / ]".w ..
ea:.",
_U"',
/, · 1,,4.
NOOI' 1M E E M . Then T ( ] anl",oiUOio:iet the «Ier relMion; i~ "" j , .. ~..,. 10 ., /. locally aI"... every"'1\en:. ThllOl" lIIIqutrte:e Ci.).» In ~ (V ,,) ha$ II oWl it>! in6~", if and only if inf. ;!:! f." 0 ioo 0, I~ ~xi>;U 'i'. E D 8ud> I"'" "".(Is,) lhal. T .. T, + To' N(>W T E L~ (I' )' is di$>im fro", G io L~{P)' if and. only If 11'1 is disjoint (tom C .. in L':(P)', tht.l. is, if and ooly if ReT and 1m T are dioijoiDi. from GR in L;'(P)'. Each T _ ReT +; ImT of L~(P)' eon \""",Iore N "";tl
L:lu )~
-
to"""" L\i'lu)'
L;>luJ: _ {OJ. Then (N.. U. )) • .!,
for..m ..
for ..m
"'"I"""""
For ....,.bft p;vol of Pn:>poeitloo 10.6.2, _
,7.,
E"u cioc \3.
••
10. Radoo-Niloodym o""i.... iwII
214
Ext/Lilu fo r Chapter 10 1
c;."". .. aDd b in fi o.od:t tlw B < b, deoou by d'z Leb.o.ie:"" ....... ure "" f '" )et\ _ Q,1",u) .. ith in11ft' p:rod\lC'l (g,h) .. 9/10,, :etcioo II),
t:Ot
(p.),.~
I. p....., Iho. Po ... onitary pOIynom .... of I weIf»:"",t •. 2.
For - "
)." .. !P--,(-'. Po -
2: 2, obow lha. Po .. (r - ).,,)Po_ , - ,,",,"_., ..1Ieu (ZPo- " ,..-o) aDd " . .. 1,..-,1'" )p._.1-· I'~ '" ,ha,
0_ """'_. . .,_ '"' " , . .
!~_
n
. " rp,, _, ••
bj'-'
,Po_' ) '
3.
i~'_ n ~ I, let '. be ,be codIicitM 01. ,,'-' iD Po· SJ>o,o \ho.t )p.r' . (Z}'o, Po) .. " - •• ~,. (Ol a , •• ~ .Iw "Po ""'Y be .. , .. IOn ..."" + ... + .... ,,...,, aDd «>mpu\ ti:o m ).0,10{ &nd .. _ «don ollDllltiplieCioo'I.
r.,u, Coo
d:eIi~
I, &nd In>m put S,
io otrictly pooiti"",
.... at' int
F\l< oil 1 5 q 5; n - I, I
,bal.I
and I.. fi ....
I . S",,", u..t 11>0 uniQ ... polynomial of ~ _./tan n _ 1 ,bal ........ al r ; ......11 .. i.. fi .... (... _ 1) dOli ...;.... (10< all I :'f j ~ p) ... W ~O.
2. l.d .f(X) _ L.s' so _,U/ Irl)a..X' bo • pOIyDomiaI el dqreo _ t/tan n - I . Shooo- tloM r.net itt firat (n, - 1) oX, ioatl_ an "'I""i at:Iu ... nloe) ..,, : , _ 111".,.1' (' .11J(2) .... runo _ 8. p ...,." tloa& g lwIc_p to Q,( .. dI). (Otl .. 110M 0\1\- I'll 5 2.. ' - . \tI.) .. Lo:t I E Ebe,,"~ •• I, ;D E, 100,1000 ruO>tt;...r (n 2: 0 ). 1 (1) . ."p(u..). Gj>{ - I' Ill '" Ii """-pNc I" C . _ .bot oil II< oor..w , , _ ;B ~(dI). ob,- .... _ I ...... \"
Lh{4I) of tho...."........ (I" · o.PI - :O:I' I).zt'
·b.
E'). s.
ld. V. boo tho _ ......1»« "'- E _ i{(I ... dr) _ _ oed by 'M fII .... tiona 1". 1' . , ... 1", S _ ....... P.(t) . ,' . oo(. ) dl _ 0 b oll lDt g •
f,
OS t < " . CU.. inlql'''''''' by J>6tU or ordot n;
[ r'Ij:..M:a-)u_
L
(-I)'" · ",-,- ' 1(,,).
,(o)(rll: • H
J"
o,s;rS' - '
if /. , .... "".............. """,_I ~ 1-.1\ __ b.o... " COD!;. .... Ded_t"".II.. P, ., , . . P. focm ....................! '-io 01. V• .
~)
4.
Shoo< that
1_', P.(I) _j" . tu(1) 41 .. 2".~.".' . na +- n" I )· r {8 + n" I)
'-
r (a +tI+ :lr!+l)
.
........ r .. tho E..... pmma """".,., Conclllde II..... ... all .. ~ o. " _[ "
r +'"
, r«I+" + I ) . r(t.l+ " + I)I- I" . ~~
Q+ I' +lIo+ 1 I'("i- l), r {a" " +n + l )
obC' 0 - 8 , .... IJ + tn " l po.,CI) 2' .. .. tl. " 4 ft - I) a+ lh:m _ 2 +2n)(ai-n -l)(" ... . +- C +Co.II+8 .. n)(a +8 .:m _ 2) II ~i Oft I. 8y u.. ... 3 eli .... ..,(.. ) . E.~ Q.(%)w· kit ~ .. E T. 5"""" tbat
Let
%
(1 - 2>0% -+ ~'Jft)' I n< n 2 l aDdewry r
,
"- J- I.I{.
_"'''P'Y ...
LoIo S be .. "''''irine in .. n. I' ..
4.
0 •• Ok ~ l' u..'lo ld . -............ 01 1'"' _;"";1\101 , ' - ........... t.o O. fbr _ " Ii t mil"'__ ... U:ft .
Fr -=h We" . Itt ... M ,1000 contin__ U""", form (~.).~ - t:.~~!/o Oil l'. Show that u.. IIUlpplDc , _ ..... ~.....,. ........ .,.,. 011' ""'" ,ho _,DOd dual
(l'y of l'_
0 . - ~ t;' (I LIi_i""ly. & ) l be _ 0111........' , . (".J. ,.. of 1"OOil ...",ben .. hid!. .,., ... _ (I't' ;:ocll...ty, .. biclt «>D'''~ to 0). \hi~ (1;:)' lor int S...u _~ Ie b,. c..t.l lbe uruquo """'pie>; Dumbet ouch ''''''' ''' d let 0 _ (a' •.... o .) be an .le"..... "'_(IV }. S,-- .bat V .. w n . -'{a ) • <XlI""""'boot of L;:(p.). heoce bao aD "",rem.oJ polnt, (whK:lt _ . . . . , Ok -,w"oblo F wilh val ... In 10. Ill.
s "\ • """''-'
• rom_,
2.
> O. The .. ",Ct EIO,I D > 010< ...... .,.. l S i S " . utd. 100- oimpliciCtion ~ ... . (lzl ) deII_ • -... .... E . Su_ E io «M>I.~ no ,m. _m. T1oen • .., ~ _ tbal E io o..ddohod oompiote (Cb&ptoo- I , Exercioo 13). 1ft ,be __ ' ion of FY Id"" 12, I.
s- , ha"
Exe,,'"
x.
""~ito,. lor .100 IUodon _~'" Ji " f _ ~ (.- I(IJ) on For _ r E E+, ...,.n that . . .. . (z-l .... """\.in ........ IuACtiooo Irom X imo IO. -tool· """",.bat tJ;. :;: . (z ) .... t bat •• io J'"intep1obio. S_. ito f..,., lUi If ,. .. toI %) b 011 z E E .
r •.
J• •
3.
p """, tbat
% ... .
r..:o
•.,. 01 E ORI l,.:"!.u) and on iooomorpIoiom ("' .... "' ••>1 .. ,.... . .. \&liM 11oorem).
(zl io ... _
of o<de.od _ _ . _ ~.
I.
lor ! E C _ 13 of Chapt .. I), ... lha• • -1 (1f ") .. _Inir .
2.
C~
ofJi(X . R ), _ . r o w , in . (E). W~ g lor ,ho ... p<emUm of H In . (E) iUid f bt ,be upper _Iopo of H . Recoil thM r io lbe u_ ..... ioon. int>OUO """Iari ... tion of , . Show lhat u( . -'(fJ) .. tul'lo~H"( . - . ( ~», iUid ronch.de ' bat 9 .. I ,.." """" H be .... upward-dITCh thai tho "'I~ .... oil 15). By put 2, - " , E H """ be wril_
I.PI ill ... tl1aa 1/ ... We put 2, of Ch&ptot I, ..... band
p-;"t~ a»d ;. _
"- 1I 1I1-1I~1'I!I S 1/ ,,- By Elon'Cioe IS, by" lit M 'lS, R) ill . . I 1m M ' lS, R I. p,.,.. that
...... at"" "
"
' un with _
~
ill •
,.
Coach ..... h.a!. I.... 6 E H "".. . ~ t-.
'I'
ate
11 Images of Measures
I be R ' II (~. ,) _ (~ _ J,,' + ,..1). ......... , ... _ _ oftbo ~I 0( ,. + .. 110 10.::.(. w. 10M-
'5,...
to 1M "'" !map oIl.Jt~ ...dot / . t{,o a , _ . 1M. """"'pio ..... 1«I!y _~ &ad o . loOer ,., cIeno!.td by .. (JoJ. Ip Ihie
In the roI\oofIns , h",:.. ,,,, . ...., _me lha, I' .. posit ive.
"'. lIwa.t (...S' ) II " ....iW, GM p": 1" "
".(Jo). n,,,,,
Theorem H . I . I o4 ....
(a) 1"(1 0 .)dl' :5:
1" f dJl
r f r 0>11 f : n' is .m~;
JI · m"", w
(&) , ,, I' is I' ...........r'1Ibk, fr>r .... ry " ",.lrifIlipblt. the "'" ..- · (C ) "' p-mt.. urtb!e. Now fi~ .. "_rmuurable iot\ E' _The lIeU A. E S tor which A. n .. - '( £') iI ".~ folD>" P-""" C. II A. IiflI in 5 , tbere exlsl a 8eq~ ( A:')_~, of S'-eeu and .. J>-ntsJjp!Aete\ N such d'M N U( u..~L ,,- I ( A.~» tonUi ns A• . ..d \.ben MI1r-'( E') .~ __ \lI'abkbeo_ k li lhe WlioDof AnNn.--'( E' ) and the A n " -'(A:' n E'). H"""" C .. S Wt ()QClude .h&t E n If-I (E') iii I'"Lntecrabk lor all wl~bIe feU E. So ,,-'(C ) iI Nat, let p boo .. p'. ~ m.o.pplJIf; from 0' into ~ rnetri ...bIe..-Gi...... I'-inucral>le IO!t E, ,,,-, en.1 .. ~~ (A!.)_.l: ' of $'- u at>
,.._11IbI8.
""-~ Now the f\ll>Ctioa
,,· C" - ' (A')
" . ( ,,-I(A'» II ,,-additive on 5,..,,; H" {"-'( A' )) ill 6nite rQlr 011 A' E 5'. Then ..-I( A') io _i&l1)' ,..1ntqn.bIo: for &11 A' E 5' . ....1lI A E 5 and, for each n > 1. thoooe ". E If flO that ,,( A) - "~{A) S 1Jn. ~ & teq~ ( A:.. .• )... ~1 Qf 5 ' _ ,uc:h P JIOOF:
n-:
that
A n (u..:;tl .. -'CA:.. .• ))' ill ".·nesligible. 1£,..., put B ..
U .. - 'CA:....).
""...
W:n, lOr allp2:. 1. An B" io ",. nqlitible- So Ii( A n 8") .. {p - ",)(An B"):S: {p - .,.I{A) 10 loris t he.n 1/ 1'. ",bid> pM'" that A n B" 10 I'·ooclilible. 0
Noo,. 1et (~ ).tl be. lummable r"",Uy of positive measures 00 S (S "'mlri"i In OJ. Deli"" 1' '' !:;.;, ~, Let 1r be a m&ppill,fl: from 0 IDle> a ... n' . and Itt 5' be a semirins in n' Then (... S') ill ji"6uilb:! if and ooly if (... 5') '" ~..w~ br.U j E I and tho family ( ..(p ,))""
• (p) -
.. ,"'limahle. In Ih. . . . .
L.u "(1' O. " -e call lind K. E C ";Ih K. c B; II\ICII thal ,,(A - K; ) < ./n. Heooo ,,(A - U,s.s. K, ) and
:s '.
,,(A) _ sup{,,(K l: K E C. K c Al. to 0\>001 ,hAl, il ( A; ).~, to •• ,,(A,) «>tI,~ to 0 as i _
is a ~net"InlA and ,h.at 0 0(;1 ~ i!I lhe " . riI>g 8"'etatM I:.,. S (aDd by A ). Definition 11.2.3 For...:b J E Y , Itt I'J be. ~jl;"" II\CaIiIlle on 5J "';Ib !nUl I. If I'J, _ %J,.1o"'J, ) £or aU J , aDd J, E F ruclo. lhal J, C J, . then "'J )J~F Ie said w be • projed.i"" .)"Iitem of meu\I~.
Tboorem 11. 2.1 (K oJmocorov) kl U<J)JV /It G pooj«fillC
"oj
iii
/
~.
fa) The", ;, an UJili"" /-(1iD .. " .... A nodi IIuot ft>r.J1 J E F aM.!1 AJ E !h .
vlq; ' (AJ»
_ I'J (A J )
(6) Write I'; - I' ( , j For.J1i E I. I/. Jvr _ry i e I , ~"';, a (mapld do.u C; oj S,-MtI nd IIuot 1',(04;) - !lUp{J'; ( K,) : K ; E K ; C A,) ft>r .!I A, E 5" 1/1"" I' _ "I' ;, a ........"".
c..
be tWO A _ and let J , K EF. AJ E SJ , aDd BK E SK be"""" Ih.at E; _ ,t{A J ) -=I. F _ ",..' (8K)' Then, \n our prcoiou8 noutloa . ,; ' (AJ) - ".Mr{AN "), '" A ........ _ q~,,(E;) and •• imilarly, 8 N " _ {F}.
P *X>I'": Let E . F
'.IUIot.>too .. ...·,
1' .. (8,, )
~ I'JU,,{BN K ).
If S _ F . l ben AN .. • B N .. · Thilsayo , hat I'J (A J ),. I',, (B,,) , and.....,..... thM. II 10 wei! defined . ~ E IlOd F are dilljoiol., ... are AN " aD 0, t ' - allu, for
J , K; E C; tud> that K; C A; and jJ, ( A; - K. ) S tJn. Put KJ _ n ~J K ,. S;""", U, O ( A.i - K i) )( n ;,s/-,/I 0 ;) L, every (:t:; p ... ,:r •• ) in 0. , )( ... )( 0.., and e¥CIY iD~r n ?: 1, denote by E..{:r;" ... . :r.. ) the lJI!CIioxI of E.. de\ermined by (:t:; " ...• ll" •• ). By de/lnl'lon. E.(ll";, •. .. • r ,.l ' 00 ., al of lhoooo (:r, )..e(l_l;, •... " )) In (lIlU.illS J~. E. (z" •. . . , z'.) is d. b- tw"ry ,, 2: 1, dtfu>e
B. _ {Z', EO" : .J')(E~ (z, ,)) 2: t / 2} .
.J'I{E,,{z ,,)) - /J J. (A,. 1 = ,,(E,.) 2: ! for all z,' E: 11;" wben,,(z,, ))41'" for E.. 10 fu;.j %" E: 11;, with " I') (E.(%" ,z" I) 2: tl4. By illduction, ~ , define .. """!,,,,ncc 01 pointl {z .. I.~1 E n.~l n" IIOthat .,{.)(E. (",,, . .. ," .. I) 2: 2-'t lor a1lp 2: and all n 2: I. Let, E ('I be .. poinlwhol;e .. ®;V(®"," 14). For all j in 10m! finite iuhilet K of / lei. A, € 5;• .00 let A. .. n, III E (I - K J. Otnott by h< 1M finite tet U € J, l, nK" Then fbi" ew!'l' i € I . let
"'''''''*'
II;
t,.
, (11 A.) ":1
• 11 ( 11 ",,,.) .. n 1>,(.4,) , o~
. ~ /,,, ,,
"' '"' n
."
~, I> ..
I>.{ A. ) .
® of,I 1>. ( te,ociotivity of prOdua 11-"""), ThillMl. """,1\ »UsioU 10< &nite I w'- lbe 1>, an! CDIIlpiex mo:eoIu~ defined 00 Iimlirlnp.
11.3 Change of Variable Let I ... Ca,.) be .. ~PCY inl..-val of R, ...IUI Jo.f\ endpoint: a aod rilIbI. in it. L« .I. b4 ~ _un: on I and let F be .. n:aI Ban...:h
"""point.
."""
No--, li""" .. IoxaIly .I..intt&t"&bio: fune!,;on , from I
I:
intO R, _
G be an
indefinite intt&ral 0( 1> " ,.1.. So G(p) - G(o) '"' g(1)dI for all Q, fJ € I . S""", G;" cumin...... (Corollary to ~ 3.2.2). G(I ) 10 an lot........! of R, and ... let" be ~ " w " " , on the M\urai ~Iri,,« S' of G(/ ). ~!Or tn., n_at that g ill continuous. If I: G(/) _ F io t:ontin_ thea J:(f GJ, dI ... ~ 1(.. ) dto lOr all 0 , IJ in I . Indeed, for eoocoy " € I , tilt fullCtioooll: ~ ... J~(~) /( .. ) du from G(/) into F Is .. primiti ... of I; .. (Il C)'e.,) '"' I (G.,) , (.,,) for all '" E 1, from which _ obtain (/I o G )(P) - (Il oG)(o) ..
The formula
[(f
G)gJI.
I: (f oGndl .. ~~ f{~)du " Mittd. ~m '_
."
G(.,) and
G(~_)
ui.ot. .. nd il J;' ... in' • • :J
G(p) _ - I J
· ".
PI\OOF: Si~"
...w.
G(..·) and G(~_), IMn G(p) .. IJ ." or tuG( .... ) < G(b_) orG(a") 2 G(b_).
indtuUd in G(l)
~ndporinu
.Ix>undo>d.
(G. S') III ,,""sulU:d. By tbe domina-told coo·.u ", '1O! theorem, G(a" ) &Dd G(~_) o:xioI. in R. and it ",mai... I be proven that G(p) _ :I: 11"" ~ / E ~ (G(I) . R ). It ou!licao I ohow lhal. f (/oC), dJ.. - :I: f / · I J Ib; (Theorem 8.1.2). ~.., bot; .. 1>ow>ded cootlnlJOlll!l function from R Into R tha$.
I")
(g,, ). ;!cL cnnvergs to, J....a1mooot ~~
(b) (g., ] !! h 10. all" :!: 1-
f:'
19.),.2;1 wn ..... gooL 10 , In CliP,). If we put O~(",) ~ 0("'0) + g,, (t ) dI. wbel t 2'0 . . . !hed point 011 , the fUI>Ctioo"" G. (Om~ unilo,.ruy to G, 10 O~ I""), O. I~_) wn",,'I" 10 0(.... ), G(~_) , ""'pectimy. N_ 1(10 0 G.Ig.1 • omalIer tban 11'1 . h , for all n E N . ThUll f(", 0 G. )g,. dJ.. COO"",!", 10 f(""'G)gd). ... n _ +00. F(J< ~ R E N ,Iet (e. ,d..1(..ith c.,. S d..) he .. compooct. Iubinl.en'al 01 1 CORt";ning lbe oupport 01 g., 1"hcn
j (", OCo )g,.d).
-
[('POCo)(IIg,.(c)dl 'I"- )
,. {
'l'{Od{, C, lo.)
boc., ...
whtrt c.(",,),. G.I"" ) and G. (d..) '"' c. (~_ ) G. is """"I&Ilt "" -.ch oitbe lotenU:o (.. ,e..1and (d..,~). L0 com~"","1 of 1" ,...1 wil h .... p O(b-).
t:(:;]
o
Propooillon 11.3.2 l.d z., E / .
,.)
u.~ li""iU
G{.... ) . n.d G{~_)
n.. .... (G. S') ;, II"aud if and nIJI i/ G(b_) .
P ROOF: ~ that tbose mnditiono are w l$6ed, and lei. K be a oompact ",bid of G( / ). Then B .. e- '( K)n[zo,~) is cbd ill (..... , ~). IfG(b_l ""~ 10 K , (" o, b) io I'""in~, and the WI>/: is true of B. On the odler hand . If G(~_ ) ~
"'" b 2"0,« to K . b ~ DOt lie ill the t\')iWlO of B (.. Ith I"OIpooct to R ). hillO: B ill c:ompod. Similarly. A .. ("."olne-'(K ) is I'"i~. Thus (G,S') ill ~ted. COD>uidy, -..mo tMI (G.S') 1& I'""wited. We arsue br o;Jatra
JG1 .. )
(Pt .........tlol> 11.3.1), th.t.t is.
.
t:..,'~' /,w .. JVo O )g ·l .....
) 'O' .
'" both CUQ;, tbe IlIIIt equality 10 \rule. Likcwioe .
{'t..) / dv
JG(o*'
_
JU GG)/I' 1(.....1
_ tliall _
11.4
in Chapter 22, tb/:8e n'!$ultl CM be refi!ll!d.
Elements of Ergodic Theory
I" ttu. _Ion. _
~
Birkholh up>
fIl.o"i~,
+ 1, ...
60< • ji.we _ " ' " of r<eI n.Mb.tn, aU /d :S. m :s; n. ~ b., L ... ~ .ttl 0/ ,,..,..,.. p,up..,.l.., /lien: ~tr "" 0 :S. , < m ta.co\ tIwot
~ I). ~
Eot-t- I i ;,,.,...,.....
in.,.
PIU)C)Pc Let i E L... and In , boo tt.. smalitet imoser in (O" ". m) ouch thall; + .r. + t .... ~ 0. For 1 S q :5. p, _ h.t.~ I, + .r. + /..... _ 1 < I) and "+ --' +1.... 2: 0, + .. - + I.... 2:0, "hJcl, .... O0:' :"1 tQ L ... If L .. .. """""')", _ can ~ J, •. or J. IT 2: I) 0( ........ ""'ptr inten1l.l8 01 (0. nl n z + oueh IMI
be...,.,_
(a)
P(J.~ , )
_ruct.
< o(J.) 10< all 2 'f J: 'f .....
(b) J , u" ,u J•• jO,8(J.lIn 1.- for all I < k 'f r;
(e) E",I. I.,
l ri: e~ ••~ "" ha.... obt..h,.,j noroo.mpty in~"'" J , .... , J._ I (.. ith • d (e) bold for all "# L".. U i . .. lbol $m&I1ootit inlep 0( t 'f • - I, and I"'" J , u··, U L". - (I, U"' U J._,) &rid p, II lhe ..,...:Jl$ of lhoe. ID~ 0 'f , S rn """"
J._,
II,~
that
t. . ... ...... t ; • •• ;>: 0, ~ ul<e J. ,.
may !I"X" j .
Now " _
thai. E,~ t... I, ..
Flomonu of Fzp:Iic Theory
231
{i" ... , i .... p. I. Tb.. the lnductio maPJllnl: 1 from 11 InlO • _ rt' . put 1. .. 1 .. .. ' b all t E Z+.
'v.) ,.
PropoolUon 11.~ . 1 (Maximal E'&dic Tbeo""n, ) Ld 1 : 11 ~ R Ioe oe-tiel/, 1'.;'11.,....we. and,1Il
A .. (r E n : 1o(r) ... ···
+ M r l ;>: 0 1M GI.
U-
It.u.I """ p;>: 0).
n- f" 1 dp.;>: o. P ROOF: For &IJ k E z·, 1.", e!IIIenUally ".ln~bIe. Fix m E Z·, and Ie\ A_ be the -'" of thoee:z: E l1 .uch that 10(:z:) "" "'" 1.(:z:) ;>: 0 tor at Icu\ ..... 0 :5 , :5 m . Let ... ~. n ;>: 0 be j1iYeD. For -.II 0 ::5 t ::5 m + .. the let B. , eI t hoee :z: E 0 rueh that (1. + ...... 1••• )(:z:) > 0 for &1; bH one 0 < , :5 Inf(m, .. + m - k), Is I'-",c·urable. Moos,.., •. B• .. ,, - · (A .. ) lor 0 ::5 Ie :5 II. Now, for """" z E 11 , Jg 1... (.0:) be t he !lei eI thoee IndW. o < Ie ::5 m + n for .. bich l he ... exillu 0 :5 P < inf(m. m + .. - Ie) ouch that (1. + ... + 1. ... )(r ) 2: O. Then, for all :z: E 11 and for oJl 0 ::5 Ie :5 m + n , " """" 1".(.0:) .. 1 or I",(:z:) .. 0 .. k beIonp 10 L- (:z:) or 110\. Hence Eo:s.~_ ~ 1.(:z:)- 1".( :z:) .. E.u _(. ) 1. (.0:) for oJl:z: E 11; howe""" the ri&hl-)WId oide is ~UYe by Il .U , and .. Eo:s.~_ ~ f ! • . I". Iially ".~~ eel-.b 111&1 b < limsuPo__ (I/ Il) IAs' s.-, h (~) to0 let of 11>00II: ~ Ie n for which \.hom ........ P 2: 0 ouch lhal (10 + .. , + 1,)(..:, -" .. ~. atd(O:S J: :S p: "'(~) E C). For each Z E C and each 'I(J 2: I , ~ caD find II 2: 'I(J .., \ha\ ( 1/ 11 ) LoS'SO_1 I. (~) 2: b. Thll'l A 0 mpimal ~Ixodie IbototClI', 1...(1 - ~ . Ic)d" .. I ... I dl' - ~(C) iI p()8!tive, wbkb leW 10 bJ«C)
:S 11I14j.·
Now Jet ", b E R be , uch lhal " < b, and df:rxKe by E !he r E fl ..xt. that
, + I»
~.s o h (z )
:5 (1/(1l + I) !:OS. SOt.(z) for
in-
...d " (E) illDcluded In E , In rod , ,,'(E) c E for all J: 2: O. WrlU! A for the eel of tJx.e z E n ...m lhal there ex~ " 2: 0 wilh IV- 6)· 1.1.(:1)+ ' . . + ((1 - b), I.I,.(z) 2: Then An E is lhe let of tboele z Ie E tum lhat. / o{r ) + ··· +/p(z J2: {p+l)6 for at Jout _ " 2: o. n... A n E .. E In ...... of \1>o detlnilioo of E. Now , aJ1P11in& tbe maximal dliQdio: the"'_ to (! - 6) · I. , ~ ~ /!Jo(El :S fl· I II 81'. Similarly, 1 I · I. djl < ~( E ). So I'(E) " 0 and E Is k>caIly I'"necligible. Foo- uch (a , b) E Q )( Q Judo l bat " mplex.""'....:1 !unet"""'.
Theorem 11 ....1 S1IJI1II( i' ;. bo~n.ded. Let J : 11 - C 10: ClocnIioI/J w ~. end let n - C 10: .lOdIlhct ( I/ n) !.).~ I """""'iC' to Iooolir ,.·almcot ..."",,--meuurable
and l'"invariaPt funetioo from 0 inlO C '"
OOItant Ioc.o.lly
1".&....
Pl"OpOIOltlo" 11. ... 2 " ;. ,.· .. lIoJil1owa. [J
•
:uo
I!. Jmng.. 01.~
Ert:rcUu Jqr Ch(Jpl(f' 11 I
Let. I' b'> I.or:I>
r l' )- I
4.
3
1"(·) ~II.w ........ ~_ thM. lo . _ pic.
Wri",). for \..obaJ- "liP"'" "" R. Let I ~ .c~ l l) ... tw:r. F;,. Bon",*, _ , ADd III!. g Iii .c;;'(,\) be .... , Iudic ... ith period I. Filially, lot Ie. )"~I be & ........ .... to (J / .w) .(J g J~) .. , - -toe>. o..J_ t b&l 116- = ADd I"",,eb_ tb&t, .. J
o.
4.
c.-Iude
0
aDd • .oq""""'" (~. ).;>:, 01 lirictit paoiti,-e numben cooverp", \0 O. which alilfy the foIlowi", conditions: (a) FOE ..I
~I
i ~ I. fI contaillll the c~ ball 8'(z. r ;) of l'Iodi ... ~, oen~
z.
(b) For ~l i ~ 1, £; • included in the cloeed tJ.l1 8' (z. T,I .nd ,1, (£;) a · .I.(8'(Z.T; ».
~
Dell nltlon n .1.2 Lot p be,. mmpJa ""''''''''"' on the naturallt:llliri", 01 fl. aDd let z e R l . lf A e C . ouch tlw li""_ ..... p{&)/ .I.(E,) .. A lot e>'ery lflq\Wenot (E,),;>:, that "'nnb \0 z nicely. _ call A the dm_ive 01 I' at. z. aDd we wrile (Dp )(z) .. A.
The princll*! rtotult 01 th • ..ction .. n...or..m 12.1.1. The foIlowi", Len>..... 12.1.1 aDd 12.1.2 win be r.eedtd lOr tho: prOof.
..no... •
Lemma 12.1.1 II C if • t:Ollt..otibn 01 J - lt . P ROOf: C'-,. compoct!OK K"" that K C IV aDd .I.(K) > I. Sinot K • ODmpKI. . It 10 00, ...,,;1 by 6ni~ly tnalI,)" cLemen" of C. MY U, . ... ,U., which we
....y order 10 lhat their radii .(U;) .... mfy r (U;) ?: I'{Uj .. .) for I :S ; :S p - I. Put B, .. U, . Diac&rd ~I UJ ouch that j > 1 and Ui tn\e.--:\l B, . Lot II, .. U.. be lho: ft"'t of the mnailli..n« Uj ( if lhere are ""'Y) . Diac&rd aU UJ auch that j > i . aDd Uj inu-me" ~ ; let B, .. U" be the ftm 01 the mnainlOC Ui' Rep et1 th. ~ .. often .. """,tllle. Thio ,h~ tho: diltld Ir>r ). • ..mu..c "'I .I tilt- ft<m/i<m z _ D,,(or ) (.uji>ltld ).·"""...1 pwk:r.:) "Ioaa.llp ).·inl 9' M . JliQ'W '(" tilt- LdeIg'Me mp»Uion 01" ....lGli"" '" ). " " _ (D,,).I. + ,"" .mil DI'. (r) _ 0 )..",....,., ~m!.
"'fI ......,
"w'
PItOO" It ru!lid->i 10 j)rO'o"OI the lheoo"em !IePI'",1dy for" 1. ). and lor I' < ).. AIoo, we t>«)(\ only obuln the ruu1l for ~ ," . I f " 1. )., then ,.... .1 )., and thern" a ""' A C fI "';th ,,'( A) _ 0 and .l.(A' )" 0 (P'Oj:O($tion 10.3A). 8 )' Lemma 11. 12. (01'+)(" ) ., 0 .I.-&!mI)6t ...--ywheft. Tbe .. me arglitnelll . . . . ."""" that (D,.. -)(r ) _ 0 ).·a ." . lit .....
(DI')(r ) _ 0
).. a ....
N"", ........... me IMII' < ).
and ,.. ..
,.".1.
;" a Ioo:aIly .l.-int~ funchon I from fl lb\O R ,uch that I' ,. The: theon:m will k>l1ow 00wilJ&: "hn':MI~ """'>'
%E
n lAtis6ell
~~!;!,P ~~i~ :s 1(%) for I!YO!f)' _ (E;)."i!: ' that shrinks to % oioely. If...., ~ I' by - I' . and I by - I , It fl~ tbat &lmoM ..tidlo:& ~
e¥O!r)" %
E
n
1(%)
"" - r ~ (E;)Q. that iobrinks to:z nio!:ly.
o
The proof of the \.I'...-.[U .. tblll complete. AJ>I)I.het lmpOrt.anl. ..,...It ill the IoIlowlng theorem.
Tbeo...,m 12. 1.2 1..:1 I Ie q koa;Il~ ~.;"kgm6U: mapping from &.nacII 'J'II:o) .. 11 (%0) - r l for .1,.,.,.. all 7'0. I.cI. Y. be tho: u .....:.00.1 ""' . and put Y .. N U (U. Y. ). Then ~(Y) '"' O. If 7'0 rf. Y , If (E;).~ . obrinu to ..., DIoeiy, and If ( ;> 0, t~ cxist.s T ED . uo:b that 1 f(~1 - r l < e. s.noe, 1/ (%) - /(%0)1 < 1/(%) - rI + f for aU :z E n,
...
A(~.I . E
.I I(%I- 1(%o) ldA(%) ngie .,. a ". 1'IKn
E;.o dUjo-int ..niono/rtd4ngie. p; E
z· lor ~1Jn')
].B.- P,2: 1,8;- ;!j
IT
(n E z ·,
ISIS' lSi S k) .
PIIO<W : For e""b .. E Z·, let Q. be ,hoe
(I ....
.ioa Uo. R'
2-&7
are ne\sbborhoodl of". HtllOt. lo, all liE N , tlot functioo >: .... ~ (s) II lower ~mlo;:ontill\lOllf on fl. Finally, >: .... inf~>l .0. .. (,,) .... Bard function from Ilimoltand _ _ lb __ n( _,.)· 0 ...... tlwlb{s) _ !i,.(s ) _ D,.(s ) &I. every " E 11 for which O,.(s) uIst, From now m , _ _ume that ,. .. positive.
n,. ;
PropoAltion 12.1.1 UI A /100 ~ ~I "fn ~""Q > 0 ~ rmJ ............ $""",Ie t1uJI DJ« s ) 2: Q for ail s E A. 171m Q)'- (Al ::; ,.-( A). P IIOOf"; Theft II no ""'rietioa In ...... ming that ,.-(A ) iI 6nit
i.....
A.
n
CI_ .. e N . Given 6 > diljoint
0-,
,,'
.,
have ,.· (A. ) 2: (Q -~). ),· (An ). no,.,,,,~, ,.· (A) 2: Si""" € iI arbitrary, .., moclndt tha~ ,." ( A) 2: Q • .I. " (A). [J
5;...,. 6 iI ... bitrary, _
(0 - () . ~" ( Al.
Pl""'>PWilion 12.1.2 Ld A /100 .. ,..u.1 ofn 4"" > 0 4 rmJ n~. $"PfIOIe t1uJI !ip(s ) ::; /1>
u.- !i,.(,,) ...
+00
,. "~I ("'i"~'
P IIOOr: Let N be .. >.-"",Iisible_ .. hich C&IT"ifs,.. For-=h i~n 2: I, put Eo _ (" E N , n~,,) :5 n ~ . """'" ,."( Eo) ::; n· )'"(E.) Je.ds to ,."( Eo ) _ o. ~, Is EN : !i~,,) ... +00) bOil ,..~bIo compl ' '''''nt . [J
"
.
Propoo;ltJ.o" 12.1.3 /j1J,.(Z) p~
+ 1'.
Let ,. .. fA
"""
< +'e.od b7 finitd)' maDY ol u- S.tcU E. .. If IIOinC pOi'" at rl !ite in th_ S-«u, ""'" 011'-1 .... In lhe union of the 01"'" 1_ and can boo ,t" ."td wit hout ~nc the union. In Ihifo .....y. _ ,,,, •.,,,, lhe ... periI\lOtlOl S.....u e., and ....,. _u_ that no poIm. IJoo in more than 1_ d. the S - E••.
n.....
Jillm ol It' . the poUr (,,- '.5) II A..... iu,d and _-I (A.) ill Invarlant. W>dtt InMl ........ lienee lhere ~ .. """iii"" number callfld the .-lulU!! of II. sudo that ,,-' (>..) ... mod(a)A..
-«_),
••
f lo(~ 4 (tI- '(A.» _ f
lo(E) () v- '4A. =
f
111 "("°," )- ' 4...
for all E € S, ",hftooe ""'..., tllA' mod(!>" u) .. modltl) · mod(.. ). fOr all i. ; E {1. 2.. . .. tl . 1e! E' J dmote the t " t nwrir: tbat ball tho: ekment in tbe (i .j ) placeequal 10 I and all Mher eimoenUi equal to O. If ; ~ j and " E R, put B..; (ot) ,. I. + (l E•.j, ",here f. Is tbe unit mIItrix at (N'der Ic. For any murix X of order k. 8;J «I )X ill obWned by addl.r>g (I timoo tbe j t.h row of X to tbe ;.h row of X . FUrthermore. B;.;(ot )- ' .. B;J( - (I ). PI"1)J>mI,1oII 12.::U Ewry i,,~ibk t • t .....mz " A prod ... , of "",/ri€a of 1M form B, J« I) And G "",/..u of 1M form I, + (a - I )E ... .
P IIOOP: Consider in....n ibk maIne... of tbe form
, , 0
0
x_
.
0 0
0
0
,
0
0
0
.
( , ,, _h ~U_ h
,,,,,,
..
,..
.
{.-.-,
('-' -,.'-" ... ( ' ''- 0
.. ~ 0 S " S k - 1; i f " .. k - I . tlw:n X "' an ..-bit ...". in....nible "",.fix. T be proof Is by induction on " . If " .. 0 ... m .... have { ... " O. tI~ ..... if ... multiply 0 1) .
__ henc:t Iff> ded~ thac. A. (IIt )O , I)' » .. A. I )O , I)' ) and !nI)<j(II) .. I. Finally, ",bee> II. arbitrary, "beoo~m IZ.2. 1 rono,.." from Pf'OptlIiIlon 12.U.
0
l"<w. if II if a Uneat ~Iim ol R' ",hkh" nc>t bijt:rti,'" &nd if P iI tbe rank olll, tbore .....,. &II (l...,
~ m ' !PI'"
"'ftY IIOrIempey
opal
on lhI: tatuntJ...rurin(
s.,
oW. P~ll lon 12.3. 1 Gi........ .".... • ..but V 0/ R ' . kl T be .. 0II'IIli..""", .".... ""''';''' from V inlo R ' , alUl • .."...~ IMI T u djff."WN0"k a l _
I
J.
"","U E V . Po4 6.(r ) ..
A~~»
5 (I
+ 26)· 'i I + f,
.. dosIrtd. NI!XI . _ume lbat II .. OT(O) • • Ii".,... IUlomorphiarn of R " . Then, D{.~' 0 T)(O) .. Id R•. B1 .. Iw .., haw: ~
an imep:r n
illSt
1!bow1l. to "'di f
> 0 IIIne
> I oucb that
, I"--'(T E» - I OS ,'(TE) A(E) - 0>00e proof Ie moce difficult .. might be expected. The foIIa..;ng ch.n~e of variablffl formula .. . orntral _ult of this book. w~
,\IbM"
Theorem 12.3 .1 Ld V , IV lo< II... """n 0/ R ' "lid T "hom..,....,..... )Mum 01 V .,.,.to w. A ..""", lA4t T io tli/J.,..,nI.~ III ...a. po>inl. z E V , G..... p1II J(~) _ det[DT( ~)I. n.. .. IJ I ' % ,... IJ (% )[ io 1«oI1~ ~ v ·;"ltgn:Ib-Ic ""d .l.w _ T (lJ I ' .I.., ). PJWQP: Put I' ., r-' (.l. w ). By Proposition 12. 3. 1, n,, (z ) .. Allile for all % E V. Hence I' < .l. v (Proposit ion 12.1.3). Now IJ (%) I ;:0 Dp(r ) for .I.". allDOllt all %. by Proposition 11.3.1 and Theon:m 12.1. 1. T his ~ that IJ I "' Ioeally .l. v·i~ and " _ IJ I· .I.." .. I>o:nce ..., deduce IIw. T(IJ I.I..,) .. .l.w (P ropooition !I .U ). 0 Obee..~ that
12.4
f / J),w _ fu oTjIJld..\..,
£or.-..ch /: I\' _ 10.+< an i~ n ~ I. Writf: S~ 10< It... unit.phere It E R H ' , Yz i - I) in R~'" (...to.re 1·1 Ittt... Euclid..." Il0l"'''), and 8 for.he Borcl " . algt:bnt. of
S·
A=
AI ...
(b: ; I £ JO. 1),,, E &,..,1 .,. .. ,( ,,) _ : ltb,.".. of M(I) (1::; k ::; 71 + I). It 11 easily't>own that del. (M. (8)) .. (- 1)" - ' (¢(Illh )«(>$"- ' 9.)(0::06"- ' '. - ,) . , , (coo 9, )•
.. bere (*18)1• • ) is the .... 1• • product of 16(B) and • • '
0
borneomorphism from P (IJJIO ob( P l, and tbe inwrse ~'" II, from ,,(PI in\. R " , iI 8 cha:t of IDe manifold S· . tbe canon>cal be.Fix 6 E P ....:l put,. .. ,/>(B), Deoo~ by M of R O. Si""" de(.(J(I ») hM ei&n ( _ I)·, "'" .,., that the (" + 1).tu!lle ('" Dt(B)e' •... , D¢(9)c. j is direc\ ill. R H"'. Lc! j be lbe c&IlOnico.l im"",,· lion of SO lll.to R " '" and. for &II 1 ::; k ::; ", lei be the uniQ.UO: _ 1m in By P ropooition 12.4.2, '" indUCftl
&
«(,.... ,(.. )
v.
12.4 Polu Coordl_
tbe ~t l pece T. (S· ) l uc:b that d.i{~. ) • D"Ol-(8}t!• . n..-n tt, 11···11 u" ;. in tbe orit:nI.allon oCT. (S· ) (if S~ ;. ~tM "toward tbe outside"). For eacb n e N. doeoot.e by V" tbe volume ~ ( B'(O, Il) of the unit t.J1 In R " , .. hoeD R O ;. Ii...... ita E uclidean oorm. and by the ou.rt.ce ..... I dS"- 1 of S· -I. Siraoe A" " lhe imqe meNU", oC 10 - ' dt 0 dS o- 1 under
n..
f
(1, %) _ tz.
n.. _..Vo . Now
V.
-
[ ', u"
_
V._ I
I··· I I B·(O.I)L"'~}d.I:" " .u-._,
j ' (\ _:r! ){. - L)" dz.
-.
/;(2
/:(2
lor eva)"" 2: 2. Hence V. _ 2V. _ 1 COI"8d11, and ctUo"8dIJ ;. ~ to compute. Fbr all Il > 0, put r (ll) _ / ~- • . r"-' u "o ._ I' "The function r : Il _ r {_) from 10. +o:o( into jO . +o( ill called the EuIotr pmma fUDClion. Clearly, r (_ + \) _ .. . r (.. ). Siooo r ill _ I, ~"'"' that r(n) _ (n _ I)! for all .. E N. On t he «her band, r(.. ) - / lap( - :r'j . :r;1o- ' u /lO.+ 0. N_ Jet .. > 0 and ~ > 0 be t"", rNJ numben. Md lot / be the fuoctlon (:r;. ~) <XI
_ 2 ap( - :r;'J . ",10 _ ' . l a p( - v') .,a-I
10 , +o:o[ " JO . +0:0[. n..-n / / f u /lO ._ 1d~/lO._1 - r(a)r (6).
Buo., Ii"""
T : (r . 8) _ (rcoo8 . r lin8) ;. a diffeomorphism from jO . + cc[ x jO , "'/l [ ooto jO. +oc[ >< jO . +oc[,
II
f dz/lO.+tV.• C~ if. Ior_ • #- 0 .. C . oM be... (. + p) n p _, (..... pOd i ~Jy. if R " _ U .-(!>(. + C)). A .. _ P .bid> .. botb • G·podd", &ad • C_inf: .. ealIed • (p •• : ' . . \oil. IA:t ~ be I.c,-,," . 'ra OR R '.
I.
c·..,'";''' &ad P it. ~ ._~ C-poacti..,.
II Cit. ~.~ ""- ,Iw ~(C):!: ~( P).
1. Ld d (C)
be tIM; iert ~(C). "t.e d (C). ""- tbM tI..... - . . . " 0 i~ G n (A - AI·
l.
C-u..
SMPI' l ,J.or. it an imqtablo G·~l&tiool p . !At Go be. ouboJrvup of bit. _ ~ 100 C . ....... .... . •.• • • .,. ' .... ' ' ''''''~of 'hoi" " to 01 Go ill O. st- that ~ • + P) ... C ...• .,' ,Ila, ...... &Dd """"Iude lbI. d (Go ) _ • . d IG)·
U,s.s.("
2
...
1m .... of ,be
Mq .........
..-.II ."",.
.k _~'~ ' 'Y
IM_ ';" 01 """""",,.li~
~
A """""",... i~ P""P -.r be rqatd ......... . _ ,bat ~ ( A ) ~ 'l' . d ~ 1... 101's~ boo.boo boto""",phiom i - (... lij) '~ ' 1~ 01. Z· Into Z~ . If Iooernd 01. lol-ol l s,s ~ , .ha. ~IGoj" " dl ..... 01. ,-. Dod...,. from M~ '• • lMoor
(io &" .... 3, ....... lot A .boo e _ boJl Uo. R ' .n.b on 1- 0 ao>d radio.
(....,(4/ .. jp)"'). 1.
01. 0 and • iD (O. L. .... II' _ 1)/2 ) .1Id> .bao o' +".. + Coodudo tbat. if z" z,. z •. z, are .. im put I, ti,eD
p~ ,,,," "" .."'...
1. 0
(mod
pl.
zI +rJ +z1 +:or! ", J>. 3.
Fo.hat ..... ","ridloa 01. '" to V .... too'··_I"· pltiom from V onto an _ ball cecner.d .. "" aDd """taind
r
be .. ill
.w.. t ho val ....... (10.(0). 4.• "l::1ootd ~h in A' io • pot~ ~ : 10 .61 _ A _ h ~ "1(0) - -n~): if "1(1) - 1(0) b- .. II E 10.61 ....... ... it .. pUl>" . A ~<w IP ' 10 .&1" (c.d) - A io It IItrictJ.y- ""SA'ive "" 10 ,bI .
2.
Let6 O:iO,60(. [f 6 < b, ot-tb&t
I.
1O,.s.,[ ...,. 1>(:I) :!i +«> [ .... pI..:e ..
,("~'l(') "1 :!i ,¥ .•(1-110(' ). [ 19'("),,,"(./ "
"j ' . [
.. -... .. I 2: 1_ Pro.. tIu.l . be Ieh_baod IIion owIY.be mi~ con,,,,",",,,,,, .I-b».", 10 ohow IW
,"," .[[ n*'O(.,,u _ L •
t·(tW · '·"
.r(. Show tha •. rr.. 0"W1 f""" [0. II ......, 10. ""' ( - 1oc(1 + z) + "J'" f..- 10. ....10 (0.+00(. ( -
_ &Ad .... Il1o .Iorir "" .... m..o.ppiq:l. Writ
(" [C"" " ' .'.rClj_
L ....!.... .•(Jhll{O). '-·. r(HD] O::S'S~ ('lk)1
... 10 0 ... I _+«>. COlIC ..... that
L
r(1) .. ff. ,,, -1 - ' " '. -'. [
~ ...1>*· "(0)_ 2" ,I (~ ) .,-' +oC'-·lj.
._il,
"':!O· s· 6.
r , pu.'''' .. ""~(0l. By pu00 'C1.hon "
1'0...11 k E _
L
_ _ ... k ODd pole ,bat 100 "onot..... "" OKlo. p......JIeI wllb p
"""""#0 '"
, . Oed""" from pot' l ,bat
U
fA'"
0)0/.:0 (..) _ I lor olI :.: E 8 (0, I ).
S' _ ( .. E ft' : I~l ~ I) io...,. eqwppod witb.boo p-oup otnoctuno lor _bid> S ' 100. oubs_p of . be mul\ipj",*,;.o, VO
2 boo lUI ~. If
Ii
fu_
~
I . 01_ .. E S"- ', let "' ..... be fu nctiooo" _ p(% , u) from B(O,I) into R. Coml"'~ D'op(:.:)(:I , :f'"), fD< ~ z E B(O, I ) and oJl %' , z" of R ". Dod""" ,bat '" is harmonic 8(0 , I) .
in
2. I.d f be • ; •..JLioO 0 ...... d!al 1(6) • ,(~ )oIcr{.). p."... II,,", f io ","'10' i" in n. (S 0 """" d!al fI .....at.. 8'(0 •• ). doli.. , ...M
J
1).,_
...
_ 'rwil r
s.._
.-"""'-1,
--.ai_ f......,., " - 8'(•.• ) 1_ R wtoiclo iI h . """,It;,, O(o. r ) ...d .. " .m.~ Jon""' ....... +rS~ - ', ""I~illll by".." .....""..... . - t bo.t , _I 01> 8'(0., . ).) ... Let Ha{n.CI lot tbOl _ _ 01 \. ,_,It fu .... _ f""" I) Imo C , S __ II. AIto:r F on H,,(fI.C) ........""' 1 . fu ..... .,.. 1 ~ ..!6onnb' on ,100 ' : 2M . ...tooot. of 0. f • b.annot>lt
.!>aI.
.Iot.
13 Stieltjes Integral
not"""
Ai tbt oDd ollaot OO"'WY Stieh';" introduced 'M of "d"lflbuho" 01 T" • on an i~ . 'The ..,..ol " - ol 1Wi&tion wtoo... den ... i"" C...hidt ...... . ~ ~ .[ ).bc_ (n...:..,a 11.UJ. "100, if M 10 I," " ' ia,,.bet. M 10 dillet ' Iabio 111_ om) .bon! &Dd itt deti.....i .. 10 locally int..,.-..blo In--
.....,.1a
0 ouJK"'IIIUm of
L '" L A
I'(MI... _,), MI... )l
l~l S .
wbm! t:. _ 1l1li, ·· ...... ) 0 _ of eubdlvlliorw of J AIIIo, VI"'. c::j) ., VI'" .111) + V(~. cjl 100- aiL pOin.. _ . b, c in I .. tWring a < b < t:, bta._ V( [.. •cjl ill .ho ""fftmmn of lhe numbm lor allaubdivi$i<ma t:. of (a ,cI ((ICIwnilll ~,
LA
Lemma IS. I .1
Jo. /J)
iro I. V ([r . .8l1 "'"'rtrpt,I I ia 1 •.8). S;milan" I"'" an, ~I, (a .,8( in I,
F".,. .... _ P I )
V(!a . .8) ., :r - a vIla ,rl) ..... wrru lei V«(a. 6() ... :r. - 0 "' (0 ,81.
PROOf"; We will pI"O\'e 1 d.:oe"'"1 I , I il baoi a limil .. r - a . .,hI O. I~re exiN 0 < 'I :5 fj - Q lucb 111&1 p( M(a), M {z )) ::; ~/2 !or all z E1a ,a + '11.
Eo>.
> VUo ,BI) - c/2 for .. Suitable m.bdivisioo t:.. _ (.... ... , .... ) of In ,6]. and..., ~ IUp»Ot': 0, ::; It + 'I. Then Na.
O!:
•• • V((a ,.B])- " - ,,.
SinDt ~ .. arbillW}", V(ja , BI) - V((a ,.BI). .,.,l" IlUpp e. By ! em_ 13.l.l, VUe, " I) _ V(1e . /ioj) _ V(!" , "I) """'ttP to V (Ie. "' ) - V(jc,bJ) .. :r ~ ~ in /c,&!. NOOI V.. _ V.. {e) + V.. 10 V.. (:r) """ 'tip to V.. {
;"1". ".
PROOP: 8 )' ddinltion of (PI. IIIIC ja,/JI) " V(M.Io./JI) ro.. all o . pe l ..u.(yI", 0 < p. N_ let J .. (0 ,/1) be • ..,bintenoo! 01.1 witb endpoiUI.l 0 ..... fJ Crt f- 6I. lf J - J
M(e~ )
lar ..nce)4 , ~), ~ M (e-)" Hm. _ M (z ) .
•••
PROOF: We CII.....:I)' know lhat M .. of IoaIly baund.od vari&tion. Le\ e be In (" ,i{, ..... 1et (z; )'1::' IIe.de'tfll"l to e. ~ II("', Z~]) oo... ug to 0 .. n _ +
+ IM.. (4) - M(4)1+ V (M, Ie ,41). Now. lettlng II' ~ to z . in
~
1>:. ,..... ,{ (fr eodo 0 ~ k :S n _
IM.. (:r. ) - M... (:r. .... )( :5 V (M. le,d))
I), "" obtain
+ IM. (4) - M(4) I,
OS':S;~ - '
...
~red.
III Iohort , !of... \Ii A function of locally bounde< Slielljos mtUW'e dto/l""" by M , &Dd \Ii written dM. Now .... descri l>< II>< Radon
!IIOUIlre
......:>ciated wjlh dM.
P l"OJ>OIIl tlon 13.2.2 u.1 1/10 • mpCI 0, IN..... ....w. ~ :> 0 ...,," IMt
If I
dM -
~
(Aft:r, ) - M{:r,_tl) ' / (1,)1 :5 !
' S'S"
10.- ""II .....",,;.,;.'" (:ro•.... :r. ) 0/ 10 .1:11 "';ou rnuh iI . ",aller IMn 6. on,d 10.- on, .,.,.... (I ... .. , I~) 01"",,16 .,uil!lling z,_. :s I; :5 :r, lor 0111 :5 i :5 ". PlIOOf' : Oi"",n t :> O. then: o:xWt5 ~ :> 0 oucb lhal I/ (z ) - / (,1)( :5 ! for all :r" in "'.PI S&tiofyin.g I" - 111 :5 U. Let (:r", ... , z .. ) be & subdivision of
..,..J1e< than 6, and let (I, .... . 1.1 be • f)'Item of poIQtI .... Wyinc z,_, :S t, :S z, for oJJ I :S i :S n. Then
10 ,.8\ ..bole msh •
1! /dM - "HQ» · / (QI-
L
(M.(z,I- M.(z,_,I)· / (I.II:s:
1$' $ _
~·I"I( \o ,.8I I·
,,,.. ,,({o )) · / (0 1 +
C"
L
(M. (z,) - M. (z,_,»· I (t, )
- L
(M (z,1 - M(z._d)· I (t. ),
, s·:!;_
e .. ,,({ol) · / (0 ) - (M.. (ze) - M(zol) -I (t,) + L (M .(z, ) -
,,({oil· 1(0 ) -
(M. (Zo)-
1 (0 )" o;r Simlilrly. beeo,
M (z.»· (/(1,) - I (t, .. ,))
M(zoJ)· I (ft ) .. (M.("o) -
0 > 0.ben '
M (z. » · (/(0 ) - I (t,»,
M .. (Zo) _ M (Zo ) ..
,,«0» if 0 .. ...
(M.. ("') - M(z.» . f(t.) .. (M . ("') - M(". » - (/(t..1- I (ft). iweo'. I (JJ) .. 0 if (J < ~ .. h.c>: :/ 1 M. {". )- M(z .)" 0 if p .. ..
"'"c ..
n . Lmpla
(M .(Zo ) - M{"o» . (/ (0) - I {t tl )
+
L
(M.. (z, ) - M (z,))· (J{I; ) - / (I,+, ))
+ (M.. (".) - 011(" .. ». (/(t .. ) - I Ifl», and tbe...Jo",: dLat 1eI:S ~. V(M.. - M . fo ,fJ)I. f inally.
I!
I dM -
L
( M (,,;) - M (,,>-,»· f (t,)1S
'S' S' ~.
r" I([o .(Jj) + €. V(M.. -
M. [0 . .8\1.
o
'VI
ate
11. g.iol.p
In! pal
Tbeo,..,m 1S.2.3 ( Integratk>n by Paru) U I u. ~ /It I_lund>", ~f k>""'lIr 6.>ttndtd t'Grialion""", I into C . .4........., /hal It "nJ ~ h""" "" com"""" poinl (]of ditoonlin";l. OJ 1oaIJ/Ji dv·;n~~. v ;, IoonJIy du·;nl~, ,.,.J d{....) .. udu + ."...
n...n ..
-=
PRIlOII': On o::xnpoo:t ,ubin~ cl I, u ilI l be uniform limit (lEa !lequence of mp funaiono (PropOIIilion 13.1.2). n.e .. be, It is locally .w..~. I.)efi", II>/, fun - ..I"M")·
d(ug) .. dlu .. v.. ) when.. "'- not belong to I . and
dIu) - IlY poillUl or 1 . On
f .. f
la ,(d., ® eluj .,
du(:z)
· 11~ .P1(:Z) '
~
du{{:z )) .. 0 lbe other band,
f
l lo.P1(, }du(,)
l lo.P1( :Z) · MPl - u(z) )du{r )
..
( ..(,8) - .. (0 ))"(,8) - (~..n.1O'" ,lJ!).
=
f
Simil ... ly,
f
l ... , (du ® elv)(jo •til) " Oo .tIl) .. ( .. (,8) - u(o ) (,8) + ..(,8)(,,(,8) - "(0 ») - ( udu + trd.. )()a .lI)) Tbio ......... that
d(orul(j .. . OlJ .. ('''')(0) - ( " v)(o ) .. (udv + trd.,)(]" . ,8!) , N",,", Olio«! cI{..,,) and ...." + t>'/1" !.eM"", ....... ....., "" t . Lei ( ..... ) _ "" bo " fI.ilill(tlJ" .....",,"9 from F" G iftlD H, ..m.. F , G. and H """ thrtx r-...I 8o>w>eIo 4 ...... AiN, leI t - F .nd g' , 1 - G bo k>t:altr >..~ /o
[!.d>. -
' (oll F (o,) - F(.... )) + ... +
.(n~_,)IF(.... ) - F("~_,lI
F (", )[. (4o) - ' (0,)[ + ... + F("~_L)['(""_ ') - '(Q~-tl)
+ F (.. ~)6{"R_ ')_ 51""" lbe codlid.,,11 ohlle F(",, ) on lbe l'i&ht-haDdIide am pt8iti.., will> own
*), ( 1/ .e.. ) f I,d), lio:oI in D. Now ~, br the Jtq> fuDCtioa
..m (... _1 . ... 1( I :5 ; :5 ..), .. tv:re 0; -
.~
o:quaIl g(6) at 6 and I g(O;_ I) 011 .. + (;/.. )(6 - .. ). ~ .... uroce ( •• ). 2; 1
10. at
ate
OeflnitiOS> 13.3.1 If / ill .. D>8ppi"i fu)m X :> M ( I) into a 811nod> Sp6OI: such that / .. AI is dM·in~. then JU " M ) dM is w~ the i,,~ra1 of / ~r M and ill writ\.en J~ J. Now , WI! " - ' that the li~ integnol J~ rameteriz.al:ion.
/
ill In,...n..nt under
"'''10'' of PI"
PropOaition 13.3.1 Linternoll Q/ R line! '" " o:>n.h....• ..... ~i"g from J Ill.. I . Thl~ (",. S) is d(M .. "') •• IIiI..t line! dM .. ",(d(M 0 ",l). ~. IIIMI = ..,(ldiM 0 ",)1) m....:..... AI it rig/lt-o:m.tm ........ "" la ,hj &t' '" is ~trictJ~ mc..n.;ng.
""'eli""
P ROO'" Clearly. !of .. '" i& .. funet;.,n ollocaUy bound
_
( M 0 ", ).(a') - (M o ",)(~') M (a) - M (o) .. O.
Finally, dIM 0 ",)(,..,- ' (" )) .. d{M " ",)({o"H ill equal to M .(a) - M (.. ) .. <W({ It }).
'I'
ate
Ib short , ..., Iu.,.., 1"00(11 thaL .,,-' (8) i6 d(M o ",,}- Int~ !'or eech S...,.. E and 111M dM (£ ) ~ d(M 0 . and 'Pil) _ .,,(.). So. i£ I € (e .d! ie not &.Itat;ooa,y poilll. then (M 0
iliao than IdMIOa ,81). We (t)"dude that
141M ., ."JI(",-I (]a .P))) S IdMjOo , ,1.1]). AMume ---..Ily that If" doO\OI "0' lie in J . toO Ih&t
IdM10a ,fJ() -
.-
(j _
b.
T~I
lim jdMI([a.:zj) ••
M (I) inf a EJa"od, rpaa. Thm 1 0 M ;.,dM·inlqrabk i/.....t O!Olr if f 0 M, ;., dMi'~ J",. oJ! 1 S ; S 2. In tAu ......, 1M I ..
,100
1M, I + lot, , . P ROOF: C\ovly.
dM, .. (dM )/ " '» - (M(c) - ",+ (,,- * • •
w!>ere I I (retlI"""'i>'eiy, h ) "' tbe u 'p ',, ~ O
:r E J 0i0>d> Ih.t D( r(p)j(:r) ",,1m. :if ...,. deri ..u!oe D( ... (p»)(:r).
poi"'"
'l'btor8n 12.1. 1 """' ........ lhat . at It.J.III",,* all % E J, M haoi deri_ tloe j (:r). H..,."" at A.al",.. aU:r E I , M haoi deriVllti'"l/(:r). a
n..llnltlon 13.".1 M: 1_ C is oaid to boo: aboolutely conlinU<M.III wben.eoer. lor tao;:b f > O. tber& eo:lIU , ;> 0 iIUOh that E ,:s;.:s;. IM (A) - M ("';)I :SO t lOr ~ fiDit
E. :s;I:!O.(A. - 0,1 < 6.
«",..
- -'
s....,'"
In thil c_. If 1 io (Oml)lKt, lhen M is of bowt.d«I VlIriatioq.
Definition 13.4.2 M : 1 _ C i0oi oaid '" be io>o:IIJb' &boo!u""b' CODI;"'- il IU reotrictlon to ~ (Ompod. ",billl..-vai 01 I ill abooIute1y _iIl\W)O. In \h\o cue. /of i0oi d Ioa.Ily ho"nded variation.. T heorem 13.4 .2 Ld I' ..... e»mpkt: """"~,.., II 5 one! M 411 ;~finiu. ;"h"wl 011' . .4........, ~ I' • ., ............, o. wn a Iiu in I . n .... M i. /.oaoll, uNlliftS, .....ti.. IiOIII if ane!1nIIt il I' ;, a m.eouun: ";/h lwm it. P It(IOr: Lee. f bot .. real-...Jued !t>lute1y CiOnUn ....... II and only II M It. By n..".,tm 10.:1.2, \Ill o:IM variatioa and hM deriV8.ll"" zero ),.~.~. Theorem 13.4. .3 Ld I' 100 4 """,pia """",...., on S aM At an i ruh/iniU inl'fNI oj 1'. "111 ... I' it fingulor (i.e. , /fujinl fram ).) if and onlJ iJ M it
riIIgtJ 0,
i
> -.
•
F\>r e.ery V E D , let t, be the inte.--,ection 011 and the .--~ and /'. ill d~nt from ).. Denote by AI, the indefuUte inlegr&l of I). dw . takes the .-oJ"", M("C:b _ _ ure. then, "" e-.;h :t e I. 'fI'e defi"" Qp(:t) and Il,,(:t) at. foI:o::.....
1;
Qp(:t) _ oup inf " He .d) ) ~l:' (~-"I 4 - c Il;o{:tl -
In!
&up
0,/:' «-"1
I'tJe.dll, - t
where (t,,,! runt Ib""",,, lbe d-. Dooe S«u 0II1 all poi"'" of '" • b), and it hao ri«hl-hand d/:ri..uive +0> .t 1""0'"", all r E (.. ,Ioj.uo::b that jJ({:r )) '" O. In particular, _""'" 1'".018 pofIItiw, ~ and 1IiJ>«uIar, 111 bat derivalh.., +0> &t ,,~t all poinu of J , heot! all:r E I ,
--'-- . [
II'-:t
•
!/ (I) - !{zlldl
o
PKOOt": Th. foIloon from Tbeot em 12.1.2.
Coa!leq"..."ly, for A-a1rnoss. all :r E 1, (IICw - z »)·
Ifr )"""' :r I ...... to :r.
f: / (I)dl ron'bPS
tQ
282
13..
~iohjeo
InlqI'o!
Upper and Lower Derivatives t
13.5
Lee 1 _ [a
,bI ( _itl> a < 6) be a (Ompact 'ubin~'II! in R. Md refer
I.otbMcue ""'''Ire.,.. I .
Lec 1 : 1 - R boo OOQUnuouo aad let
\(I
A ilr
I _ R be regu!&led . SUJ>P lhal, a~ ~ % E 1 _ D , I has dcri\lllli"" r(%). TMn I (s ) - / (") " g(t ) dt fOr aU % E 1 (Section 3. 2). The following tMot en>. due Il function, from / into Ii, Mrklty than /' al ~ pOIbt of [a ,b{ - Po ""'" tha. t. g{l) dt S t.1'(I)dt +c . For a bed 6 > 0,....."k\Ction FI:
,_tel'
1 ~ %-[tI{t)< ....., ratior!.Ul .oeb t ha. 0 < " < ...,. om ... I'"" .,,) .. E;: 01,," . 10) _ 5oI1ow..: E; . )a . /o) &Dd, lor owry " e Z' . ~, .. tl>< oeion d 1- .8! "'.. 'h"""h tho eluo 01 011 "",,_ttd """'pane .... 01 E;:. SMw that .l{E: ) «, I...,," . (~ - ..). Oed""" that t ho ... (" e la . />( : ,D(z) < ,., < ..., < D" (:o:tioG I :;u I . 1Io.oj tI,o.
J
I.
FiJot,u· ....... lhat I 11-' .. liromiDIIiOUI.Let.(I' ) ' ~l bo1 ... iDa i", ooq_ 01 """,,i~ h",e,iono from I into R + .. _ "PP"I" on. I ....., II / . Fbr -rr n E N . - . , by ,ho funcl.ioG .. ' !Io .", tIp. FiDaU)r, i« 1'" : I _ R + ho....:ll.hat 1'"(.. ) II ,be deri..,i.. 01 F .. ....1 _ 011 psK_ z 1_ F .... 4). 5 _ (Id in R ', Thio wiD boo portlcwady ..." .. u ...., ... ben .. """",
•
2M
I.. The Foutior n....fonIo .. R'
P IlOOI': For € > O. let (B,),s.s'" be • .Iini.~ putitioa 01 A, u...o S· ....".. nps wch ow. V,.(A,) ~ r; '$:O"!i '" W8,,)I+,13. We m.o,y "'1'1"* tIw An 8. ~. for.u I -me ...u"' ..... kJ.... tl1O.t E.~I V,.(A.):S VP(A ). Let f > 0 be ",..ea. Applyi~ LemmA 14-1. 1 t(I Bee tbol there "' aD S'reclanPo A' I<X:fI t hat :if C A and VII( A) < VII( A') + t/2. FOC' """"
V". _
•
n 2:. I, t~ ",,!ttl an S-n', '" all n;l C ;, 4Jdit>«..
PIIOOP; Suppaee lilat eod> .ode ).... b.) .. I, ol an S· reet;&llj!be A is panltlon.OO inti) n, tubinterYUi J;.j - 11.... - •. 1' J] ( \ 5 j 5 "')' ...·ho:nl ... - I ,A < 1;•• < . ,. < I ; .... .. h, . Then tbe n, ... ··· ". ~ 8 ;, .... J. - J. J. ,,·· · ><J.J.
(ISj, S ........ IS}. Sn. )
(\)
parti\.ioo A_ Such , pm.it.ion II called rqular , We fine .,...,...., that " is additi""
10. "",,13, ~"il~
p.(A)"
L
,,(8, ......... )
(2).
UO .... J.I
The ri«!tt-hand !lidenl the (>Ute< tum extell:tends ~ eod> " thlu is & 01 the ~"'" B. and. fo, fi1«!d :r. tho: in...". BUm
tbe right-hand ODe
or _
~
all 8 ol whlch " ir a ","ex. l'ow BU_ that" io a • ..-to; ol ODe or onore of 1I>e """"fit"" B, but Ie 1"10' • ~"e>o of A. Tben there mUlt be an I (I SIS t ) IIUCb that '" is neit""" ... "'" 10, . Fix one I (there IDA)' be ~",a1 ouch I). Tben '" .. I,,, wilh 0 < j < .... 1l>e """"'n&Wo (I) of whlch :r is a """'" therefonl OC.(
n ),,>'))~ n ,, -.)
'S ;S'
' S' 5 0
meu~ OJ> O. Bf """""""'" on S.
for all S-""",*,,«I6!. that is, .\0 "' Lobftig"" Now..., otudy ""3""
"""""rwe"""
FOr 011 1 $ ; $ k. ",rite Po foc the function (:t", . . . ", j .... ". on O.
Propo<ion 14. 1.4 ld ("" j~;;:, l>e " ~ in M "(S) ~nd 1''' """"'" ........,.. on S. If (" .).;;:, ",n .~~• ...,...1)- to 1', 1Mn1'.. (Aju.w to ,,(A) .., n _ +00, lor 011 " .~ S -t'I'. DI':/IMOCl R' the DOI'tn Izi - OUPI£.s,. I- 0, t~eriIt>I ~ > 0 web that 1od .... furu:ti(m 01 Pu , for all n e N , GmI F 1M dwrib1Llion fMnd ...... of p, Thm (P.J.:>. .b'~ to P if "u an.Ir if ( F. (:cl). ~ l to F(,,) fM' ."";; of ..",linllitv,:z, of F
"""""''lOU ....
"""lit
"""""rgou
P ROO'" By Tboc:nm 7.1.2. the condit;"" io ' .... e " y. eonv".... ly. IU_ it ill sati$Md. If A .. n's,p]a;.M "' an S· ...... · "81o> onxh that ,,;'(G, ) and ",' (~) an! lI-atgl!&lhIrL for aU I :S i :S t , """ can .... pI P~I A) in term!I of F~,
and !llol.loor1 that ( P.{Al)~i! ' ()(I 0 .. ~ O. ~f(l~. N,(m.,,') H, (m,O) .. " :> 0 .. ~ O.
Io: FOIl';"" tr ....1orm aDd 11>0: ;n ___ FOIlIier Y"anIiJorm of P. ' ........ Iimy. Deflnltion 1..... . 1
p ba ..
l.et. e' (R '. C ) he the norlMd""""" of bounded conl inOOO8 func&iono from R· into C .
Pro_itio" 14.... 1 Far 0>11 p e M '(S, C ), ;Fp if ""iJarmlwamtin...".. ,,"01 """'.......,. n. Ii_T ....' p .... ;Fp from M ' (S. C) .nlO CO (R · .C) ;, .,.",n".
-
1
PROOF: By I~ dominated COIl~ Ilteon!m, inf(2,61' I) dVl' (:) tcDdI to 0 M 6 :> 0 ..pprooclIeo O. Ci""" ~ :> 0, lhere lhuo! ""w" 6 :> 0 wch lhat lu.f( 2.61%() dVI'(zl:s f . Now, if l:, \I € R' S&lis!y jJ: - , I:S 6['h, the!> IF14") - T"MI f, 1>«:&_ It- - I I - I 10· k " dtl 1" 1 for all 11 € R.
:s :s 'fl>oertfore, 1"1' _ unlfor",lJ' corui .."""". Nut, IT ,,(zll < 1 1dV I' for all z E R., "'hiob jXo_ \be oecond aq.tion. o
1.01;~. or dz, ~
Leboor;ue measure on R · . For all g E Lb{dz ), t~ fW>Cf,;"'" 1", _ 1"(14) and 1'g _1'lgdl:j o.re called 11>0 Fourier transform and I~ i l l _ Four;.r lransform of g.
For o:xamp\e", :, -
T ,, (z)
ap(_,,1, 12 ) is dy-in~, 1"Md, '"' I, &nd
0
0
II 1~-""'.' .ap(-..~)d\li IS,S· II ap(- "",,') 'SJ~ '
0
,,(,,)
for all '" ;" R ' (br klion 14.3). !II> T" = g,. For each" :> 0,
. F9« :) _
j ~_ .'u. dp(z)
-F9,, (' ) . Fp(z)dz
for WE R ' . Th ... p . g. '" :J'(F9- n .p~ n. W~ """"lutk lhal '" • g.. )~ COD>('gt6 ",....-1), \.0 p.. Since p. • 9. = :J'{F J• • F p.). _ M~ lhe 1OI1owi"3 II.oot cm.
Theorem 14.... 1 The ""'wi"f F : I' _
u illjecn..o..
F p. /n>m M '(S. C) irttoC'( R ' ,C )
Up \.0 Ptopooilion 14."l, ~ I E J:hI~l "" .9~. For uoh:t E R ' , ~ by -,(.:)1 the !un: f'il'!ll., ~ tll&l I io amlin"""" wilh rompoocl.uppon K. Lee V. be. compod ~borhood orO. Sina: I io W1ifonnly CODlinl>Ollll on R~ , glvn> ~ > 0 tben: o::UsII . almpooct neighborhood V C \00 of 0 such lbat.
1/ (,- :.)- / MIS A(K: V ) o Nth{:.)! - !) '"
for all:. € V and o.Il,E R·.
f I/ {, - :.) - / M Id!!:S;
br &II:. E V.
t
Tbaebc, tbe mappilll( :' _ "}(:.)! io amtinltOUl at O. NO">" ~ tbe ~e"",&l case,;n ...hldt I E Ci, P,). IfU. ).~ , io .!tCquoeno:e in H { R t , C) con-stllll" to I in tbe mean, tbe ...wIQII
Nth(:.)/. - "}(:. )I ) - N,{f. - I ) ahooos 111&1 tbo: 1Iequoeno:e '" futtctlons ~ .... '1(~ )I. amvervo unifomti,y QII R t to:. _ TCz )f. 111... ~ _ -,(:.JI ;. CODI;II"".. at O. The propooilion roIlo::oon
euWy.
0
Remark d w, for """')' COffipoocl neighborbood V '" o.
/.v·
M.d!!-/.
v'
""".etp5 to 0 .. " We know thai I oi l .., thai
"-'g, (!~}i'-l g,(~) d!! " (v/ . 1'
> O.pproodteo O.
. g. •
(f.b: ) . fH 10
A. in~
for ted!
(7
> 0, and we
f fH(,- ~)/{~) .u - f g..('+ ~)/(-z).u -f
(f * M.-IM •
/(, - z )g.(z).u.
Propoaitiou 1f.•. 3 I ' 9. "'''''''~ to I in Li:{A) "''' > 0 .~ O. P _ For &II
(7
> 0,
N,{f ' 9. - Il -s.
r
d!! f l/(II- "Z) -/{~)Jg.(Z) 0, let V bo:. almpooct neighborhood of 0 oucb that N ,(-,{z )! -Il -s. t for &11 :. E V (LoIWII& 14.4.1 ). "0 > 0 110 thai. g. tlz :s; f for all " E ~. ".J. Theel. for all " e "0).
to ,
crv--
f., M.(:')· Nth(" )1 - I) u S
Iv.
€
....
'VI
ate
Next, the funn.ion Iz. ~l "" I III- z l ill (",)')@,I,..int~ablc, Ivoa,"'. Iz . v) f{~) ill (",).) ~ ,l,..intogable IP'vpo;eition 14.4.2), I'Urthe,wore , / fill /
1/1, -
z l- /Ml d(g.).)(z ) -
-
/ " (,g.).)I:r) /
/
[/ (w- :r)-/(w) ldy
~(") ' N, b(:r)/-I) 0 approoochal 0 (Propoei. tiot>o IU. I &I>d IU.3), &I>d F/h ~ at infinity, We may <x>eon!m , F g.. . FI' ron""", .. 10 F I' in Q,(d>;), lIS (1 > 0 a~ 0 (he 0, and b E R O, tbe fuoc:tlon "' _ fJeo:P( - " ls l' + Ioz) can be .m.ten Qg,.b-), ';th If .. ,filii, II' .. 6/24. and 0 .... Itably c:bo:! : 'Tia;'elooe, tbe _ _ !I\Ibspeoce 0( C"{R " , C) K""", alt!d t,o tbe "'(11'-) iI &II
~ A (wltbout unit). MrmIp "" wmJlOd
.d.,
J
PII:I)OI' : Fi~. iUppGI!Ie that (.1'.... ).;: , C(Itl• ."g... pOinlwieo. 10 f,.. I 111'. .. f I'. {O) (1)D~ 10 .1'I'{O) .. I III' lIS " - +00, an sup,. 11'0 I II Iloilo!. Now. "" tbe dominated C(Itl'" ,"gu_ t""""~m, 10< ~ tr > 0 and lor every- ~ E R ", (p., • , . ) (, ) .. ~~ e>;p{ _ .. tr2 1~1 ·1 . .TI'~(~) d.:i: aKlU'VS 10
J
J
w ' ,.)M "
••
fe ..... ""p(- "tr21r1') . .T..(~) th
"" - +00,
Equi_olly, 1'0 ~ (,- )] \ends 10 I'[g..( r - )]- II foIJowa tllM (p., ).~ 1 C(Itl'-"' ga& wee k? 10 I' (P "'!'OO11oIl 1 ,1.3). ThUllI' ;s poo'Clad \0 1'( 1) . we IDIIY mndude IM.I (p.,). 2:' IIw "'pp o
'VI
ate
JO'l
14.
n..~ nonobmin R·
Theorem 14.4.5 ( Levy) Ld (jI. )~~, be: ~ ~ in M ~ (S) Juch Ihn)'" to I'~,
""eft u.~ ).~ , """....-,eo 114m >tdr I #.
PROOP' 'Ibcre is a .n) ql>eDOl! v.,..).~ 1 ~ (I'~).~ ' 01 v.,.).~ , thlLt ~ w" kly to I' E M ~(S) ( P fOp(IISibon 7.7.5). Si""",., t..,1onp to ~(d:I: ), ,he function ~ _ ..,(:.-). ap( - ..I:.-I' ) is d:I:-integrabl.. Now, by the dominated ron~""" t~,
I'~(!h (~ - »
~
v.~ · ,')M
..
!~~" e~p ( - "IZ"I') -.FI'~{:'-)dz:
to f .';"" exp ( - "1:.-12 )tinuouo at 0, """ only point (" , ·· ·,II~j in {R· j~. deno:M br T( .. ... ... l \I", fuDdioo (11, •...• 1I~j .... (11 " , )··· (,,~~j rrom (R ' j" IIIW C . n.a ruappillfl (", .. _,h) .... 1(., ...... 1 from (R 0)" into L "( R '. C ) iii mwtilino:ar. Ii. ",,",. for """')' l: € R ·. ~ .... i" . ~ .... T( ....... ) iI Now, by n..o.em 3.2.3,
. For all 1 i:S n, the imag'
/
~Il .t,.{%) ...
L
';s's_
/ :r~ do«" ) -
L / zl 4",(:OlLk:al boosioI of a ". Ao n" n_malriI D. with ".,.J enl"'" C<J' it IUd 10 be Iymmetric and pooili ..... if the biUnear form '" on a " " a ". such lhat V{c"c; ) = "'J for alii S ',j S. ", is 'ymJIloltri(71,) ,,) .. ~"') for all z:, ~ e R". Clewly... is lIermilian (witb ,upect 10 40). There", &II Orlbooormal b.o$is (f,.- __ , I") in a " (0( ']sting of ~""""""'_ HeI>oe (flo- ... I.) is orthooorrnal for 40 and onbopuJ for t/J.
h .... ,I.
if )'ll''' P,),• ..., may IUpp
o
.'•
c. o
o o
IU Nor....!
a.-. In R "
LH\ins" bt Iht ~"m"f R · ruoh that u(~l- I. for- t.IIl SiS n, i ... "'""" A wit h 1t..""",1 It> (fl . ' . .. e.l is orthogonaL Th", A' . A .. A· A' • I•. Since C ill the malrU:: " r ¢" (u )( u) with re;1'fjC1 to (e, .... ,f.)• ....." _ 111&1 A'·D·A.C. N....
Wi
. "'"
the meMll. e
.... S. , where to II the measure on 5,
defined by the untl maa.t O. 'l'hen
E(,,) .. (0 •.... 0), 0(,,) - C, and
jcha{,,)){z) .. exp ( - ~,,~z:) )( , , , )( cxp ( - ~,,:z:)
.. n:p( - ~r'o("l") br all z € R O. 1'bercfon:. " is C.U8IIiM . The irno.g
Theorem 14.5.\ , ;\
A' D- 1A .. 04-' D-'(A' )- ' .. (A.' DAr '
)r
ale
o
c-' _
,
o
"
r · A'· V -I. A · % .. ", 2:r~ " ... ""; ~.r! I'or aU r E R -, "'''''' Jet 11 : R~ _ 10 . +oc>( M .. Borel function. ThO'"
lienee
rh"" ~ j"II(m + U(%)) du(S) Ie "'Iu.oI to
(2Ir6n. del(D)- '"
r
hem +
tI{"»·exp ( - ~"'A'D-l Ar )d;r,
Thor: k.nnullt. for W,...e 01 "". iahlc8 olKIWs I bM
r
II dj. "' equ.ol to
(h )-an .doc (D)- 'flI
x
"II(~). ex!> [ - • J
2(" - ' (11-
m»' · ,4' V - I A · ( U - I (~ - mIl] dv_
Slnoe .. - '(~ - m ) "" A -I (~ _ m), , .... ptOpOOlloo I..,UOOO$,
(r , .... , z:,J -
a
(a,,, ,. _.. _
A· C - A' .. O.
FuRI>crmore. the raak of B ill the rank p of D eo.."""""ly, let B be all arbitrary n " ".matrix of rank p, oueh that B · 8' .. D. If II> is the Jintat mapping from R' inl(l R " wi,II matrix B and if ,. io ,be im>«" 0."'-" lho.
~j'_. "
:II
-- "'~(I ' ~ 'fol~(lI~ - ")) 1I!,~. (I II " - G)
1. DId _ _ put 1 .hat
2 I. the -.tloa. of E"tEdoe •• let jl bot.be I"."ure""'llqalat ~o.k: funct.ioII 0( J 4 .
g.,_ .hai,/,: R _ [O.+co( ....... ,,,_ ..cO) .. !. a.d ..c-I) " ...el ) for 011 t > O. I.
I J d.z ..
I. _
(PoIyr.., en ....
'IF
ate
Part III
Convergence of Random Variables; Conditional Expectation
•
Copyrighted material
15 The Strong Law of Large Numbers
(fl. F . P) ...b.,., fl io .. _ooa,pty oft, T .. O'• • bra ill. n, and P .. proI>oobilt. y, . ba• ., .. pooi.i>e _ ~ .. on F ""'" tha!. P{O) • 1. A ...... .10>0Io It .. _~.oblo fu"","",, from fl into R- C _ .. __ ; "' 01 iado .... Kk ...... ndtwn variabIL So. _ X, + ... + ~h-,,,,I pcoI>lom i~ probo.bibility.booory io to otUodJ" S. / n aJ>d dil!erent 171* '" """"OOK .. n _ +a. TIw " ""'I! Ia. «!up DB ...........,.. thai. U'tbt X. ba... lbo...,.. dlotributioot._ $. / n
In UUt oet\ .... tho ["ndor
We p.e bIIitJ' .I>oo
Definition 15.1.3 I.,.et rr be .. """,,""loll:, me\riuhio: uniform space, and let. F he t!toe Borel "-a!«ebra of rr. ~ (0 , .1', p) he .. probabilily l<JIA« and ( X. )":I::' • geq........e (0/ .o.ndom variab1 ... from (0,.1') im.o (n', F ). Then {X")"~I io said to mnlOlrlle in probability to • random variable X from (O.F) In'" (n'.F') if it CO' ''dV"' '" X In P_IIWIII\II"(!. Equiv.)emly, { X")"~ I mn~rg'" to X in p." and any densily of Px,. iI equal >""Le. to.he Bono;I function, : R t -10. +cot. In , h" calle, for sirnplici\y • ..., 81m ... rite Px .. g>." .
funttion (.
15.2
r
Independence of Random Variables
Let ({fl,. T; , P,)) .. , be. family of probability opaaw The ,,-rJpbra 0 0E , F, " tbe ,,_ring ,;entrated by tbe .. ~ OE , F, . Let. p be , be meNu"" 0 .. , P, on F, doofined in ChaptH 9 if I illini"'. and in ChOp'''' II if I _infinite. Finally. let P be the """" .. OM of" to ®'E' F ,. Ot.erve lhal " t.tId P ha"" lbe arne main probl,pIloo>.
n..,
n
o.,ftn ll\on 15. 2. 1 In , be probabilistic th&t X~ (t) .. ~ for ~ IEO, ...here {Z.).~ I is t ile _'..,.minulrlg baiie-bapoD'ion of /.. We ... iIl Ibm< t hat tbe X. are i~ ,..""om ..... riable\l from (O.S) into (ft,F ). This examplo: iUun~ the _ioc of Inoe: if & r.,.J Dwnbo.>r t iI choootn u random in 10, II. knoo<JedK" of tbe ~rst (" - I) \.OI"ms of i\.o noru.ermlrwiq: bueo.b apAruion gi_ flO iDdic&tioa of tbe term of order n. ~ ";l~ in fact. prooe ~ Fi>: ... prolwobility " on .1" ....::Ie that "H" II .;. I for all 0 ::; .. ::; b - I. Wriu ,.. .... Mil). i_HOI of ~((1I1l. Put Q . .. ft . ~... P , and ,.... .. ". for ..u in~" 2: I. Th"" ®.l!: I~. is tbt Eln=1 ,,-.bnt of the topQIoci. 00 Gu.)(( I}) ,. O. It ilIlooi'f Ihal F ill oootinUOUll. EI>dow 0''' with tbe ).,.;'",aphic """". Now G is i""""""'n«. For z . ~ E O'N satis(yinc :< < " tbe I .O::;kSb·,
ki nu:ger} .
Let n. k be lwo inl« - I , , , .J, or . . (I< + 1)/"" < 1 .00 (%o);:! '
II thiI: 1110»BllSion of (I
"". 1&
inti",.
z. ,
".. ".
P( X''''''' ''' ''X' ''' ''.)
-
p{Z, . " , •... , Z. _ ".) P(X; ", z.)
'" II
's·s _
for all .. 2: 1 and all " Ir" . , Zo in 0'. Wa _ rel:W'I); to tbe fUllC:tion F, Wbea p" .. 1/ 6 b- all 0 :5. " < l> - I. lor each .. e N...., """"
Fe;l) -F(;)_ ~ for all 0 :5. t < ".. ll-.e F(t /"" '" k/6" kit all 0 :5. I< :5. ".. Si~ F II coolin"",,", .... concl..o. ,hal. F (I) '" I for all t E l , and Gu.) • Le~ " .,, "' .1. on tbo: Bono:I",a!«ebra 01 I,
in wbat f~ th.at P. '" l i b for Bt le&st one 0 ::s u ::s b - !. W""" P N DS thl'Oll&h rr, the ~venr. (X • .. P) >= ~I!O ,w iD
c,,_ bo . [ F
/I'
( . ... )
'O'hU~ to
-F
('0)1 II'
_b" _P(:r, ) .. p(". )
1'"'(1) ... n _ +b&hllity of 1Pf« &\. "bold pl.y' for- • pmble-- ",1>000. inlt i$! eapi~ "' t 1_ PatTidr. BUlinpley, P" bMlil,.1at a &I""" _ muIIt ha.-e pmt,.biUty eil~ 0 or I. 'The dil!ieuIty ...... hereafter in dtttrmi"in,g ..hich of these val ..... (0 or I) .... are de6lillj! "-;th. prot lp m, ill t.c\, may be 6L.lemely dif6cu1t to ooI.-e . N"", Itt (rt' ,F .,,') be. probability spece_ Put 0 . ... rr. :F. ... T' , and "- ... Ii, for all n E N , and t'OViAble ("" )'~' _ "'. from (rr" .C) Into (fl •. :T.l. Lo:t A be .... invariam """'I. Then A beIongo to .. (Z" . .. • Z. , ... ), to ,,- '( A) I;... in
",(Z,
0
tI, _ , _.
Z.
0
~. . . .
J- ..(Z, . . ..• Z.... ... .J.
Ind""tiwly, .... _ that A belongs to ..(Z ... ,. _... Z Ir+ • . .. -J lOr -=II iIlteg« k ~ II. 'The.do, ,,. A ill. tail eYeJIt. and ,,(A) io eilher 0 or I. 0
Propaoltlon 15 .... 3
It
if p-.,-,ocIit.
P IlOOl': Ld A be lUI _ audt IMt 1" 0 ," '" I" outside. ~~I* art N. n...., lOr .......ry ILI1eg", t 2: 0, I" .,. ... I" outAlde E ... ,,-( N ). Put A' -U~"'-'(A ) and AM... nJ ~",-J(A'I. Clcerly. I" , _ I" -Ollwide E . and I" . u" I" . outside E. On l be other hand , I,," ... I", outsldo E. Si...,., ,,- '(A') C A' • .... ha.-e A~ _ ",-' ( A~), to that j>{ A ) ... p(AM) i& cit!,... 0 oe I . Ncn , let B be a p·"",...urable 8e\ IIUdL tW 1" '" _ I" II-" mon everywhere. ,"",,,,, ex;.u ..., ~ A C B lor .. tuc:b B _ A i& ~~lPblIl.. n,.".
u."",
I ...... .. I.o~ _ I" _ I .. ".aI_~~, whicbtbow:a lhat ,,(8 ) .. "IA) ioI e;t bco- I) or I. P'OpO&ition I I .4.2 tben poo'1!$ that ~ " !' !,godie. 0
Theorem 1$.4.1 ld (O.F, P) "''' JI"'I6obilill tro«. aM l E (X, ) .. J X,J.P "' .. -0 +00. PIIOOI': Write" lor 'M pnK.,bilil)' 0 , 2:' PIC. 00 ,he Porel "..~. C of R N. Lott ~ be 1M llil" tTWlliIocI1I'1on (z')j~, _ (:1:,+,). » from RN into ItaIf. F!naIly, let Z. boa the fu""tloro (Z')'2:I:' Z . fTOOl R"1< into R. for all PI ;:: I, and pS' Z. (i l¢'l a1",,* ",rely 10 P(", ) ... P("l ) " n _ +00. a
Now , lei. 0 .. )0 , ]1and lei. P be Lebetl&uc "",""un! 011 the Hom " . IlIpbra, T . of lO . IJ. For..m n E N . lei. X. he lhe fulictiort fIIom...-iabico from (n , .1") into (O',P). Set 8 - lk/&" : n E N , 0:5 k ":5 ~ , k in~) . OeftnlUOli 16.6. ] A number I E f - B is &aid 10 he rompletely IllImlOJ (with I"1!SpecI 10 the ' - b) if, lOt e.ft)I In~ k > I and ~ry k·tuple (u" .. . , Ut ) of ~b dizj!.l, lhe k-tuple .p~ in the bout 6 e.:pe.psIon of I with uymptotic rtLatlve frtq uency II II'.
........
PropaolUon lli .li.:Z (80"'1 ) p ·almo.t aliI E f - B Grt
COIIIpld~l~
""""'"
PROOF: Fh.l: ~ I and (u, . ___ .... ). For...,b integ
FiDalIy. put
Y., •.._•••• , " We .-y .hat
(,,j .. ,,' _ 1(Y.,.
+.,.,.,(,,)
"" all I :5 r :5 r, .
(y, (,,).... , Y.(,,» io lhe " ..... .,f c,clic '
n .n E N.&ndIot(o'7'. p)booM;"£ " " I. fo< _I:S ' :S n. if
,
.. (2)
v
,
En.. ' " for Chapter
I~
... defi.... X. (~ ) .. thor u""""'"' 01 do ,hal i. belonp I
1.0.) ioo. toil ..._
;.0.)
~ I.
for (X.).~"
II ".,..Ita n ..... 10. I'. T ..... he I'I<weI .. ,""«-beoo'oa> o:od i\o ....,.....u.alion, .boo LII>dobtoS It ' '' ......... r """'Y "
uod
s.. - !:' S' S"
_,."..,uoJ
ti,.
I , ""';te ... ..
[L' S' S'. Var( X• •• )1 If>
5.,. .
_it. .
X..... n..W..... t>eo.OIditlon io.~lIId""t lor t OM., .... rn- . 16.2.1) , Nooo, i&cide,,' oIIy, l b • • hio .-.J. c-. (. . . '" 5 ough ~l/,tllO!nt&ry, io usefuL
Propollitioa 111.1.1 Ld (IY . F) oM' ~ X . lie 41 in Dqildritm 1'. /.1. C ElY . kl ~ 1I>'l! "" F J~neJ ~r ~ ... il ...... at Co no.. ill lI'I'"IIer tluU (X,,).~ I om....,. in lei. t.. le. it if " ' 7 " • .,. ...... ~'" lit'" P. (X" j V) """..",. to 0 ... n _ +00. ler «leA """,pod ~ V /
'< .. ......
e.....
PIIOOr. SUPI"'* the ooncIil""" boids, and let I E H (tl'. C ). GiY(ll f > O. the ... '"''''''" comJ*1. nel&hhoLl-.:\ V of e sudl that 1/ 1:r) - I (ell:s e/2 for
.u"E V. Tben
j l o X"dP,, _ I (C) _ /,
(f 0 X. - / (e))dP"
( Jr.{Y)
+/,
(JoX,, - / (c» dP••
( Jr.f Y)
IJI and
0 X_ riP.
- l (e)1 S
~ + 'lI/IP.(X;'(~))
for
~l " E N .
II I o X_ dP. - I(ell ill to. 1haD., if n io Iarp ~b.
••
eon'eudy. -...... that (X. ),,2:' .............. iD wml*'t
~borbood
Is equal to !
on Y'.
P(X.,. V )
m (fl. F) illl (rr.Fl, aU lei X k a ",n4&rn wria.'/k fn>m (f1, F) in!. (rr. r ). II {X. ).2:' """,«i j " in f " hboilil, I X. IN... it """w:.,... t.. X in Ie.... ,~. if X iI oIml1 ",,...1, .".... 1n 01 R ". In tho: """,hoo 01 Oo:finilloa 16.1.1, let F. be the distributioa fW>e\ioa 01 X ., tor all " E N , and let P btl lho: dlltribul"'" funt:tion 01 1'. n.en eX .),,2:' Q)'''''p:e to I' lII!o.w it aDd only if (P.(:z )) .:;,. """""rg<s to F {:z) for """'"Y pOint :z .t .'hlo:;b F is eontinuoull ( Propot!!tion 14.2.'1) . On 1M other lw>d,
16.1 Coe .........
{x~~, COIl'"' .... 10 /I. In 1&.... if ""d ODIy if
ft!-· .t;.("' ) for every r E R'
r-.
m
(j ~x"' dP~ ).1:' COilVUJ1'&
(Tbeo;or= 14.U ). N.,... _ """"i
10
.....tul.
P~itloa 16.1.2 1.0 0 and h- > 0 IUd> F and all t .... FI" at,
u...t
DJ proItability Jnmt (n.. . .1"~ )
lao: a """"""""
into R O. A, ... me (X~) .. 0!I10".."u in lei_ 10 G prokbiIitJ,... Then .... X~ ... ~o fOf" tIII.dI otqOI O. cit 0 oolhllt 8 _ !- r.r)· quadrablo! aDd ,.. (R ' - B ) S 1/2. For .. i-illtably cboltIen in~ ,.. ~ I. we
...~
.. p.(I.... X"):> ~):S p.. (j X"J:> r):S (. in $bart. P~( la.X. I :> 6) to 0 ... n - +00. ~~ . .... X • ... to (Proposition 16.!.1).
'"' _ < mf
'-'
OS' S·
(I),
'"t""
fof " 2: O. 1lepIaca" to,. ,, - I In (I), ooIve lOr tbe in1egnl! "" subotltute I.bil br the iolf:g7"&l iF> (2); this si-
0"' .
'"' l iz)' '-' kl "S' 5 _
+
j" /'· (r -.)--' (o" -l) d. (" - I)! 0
n,hl , aod
(3)
i:or .. 2: t . f:M.iln..li"l tIM: inl~ in (2) aod (3) {ODrTIider ~"'Iy tIM: _ r 2: Oaod r < 0) ..,..1ads to
Ie" _
I-
'"' Ciz)' < ...
'-'
OS ' 5 ~
kl
(1z1 . . . ' 2!!!:) . (,,+ 1)1' n! o
li-"
la """ Ia8I. iDoquali!y. tbe full! term 01> tbe "11Et sharp Nlimat.e lOr !r l -...ll. the ... ... 1 an ..... ima1e br Irl large. No- SUI'P'* that , lor _h II E N , X. ,' .. .. , X~ ... ..... random >VII.bII!I from .. probability apKe «(1", T", p. ) into R ' ; «(1~, T. , P~ ) mo.y chul" ..tth II. Such • coIlealoa iI called. IrJan&ular of .. ,·h" ¥ariablioo. 0, ..! .• S ~· + fix • •1>< X! .• liP. , e.t>d 10 I~ foIlow1 by the LI»doebe<x 0 that is,
p.(I!"n - II > e) _wroed>eo 0 .. n -
+00,
""* ... """wionl of {I, ... ,n} Iut."" about kIg n cycles..
Nert, ..... .- the _ &lion 01 Cha~ ili, E"",," 2, hul "'" "';1" p" in$.ead of P, and X ... (for 15 t :S nJ iustaIIoklil~..,...,., mnd I.., Y !Ie • ru .... .tom....,.;.,Wo! oj Md(!t" 2 jrrmI (O,T) in'" R " (d E N ) ndt !hat E( Y) _ O. Loti 8 .... d x p maIrir, ,.;tI! ... not p • .noch tMl 8· B' _ D(Y ), and "'I (i , •... , i ,) Q/ (I , ... ,eoce ...... thai. Y _ 8X almo:ooM. ... rely. Ckwly. X ill . rerwlom YariabIo of Older 2, and
a1"'(111t Shrely,
D(fl'j " .. . • Y.. J)" M · D{X )· M ' M · M' _ M ·D{X )· A/'
'I'
ate
336
16. The Omtrollim.i. TI"",,<m
O(X ) = I •.
a N..... let (O, ~, P ) M & probo.bili.y $p8l:1e and (YR)...~ ' an independent liequeM! of identically distrlb,ncd random variabLo:o (0, F) into R", Sup. »OIOe l hat IYd' dP < +00 and put 0 ... O(Y, ), Wri~ k>r Y, + ... + y~_
from
r
s..
Tho".., m 16.3. 1 (C e ntral Llnu, Theoren. )
7,; (S~ - nE(Y, ») ... "'..
(0, D J
PROOP' We may IIIlpp ea0 ""'trill of D''P(O). Th....
t(:r:) '" 1:r:1-'(.,,(:r) - I
+
e' _
(1m'" D:r)
:r approoocbN 0 in R " - (0). We put f rO) '"' O. N_ tho, cha:r~'M: fo ..... ion "'" of (I/ .;;t}(8. - "E(Y, » io :r _ (0""" \lo. (:r:) -
("log 'P{:r:/ ./ii) )
.~ ""p{ - ( 1/ 2)r' Dx) , ami (W.).a' am=gc9 plin''''"''' to t be clw". acterlttll: funetion of N.. (O, D). If..., ~oror.:...!.he bypot~ of the oentflli limi. t'-rt:m, 'M oor,dusiona are It~. Th _ thioJ,..., followi .., ,...u1t .
'*' ,""
••
IU The CeDI,.! Umit
nx a..
337
Pro.,.,.ltlon 16 .:U Ld (n.F, p ) lie" ~ 'P'
fa) D iI in....-tiblf: ,,1Id./or 0'!Gd 0 < .. < infj. j., (fDC), ~ aUt. p > 0 nell Uool lo,o(l)1 :s ap« - 1/ 2)0111' 1 lor aliI e R~ ..m./,mg III < p;
(' ) lor mc.II r > 0, ""1'1.." . 10,0(1)1 < ). P ROOF: By I~ Fourier invemoo rormula, X hu., COIIlin...... denaily. lei. "J be tbe bilinear form on a " " a " .·haoo matrix ill D. AMumo thai. (\d (D) ,. O. Tben "J ill d",UNUIe, aDd Ibtte ""'''''- I Id R" _ {O) IUCb \hal ,(1.1) .. rDC .. D(tX ) _ O. II"""" IX ,. 0 allllOlt ...my, r.nd X taks Ita ...tUN in lbe h~ ortJ>osooalto I. al""""1 ,,,,",,ly, ld COdU1d'CUon with the that X hu., CO 0 l uch that loJ;{t)1 :s 1 ror III:S p, .. Meb PlO_ I. Nat, IU""""," that j(IOIt ",,",Iy. Since Ihie loon oo:!; 10 dz-nrglisibloe r.nd X hao ., """,in....... density. _....,.;"" '" ., ,uuradiction. In obon., )0,0(1)1 < 1 roo- t.II I e a " - (OJ Since V' , ...... tw. '" infinity. _
&$
n tends 10
+00, by lbe ceDlr.! 6mit lbin "tn, FUrthermore,
100 {~•. I ... ).~ , &t .• t & P""" time, then! ill. probability ~ 1- th&t all e&l1er1_ Nt.tisfied? Let I _ {I .... , n I and n • 10, II'. TIle ootoDll>m 01 tbe ... ot experiment.,.., tbe rt!I:Ordiop of c.aIIs.t tbe exchange, that ill, tbe ('H ),S . S_ E n I if tbe hit caIItr For all I -;;: J: -;;: n. let be tbe !Aut»bilit,r on ....::h that + p, and define P ... (00 tbe tbe MAbie
{:r. _
"
np + It Jni>( I - pl.
£ureUu lor Chapter 16 I
lAo. N be'M _
of otri&t P(X • • I ) . II ,. ro.- -" ~ ill N . lot ;{o' bo "'" d"ttibutioa 1\.0...,..,., 01. xto ) ~ II , ). that. foo- - , . " R. I .(m) "I) . . . _ .... &t
(X... ) ' 5°1 ".'__
bo . t ........"" an-oy 0I....u ....... ~ .... c knt foo- ~ .. E N . Su_ tbal tI" IX ... I~·· "'" ~ foo- _ 6 :> 0, E(X• .• ) -0. &nd L_...,..·. oondi.ion
~'!'-
L
'~' ~"
p E(( X •.• J'·· )- 0
.
F
3
Let" be • ~, "" R. .. ,.. _ P( I..)). Let , bo .be fwzcI.iozz
It
b Cbo.pter 16
:WI
n E Z· .
• Bez.......u; _ wit.h pmbobili'1' (p EjO . l l) b ' W' p' PCy. _ t ) .. , . - ',. ....... q - I - , . Show ......... ho X .... (I S k S r. ) ...-eiDdt... m · · and , hal X • .• ;. d~ri""'1«1 .. Y,. _ •• 'I, • . 2. Put So .. X .. , -+- •.• + X. .... .... .. E(So J. ODd .: _ V..-cs.). SIzI..bilicy I' "" R, with _ ..:\.Orio4;' fundion " . _ "",.4 " 0 , _ that
2.
Ld (..... ). .. , be._ oi' c:b&r~fwIct""_tbat ( ..... (I) . l l
'" I'" ali I;" _ - (p'--bwo.blUly"" R wbL) ooant,
T Lot (0 , 1', P) be a poot..Nb,y
""*'"' add (X. )..... ' ... ; _...odo", _ _ 01 s.. ..
nodorn _ _ _ (n,Fj iDd X .
p_.I>o<w_
17.' We iMrod....,. cht _ion ol
Li
'ia;n
ala """plt lind
p... _ _ 00 R. Si""" 0(0) _ 0 .-'(9) .. (I: _
ro. e-.-.ry 6 E (0 , I). we _ "')
"\
1)7;.. _ 1:)1"
·'(1 - 9)&- -
that .,
.. (I: - \ ).(n - .t)!
[ .- " , 0 "
-"
)- - - /Iu
.
v
,
••
"'
1.1'(') ..l _'{1 -
~. (I) - fA- _ I)'(n _ t I l 0
u)-- oIf...
Nttt, -..-, th.o.t I' ho.s • deu6ity ! with roopect 10 4z. Then F is ina ' o;n:p: and at..Iul~1y cootiauous. 0" the other hAnd , • is abeoIutely oontinuoS. It fo!Joooo-a 111M . has derlval;""
I)~n _ t )l I"{I)· - ' (I -
.... (tl - (t _
F (I)t - · 1(1),
and lbe proof is romplece.
0
PTopooillo" 17.1.' Lt.1 B _ (("' ,. ... , r. ) e R" : TMn Px .. nl · 1" , 0 1').
>, he.., in of ( X. ).~ ,. 1'Iwo M . ;.
alll>06l.lUrei y ~ .
Definition 11.2.3 E--,. reo.! number t ouclt that F (I) _ I/ Z is calI«I a modLan of " . Pl"OpOOIltinn 11.2. 1 S~ tItt'n: 01 " '" ""..... M E R ,1IdI1IW F (M ) .. l i Z. 'Then M • ...... tlCfJ'f6 aI_lI...m, 10 M "" n _ +ce . P JU)()P':
Denolu: br D the tot of t'- w E f1 !Iudt tloa.t X ,(w)
~
X Awl for all
d;'t.!nct;, j E N_ Recalilhat D haa. P.negl~ble oomplement . Now lee • ....:I I he t_ ,uiona! numhe .. ouclt that. < &II .. € A!. wt.o.o A. io P -ftOdi(iblo. Sionil..-l.y. ( F.(~ _, ,,,»).~, a:>r!""'" to F(:.- ) ""
RIo - r ~ € R, . - tloa< (F.!" ,,,,»).~ ,
B:. ",t.o.o B. 'it p-.,.tisiblt. S. For __,. ~ EIll, II. P'" 0). Z, .. XI' )'
flf, ..... Z.I 11M. de"';I)". and aompu... it.
2_ Show that Z,._ . .. Z• .... lode ... DdedL"'" that
Po • .. ,,(.. - k + l) u pj- ..(.. - k + I )f )' 1" .....I(f ) "
rc.
r«ry
1 S k S n. I .. othor
~,
Pf, ill tbo exVO",,,'ioI 10• • ith
_ _ ,,(n _ k + I).
3 C....., I > (), lot (n,F.p) be. prot..hilit)" I ,,"'" lot X " ... . X. (with ..
~
2) bo _
... ' ,., '" ... . , _ f r o m (n.T) in", R w.... ..-""'"
law 10" a (1/ 1) · 1" ~I dz. Writ. X ( I ), . • .• X , . ) 10< tho order _ ......" oltbo oo.mplo ( X , •.. . , X .). Put t . .. XII), 1.. .. XI %., ...• t •• , > z •• ,) .. ,~ I(' - ." - ... -
%. . .
).1'_
18 Condit ional Probability
~ .. de.""",, ... __ '" 0>.
...-..n.w. y
ex,••
-. 15.3
"''''pM' '"
7
" " "
Kent,.., s\ YJID) (P"-',..... 18.3.1).
...
18..1. w. oJeA ....... _itloaal _1«1 ~ ol Y IP""" , be ,..n"' .~ iabIo X. Whoa t ho "'"_ 01 X 10 &booiuteiy OODtlm_ with ","pee' to I.eboos""",...
....... _".., -"p"'" ElYIX) (.... l ion lU
~ .,.;, )
_ _ ..,....
p" .~\si bio
...
(~
I ).
1M nio ~ion io
~ to
,100 ""'" oJ. 1M COt>dit.oo..alla.. of Y ( i - X
1&.8 We _ ' " _ ~i'''''''' Ia.......... a P" io ti"-'O " 'tb _ _ to • pnod ..:t " 1> ~. 18.7 W~ _ . ho nioltflOt of condit ............ when Y io ... R' nI....:I ~ tn_ .." 18.7_1) ,
,.
....... r'
!om
18.1
Conditional Expectation
or
LeI. (11, F. P I be • probability SI*'", :D a ""1H:>4ebra F , and Ptv lbe pooo..bility E _ [ll$dPon l). LeI. Z : (I _ R {t 2: I} be ~ D. Pz Is both tbe imace ~ of P under Z and the imace mea/iUre of Ptv W>docr Z . SiDoe tile idelltity 01 R - is Pz- _u~ , \(I ....y it ill Pz·l~ _ u.u Z is P-!n\q:rahlc, or tluot Z is PfV!m~ In ohon., Z io PfVlnt~ If and only if it io P- inWcrable, and lbet> f Z d(Ptvl" f Z dP.
T t->rem 18 . 1. 1 l.d Y lie .. p.~ ..n.dorn ~from (11, F) ;1tt R· (1 2: I). ~ erUu .. P-~ ",ndom ... riGbIe Z from (n , V) ;111 R l ,..m Ih4I fA Z dP ... fA Y dP fM..I1 V · ..U A . Any","""'" ... ' ·ow., Z' fr- (11. V) into Rl h
N_.
f
he""" Z' .. Z p,.,...oJmost surely, o.od Z' .. Z p . &imooIt .... "'Iy.
0
Deflnitloll 18.1.1 III tile lIOtation of Theo<em 18.1. 1. the eta. of Z (for the odatioo of almost "'"' ""IuoJi1y bet ....... ..-urabIe D functloole) .. ealled tile R' (lJ nItr~. ",,~Jtitoou a ch "'"
,*:o...
E{YIV). PROO" IU H \0 t.ome!.rit. t 1.'(0 , P, P; R ·l. il Is eomplete. &lid hence clooiotod In L'(n , P: R " ). Tha prO)eaion Z ofY on H iod1atscuri~ by lbe poopttly tt..~ f{Y - z .~) elP .. 0 lot aU D(I/; . Z)dP
AM..
'D~ II , .. hich J>I'O""S that
. 0 IY . E(ZW))
~
£ (. 0 IY . Z]I'D)
"_~':" CIIIie ....... oboe"", that . since IYI .. the upper ocqoon", ( X~). ;?:' of poIlitive '/).simple fUIlCliono, upper e",...1ope ofthe X. '£( IZ II'D) (if .."".boo8e £ (lZ II'D)
f f
X. ' £ (I ZII'D) dP - / X. (Z l dP. X.' E( IZIIV ) dP., / (yIIZ l dP,
and the moOOlO
l. o fY.,£{ZIV lIl
S S
U. ' · (y.I·IE{Z(V ») 2n• • ·IY I· £ (lZ II'D)
a1mo&!. Itm!ly. Because IY I · £ (IZII'D) is P_in~ble. t .... dominated conver-
......... t!.eooCIQ pl'tM:S that . o IY. E(Z('D)J ;. P-inlP/"",.1 Pl"OpOlll t~
-.
P ItOO" First.. M1ppo111! that Y , Y ' .,.., random varia""'" from (fl, F) into
(0 , +ooJ such that Y :5 Y'. Si""" Y~ ... inf(Y. n ) :5 Y~ ... Inf(Y' . n ) for e\U) h"d.fIM n
"""'~ !.o E «wp~,
Y. 11V)·
By bypott-;s, Y.-
PIWOf":
Z_
r
,uP~~ 1
ill integrable for n E(Y. \V ) and Y _ .up Y. , .ben
t..:K'J t ROUgb . If
f' Y. dP _ f' YdP ~~, f.. f ..
" ZdP. w p /'" E(Y. \V ) dP_sup /...
1M all
" i!:' ..
v-u A. HffiOO Z _ E (Y \V I allllO!lt surel)·.
o
l8.2 The Converse of t he Mean-Value The()rem The Iollowing addillontJ ftOjulu 011 measuro!l will be ..-I to prove k"""Il" inequ&!ily for con<JihontJ expecled vaI~ (Section 18.3). SuppOtOne unions and int......,."ions:
(a l Fllr any Increasing
~oonoe (A. ).~,
in C,
U.~ I A . lies
in C.
(bJ F,... any de.:.-Ing ""'ll>enoe (A." J.~1 in C, n..~ , A. I;"" in C.
In p&rticnLu • • ,,-ring ill .. 1OODnlQin,,'f
P III)()F: LH m boe It.. minim&! II>(IIlOlOt>e eM """. R , the inlt.-o\ion of &II ~ cia ., CO D jo.-"!I A E "R.n<eIt tMl ,.(10 ) > 0, V- I '(F - DJ
"·jnq,,We.....,.,...,
a
"1«aI1~ ,..~.
PI\OOI' : Fi:z E E S . The eta.. 01 \I>oee A E belnr>p 10 ,.(A)D 10 • _ eta...,...,.
iI. (1/11(10»
5 suc:h that A c
f ... I dp
'R.,,,.So it is "'lu.al to 5,,,, that
f... I dp I;.. iu D br ew:h A ~ 5 . ueh thlt.t A c
n.ereen.~. ".~bIo
E and
E and ,.(A) > O.
...'-t N E Sol E.ueh thlt.t I , B- "
iI measurable
5,"_N and I (E - N ) is oepo.rable (Thc!orem 6.1.2). He""" 1- ' (8 ) n (E - N ) liIoI in 5 f","every BoreI_ B C F . Suppoooe that I (E - N ) n [J< iI """""'pty. and Itt 1»-.: m :!: I} be ..... n.. in I (E-N )n D" . Defintr .. , ball m :!: I, M tbe ~ of t,,","" real numben r > 0 such thai. B{"", r ) doo!o not mee\ D; put Z.. .. (E - N ) n r'(B(~. r ..
».
n.... I (E -
N) n C1' is included in U"i!: J 8 (11)-.... ) . ,,~, Z ....... tal .. (E - N )n /- '( D'l Fu. , we shaw that ,.(Z.. ) .. 0 for..u m:!: I; It wi!! tbon be obI-..... Ihlt.t E n r ' {IJ< ) iI ne&lisible. AIoume that ,,(z.. ) ,. 0 br """ m :!: l. n...n ~ .. ( I/ ,,(Z.. » f._ I d" Iitor in D. But
bea u? : II - ""I < r .. on
z... nus ..-e ha,-e • c:ontrao;lic:tiort,
o
300
18. Condi ....... Proboobilil)'
18.3
Jensen's Inequality
P roposit ion 18.3. 1
u , H ..... .w...l """","" . 01 in R ' , op .. I~ um;"""'·
hA"""', """_ ~tiOfI
"" H, and Y .. P·int"": By Lerrun.o. 3.7. 1. ~ ui$U an affioe fllno:lloc 9 ; R· _ R ,todt .IIM .pfH ::; .... H....,. 0
E(Y'III' ) ::; £ ('" 0
r iP')
1"..11I'I(IIIIl .. ~Iy . .o,., o £ (YIII) ::; E{IO" YI1» &lmOIIIt em-ely In A•. We dOl(lo,
t"" ",., E(Y ID)::; E(", o YIP) almost ourely.
CQn.
0
IIll .!<noea ', lneq..ality
361
Propoxltlol> 18.3.2 A_me /JI4l '" io Itric:tl~ ""'va .... H. Then '" .. E(Y IV) .. E(", .. YID ) ..s..-I ....a, i/ aowl on/)' i/ Y io, ..m..m ...... /" /IfIUIl1D a ",ndom wuwble from (n. V ) ;,w, R. PROOF: Let Z be. voen:ioo 01 £ (YII' ) witb vaI_ in H, and _ume tbat ", 0 E (", o Y IV ) &1_ ~Iy. Flm, ... WUOt ,bioI ", .. Y ;" intqrab!e, and a mal number a E)O , II. Tben "' .. (oY + (t - o )Z) '!i n . ", .. Y + (1- 0 ) . ", .. Z , ... hieb PlOItS lhal "' .. {o Y + (I - a)Z ) ;" P·intqrable. Since
z ..
r.x
"' .. Z .. "' .. £(oY
:5
+ (I -
o )ZI1')
E(.., .. loY + (I - o )ZIIV) :5 E(1a .", .. Y+ (I - 0 )"' '' ZItI»
almOIrt
.. "' .. Z
.u...ly. H"""" f "' .. laY + ( I - a )Z)dP .. f la . "' .. Y + ( I - n )· "' .. Z)dP, .., .. (oY + ( I - o )ZI .. 0 '",,, Y
+ (I -
0 )' "' .. Z
01_ ouretr. ". "' ;" A riclly con"",. _ conclude Ih.at Y .. Z alD>08t ... ",Iy. NOOI' to pi""'" lho ~
,,,w.: .1,.,....
R.
PROOF: Sinc:e "' ;" """'""", lho li",l~ G; the ..
It
Wben G
op(.) - r,o( .. ) < (.) - (1)
.-a -
,- I
,.
.
J61
IS. Condit"""" ProI:>&bi61y
and Jinally IIw 'P("~) <
rontI" ....... function from 7 into It eqw.l to "" on la,&(,
and doIine G by;
G ..
,.
{(:z,I)e1x R : I 0 YJ taking il$ ",Jue; In C . 15(YID). E{"o y fPlI ......., .tmoot 6"",ly. 115 ¥lOl,,,,,, in G. T hU9 ~o
£ (Y('D)
~
S(IPO YIP) al_ OIUrdy.
Whca a lies in I and 1'(11) > 0, whcrfl A ,. £( YIV)-' (a). lhen Y ,., a AI· ..... oureIy in A. If [" ill tbe proMhj!;,y E _ P( £)/ P(A ) on {£ e T : £ C AI, if 1J' .. (£ E P: E C A ). and i f Y ' is It-e ......ictlon 01 Y Ii> A. t!>cn E{Y ID1' A .. E(Y'IV' ) ~ a [" •• Imoat 9dy. Th,.. 1"" £(Y IP ) ~ £ (1" 0 Y IP ) a1_ oureIy in A. U loe...., 1"" E{YJP) ~ E(lI'o Y IV ) a1moM ",,,,Iy in B .. £ ( YJPl-'(b), ",hen h!~ in J. In short , "" o E (YlDl ~ E(.p o Y ID) al.- """,Iy. Now. up", Ali In P"),,,..iUQ" 18.3. 1. ...., __ I hal \I .. lUI inequalily ~ e\'OeD if 'P OY ill nolln~. f"inally • ...., note that ,·he last '''. llion loUo.... frtlm tbe _
&rIumetlt OIl ,hal in Pr-opoosilioll 18.3.2.
[]
18.4 Conditional Expected Value Given a Random Variable Defl nltLo .. 18 .•. 1 Ltl (0 ,.11 be .. measurable "I*'" &Dd let X, Z be t.-o random"4riableOl from (0,.11 inw measurable "1*ft (F. 8) &Dd (G,C) , ,..,. lpeeti."ly. Z it; sald to be .. function of X if li>ere exilllll" rancIom...uble 10. from (F, 8 ) into (G,C) sud! lhal Z .. 10. 0 X .
n...o.... m
18 .• . 1 1I-1I.en G .. R ' (fUPOCli""ly, .~ G .. R ) and C ;, 1M Bam ...aigdnI oJ R ' (fUPOCti"'I~, 01 R), tMtt Z ;, .. "'nc~ oJ X if and on4t i/ 1M (Z ) (covuti.., oJ 1M r '(C ) Jor C E C) ;, ind...u.t in ..-eX ).
,,··'st.,....
e .. R. Awume lhal ..( Z) C ..( X ).
P IIOOF: We will ..... s·,'" only tM cue FIrst. IUP1"*' IhaI. Z ill sim~: Z ..
L ..,O; -1 .." "'Mre 1 io finite &Dd the A; are diajoint .. (X~eeu. For every ; e I . tMre " istl B; E B oucb lhat A; .. X - ' (8, ). Tben 10. .. L"" 0 ; . lB. ill .. random ,'Viable fl"ODl (F. 8 ) into (e ,C) , &Dd 10. 0 X _ Z . Nw , for tM ",,,,...! CUO , ot.c ...... thal. I""'" "".. loa .. - ....1ICf! { Z. ). ~ , of limple raooom"4riableOl from (O... (X») inw R which ""'!>u p point ... ,"", to Z. Since Z. ill oimJ>le, ..., can find . for every n ~ I, .. random variable 10.. from ( F, B) into (G. C) IItLd:L that h" 0 X .. Z~. Let L be tbe oet of .....,, ","',"" or the lC<juenoe (Io.. ) .~,. Since L .. {Um,uph" .. lim iof ..... )n (l lim inf 10..1
d 10. _ 0 III> L', ttw.! 10. 0 X .. Z , as deslred. 0 Now let (0. F, P) be .. probability 51"""'", X .. random .-..riabie fl"ODl (n. F) into ......... urabIe rpM>! (F, 8 ), atod ..{X l tbe ..·...,bn."'nerated by X . Ci....., k e N . the ltiatloa 01 Px ·" l _ ,..".., equalily between random "4riabIeOI from (F.8) inw R ' ill an "'lui......""" relation. If Y ia .. P.i~ random variable from (n , .11 into R · , ""'y Px · interp.bIe rancIom ,-.riabie 10. from (F, 8 ) Into R ' ruth that I B hdPx .. Jx - '(B) Y dP for all 8 e 8 ill called ..... oioo> 01 the tonditiooal ""peetOi!d val ... ol Y p."" X . Thus .. random .-..riabie h from (F,8 ) into R ' 10 .. ~ of the WI"", .....;..w. from (O.n into IV {j > I) ..,.... 10", Px ;, Qb>I~w, omIin ....... lriU\ ruporcI 10 U,,"tgUe ........... Aj on IV , Finllp, iel Y I>e " P .'ll.IquaIW. ",Mom ..-HI. from (n,n ;"1 R· (lr ~ I ) ..... II a ...,..;on 0/ E(YIX). 1'IIDI _ 00" fowf • Px ,".,1igi&K &m!/ "" N ""til 1M JoJ./oftng a I._ /tw all % tN, an
,OJ,
Let I boo • l r 1nt" " y. Coo""""'y, I UppIW: they ..,.., .. tis6ed. For every G E . , the fundion F' 3 z _ II, (C). F" ~,, _ O . urable B and m
J. II, (C) dPx (z) .. P(X - ' {B ) n Y-'(C») • J. "
Ie" Y dP
X -'(B)
lor IlIl BE • . Hence. ror every C E • . the function F' 3 " _ II, (G), F' ~,,_O • • ....moo of E( l c" Y IX ) ( P~tin 18.).) . From!'ropoIi,ion IS.!>.! , we dIld\lCle that it • . in fact,. version of E (lc o YIX ) for 1lI1 GEC. 0 U",il (U"her _ice, we oball III""""" that . lor ci vm X and Y . tllne ex • • modltional Ia.. of Y civm X . We loet (s.m
u... .....p)'ing % -
F ~ G inl J 1(l:.,) 0I;.. (1I) .. P" ,4/mo.t
P" 'int' 1 -
if B @C";m~
% -
dP,, (l: )
f
I (l:, ,1 d;.!.(iI)·
J I (%, /I)"".{II)
if deli""" on l' and io
P",lD~ Furt~ ,
f
Idf\x .I'1"
f
dP,, {l:)
f
f{>: . /I )dl'. M·
Now , lor tho ~".., "'" fiDd. ~bIe BGC Dl&ppi~ 9 from F" G Into II wbk::h ~ witb I f\..t .I'1,a1m thal !f~ 1 os: 2191. T~ io a Ilx .y!,neg!i«\bIe Aet A E B0C JUdo
"'"
(a) (J~(:r. r))A~ 1
"""''''gf)I to 1(>:, /1) !O< eaeb (>: .,) E A':
(b) 1/.1s: 21/ 10t-opw.ilion 18.5.3 to compute conditional ""pecl.e1 .1Il.-.
Tbeon" .. 18.5.3 Ld I : F " G - a · "" ~ 8 0 C ,owl I1x ,>'1,101 •. 11\e1I z _ / I (z , 1/) ~.(,) .. Px -"""'1 ftffd, de/i.... "" P , "owl iI .. px·a/moII .-1, 0f\'4l1 E(f 0IX , Y )I X ).
.... ,!
PR:OOf': By P""""";tion 18.5.30 z _ / l (z,I/}dp.(I/ ) ill Px-a1...-...my de6ne1 on P , is Px-inl~bIe, and.
Ie
dPx (z ) / I (z , , } dp.(,1) ..
/ dPx (z )
J
I (z , , }'
..
/1 ' lh Cdl1x ,y)
•
/,
IB.c(Z'II) ~.(')
l olX ,YJdP
X - ' (8 )
ferall B E 8 . n.... lhismapping is Px-almosl ru~ly.2, the
j"1(z ,,)dP. (II)
F"~z _ O
,.
.. ~\11~
I;
r
B. MOI_r,
dP,, (:CIIIe. wll """ _ f Sene
0 Tht;n
Dellnlt lon 18.1i.2 Let (0 . T , P ) be .. probability .~. 1) a subo .. ~of T , u>d Y a rllIJdom varilt.bIe fn)m (0 . F) Into .. ""'...... ""bIt ~~ (C,C). Writ'tn X ..
...w... ...
ut
Theorem 18.6.2 X 4JId Y be tw.> .... ......wa fr< G .... (0, +oo{ Ioo .............we B0 C ond ,, 0,,·jlll~, IIIdl IAat f\x .YI .. 1 · (1'0 " ). N_ tkjiM. '1>'" IMf!mditm z _ I{z , ,) dv(~) frm (O, F) in/.(> R J a"" R ' (j . k E N ), ",.ptct;""'~. S.."po .. 1M", auto a fam'l~ (I'. ).., ,, of "",ldiliUu "" R ' triIh Ute joiJqu,ing propertia;
(a) B .... 8 ..... , .vboel / RJ.
<m
..nidl Px
;, :>nCt"IIlrnUJ.
~) Fmpute tt.. conditionalla... of Y ~ X. &i'''''' x. rOO" ..n real nurnbtTs I , I, put G(., I) .. P(X S J , Y S I). If _ :S t, sinoe X, S X ,,,,, ha.... G(.,I) _ P(X S') _ F(.)~ . If , > have
s: :s "
I.""
G{• . I) " P(X , S ., . .. , X . S I , ... . X~
s: .) ~ F (I )r--'(. ).
Now dF" .. PlF" - 1 dF &nd dF" -' .. (PI - I )F"-' dF, .,·twe dF io the Stielt;'" """"""n: nx"'~ with F . I!~, lor "'""ry I E R, l he fuDCtion , _ G (. , I) is lUI i~nit
'1_ .rI(") dF(,,) + (" -
I ) F(I) F"-~(r) . l ~ ._ 1(") dF{" j .
Fo< 1\11 or ",..:h ,hoL F(r) > O. put PI_ I
I
1
" . .. --;;- . F C:r) . ' 1-00 .. 1dF + ;;t•. ... lth !. the" " ' r
".{I -co ,I]) "
n- I PI
F {I} 'F(,,) . ' Io._ [(z) + 1/-"" ,'1(" )
ro, &II I E R ; "" ". (I - co . III dP)( (:I;) .. nF"- ' (z ) - 11_... .'1(" ) dF(r ) + (" - I )F (t )f""- ' (r ) - II' ,,,,,,1(" ) dF(r ). We UlDClude that (1'. I",,,, ie a eOI"ldi.ionol is ..' of X. gi.... n sup(X ' , •• • . X ~) . We reman tbM , lor """" I E R , 1'. (J - 00. Ii) =
E('1_oo ~1 0
X. IX )(z )
••
18.1 [.:\0.. 1 " clCoodi'lioDalr...... botz G_ R ·
for PX·a1mool all
%
171
E R '. H....,., for eocb I E R, the function
, n - I F (!} R II %- - " - ' F(%} • I ]< ._ l(%}
+ 11_....11(%)'
ilI."""';OO of £( I I_"' ~I 0 X. IX ).
18.7
Existence of Conditional Laws when G = R '"
Theorem 18."1.1 Ld (O, F , P I .. G prtIkbitify.,..ce, X a .-....10m .,..,;"bk
fr<mt (O,F)
.,..ce (F .B ), and Y G ...ndom "" ·0w., (0 , F) into R· . Then ~ au,", m""""!lanai "'.. of Y X. into
G .......
""'w.,
,.tom
n
fr<mt
-
P ROOf : For eocb I ... (I" .... I, ) E R · . I>Ut 1- 00 . I) .. , ~.~. ) 00 , I;J. Let A .. n , ~,s .)o. .ho] be I noz>empty ~c.angLe. By induct;"n on O :s 1 :S l , we ".~
E(IIo, "'I ~ • '"' "'I ~ J-.. .." • .I • .. • J-oo .,'1] and [O,T] mcb that p, q. T ..., ,..lio".1;; in [0 , II and p 5 q. n.... C ill • oounl.&hle ".~ .... and il "" .. the Bon:l "-al&o:bra 8 of [0, 1]. For e¥trY 8 e C. lhe fuPlCtion 0, :1:.1 .... "", (8 ), D , ~ ... _ 0 ;" • ....-sIon of £ ( I s IX ). Si""" "",( 8 ) .. Is{...) for >'-almo::c. all ... e n - D" then: eri.u • .I.-ntlJi&ible ... ~ ~ e 8 of 0 - (D, u D, ) oucI: thai. "",(8 ) .. I s("') for all ... III 0 - (0, u D, U ~) and all 8 E C. Now, b e-vny:.l e 0 - (D, U 0. u D:.). let ~.. be lhe - EFSun! on T &.fined by the unit ~ II :.I; "'" and t .. '«ret on C. and eo on 8. In particular, "", (1:.1)) _ I. But, Ii""" "",(E) .. I. we a>nclude lhal ... I... in 8 . III short, n - ( 0 , u ~ u 0.) c E . ..hich ... Wi&bIoI!..n.biot fn>m (O,T) into R ... _ _ _ _ law io ( II I )' I\O .,, ~ (10< .. Ii- I ;>G). Fot..-,G ",.r> - ~,
> z, .. . - . 1/0 -
r.,-,
> z o.I - ,. > Z... ,}.
Fl...tly, put ' .(1) .. Flx", ..__ .>:,.,I (A.{I» .. (,.,/ IO) .I.,,(A.. (I» """ '. _ ,(,) .. ((II -Ijl/,"- ').I.,,_,(A.._,{,» . Frnm PtopaitiDlo 18.5..2""" pUt 1 dtd ..... that
By iDdUdiDD, ronclud< tbat '0 (1) .. ( 1/ 1" )-(1 - '" - ... - h . t)- .
S.
From
put
2, dtd_ ......
p(X(I) > z , . X II) - X (I) > "t, ... , XI_) _ X lo _ J ) > %., 1 _ X ," ) > " •• ,)
" ~(I-Z'- "' - Z"' ,J~ tor 011 '"
~
• _ fu. tad
O•...• " • • , ;2: O.
n,,_ I> . ud! .hAt 0 < 2h. < I, aDd...
(II. < X, aDd
r
IZrtlP
< +00. P...... , ....
£(IYZllXj :S. [£(iYl'IX)],,' . [E(lzrIXj
j"'.
P... · , _ .unIy (~. &llo.... of IY,Z(IP""" X ). II Lft (fl, F . P ) be • 1'" 01 . '"U\J" . , ...... A ............ ( A_). ~ , 01 F ...,.. 10 O&id to be m,., 6with_..... ~0;rli m. _ 1'(A. ., £) .... P(E)boJ1EET. I.
2.
3.
'Iro&' ',..... __ J' . X tiP .. r.- (n,n im K
If (A~).,~, io "'I~I ....."-"......uno. !I . _ _ X tiP lor .wry P-....... abIoo r&Z>dom ..,iabIr. X (I.., ........., X io pooi\i".,).
.. f
,rw
T - " ,bat fI _ ,he .... Jt" bdoo. to ,,(C). So_ _ ._I'(A• ., E) .. ..1'(6) "" oJl E E C. S'- ,hat Iiooo,._ . .. 1'(Jt" n £) .. .. I'(£} ..,.. olf £ E ,,(CI (_ 'boo .._l thoooo ...1 _ ,r... f.~. X liP. J~. .!,(X \o'{c)IIIP '-" ' ''' to" f X tiP .... _ +00. lor.....,. P .illlopabio " ~ . , • . - _ (fI.1') in! R. Coatl_ , bat (Jt,,).,~, ......... i>ti ....
Loe
Iut.ly """'""""'" ..ith _ _ to P. • >q 1 1\.
,0.1..,.
s- u...t ..w .. 10 ... : r" .'0: ' ' rot _ ~ d ....1caIly diotr\lioll.Od A ..... nzlab'oo f ..... (fl. 1') 1_ R. Su_ d.... X , 10 01. 2 &:tmo. 6),
Lft 1\ be . . . . .bi!jWy (oIIOI.i.II."'. wi.h rto', r , to P . b.u '" E R , 1\Iz. :S. %) - ... ( _ Ex d .. 6).
sa.- ,ha•.
S. p _ ,bat {Yo - Z.),.~, "Thootom 10.3.2) .
"""''''id;~ ""prOboJ:iili~
1. Cooodode l~ i Polr.... N,(O. I).
10 (I ( _,...... 1 and
Part IV
Operations on Radon Measures
,.
Copyrighted material
19 /.L-Adequate Family of Measures
r.- _
oa. ooor l r f t _ '" _ ..... . beory ~. ~'2 :1) tbat of N. _bald c.... '/v ' " ..... · p' zu ~, 7, aDd 8). [a t~io cbaj>ter, .... ;",1'0:l • 0 atll." w' T1ooo. f~ J,410 1.. " 71'ft iadaood. t.,.,. 011 Y .
"
1~_2
,....... X _
~
,omi'''' of """'.,... ...... i ... rodooced Ia Ihlo oottioo>, .iU be 7 22 .....
laC« I. 1M ,..,.
""" " and .. ,.~ ndl
> 0, 11>..,., crirl> £ E tJ nu:A
(e) FM" ftoU)' com,...:! ,...1 K in A, ~ uiUI" ""'711""« (H.).2,' "f 1Mjoint tJ'loell _I· ..."" ... I< ..,dt. th.o.t I< - u..:;:1 H~ II I'·nes'i;ihlc,.
(4) A ......1 B I. E tJ.
"f A
" ~I, "'"lS'igi"'" ..h~ /-I"{8
n L ) .. 0 flX"..u
P IIOOI" First, "".111" . t hat (. ) boIds. and let I< be .. compACt ... becc 01 A. P ut (l .. llUPLol>.I.ded in I< web that Q for n ~~. T1>t1 ~fuoe, B io II· oocli«lblo:, Q .. 11(1M i .. A. wt D' joe ~ dd .. "f """'pad ~II ... .4 Ad! IMt -.-. '/ ""' m.c:A~"" in K lomo • 1'." ...... ""'"' in K . I/o.... 1Y .. ".",,_ in A .
P ROOF: l.el L boo .. oxnfW1. IUh!le1 f A. For ~ € > O. lhere 6 .' 1 K €.1) ..,mIMI I< C L .:>d ,.{L - 1borhood V 0I :r which " . p'1 on1r .. wpn'"bIe nl1lllber 0I1~ C-.." . If C is locally OI)Unt.!)le. """'Y compooCt ""bIet of (} imm .:tI only .. ooum .. b~ number of tbe C·oetI. CINlly. I~ unlon of a local!)' COUIllabie do... of 1'"......un.bIe ..,u .. 1'"" "IIrabit.
Theorem li .2. 1 S _ th4I P .. $
ole.
0
Propoooltion 111.2. 3 Let J It( Q " " ' " . " " , /rWI It inUI • "'pOlo j ' xI 'PO« F • • 004 !eIV """"" ./~ _p,'d " ,/t.M;tI K 01 It -.'I1A4J f , K .......s..-.
Tk joI~
raJ V
~.,..,
""';...!cnL'
.. wu- in It.
(t) Eller)" ............... oj I to
A
.... toNe A .. ,. . .............I!
""')!ping g
Jrvm X
into F IhGt ;, ........nt
(e) Thm: ;, Q>IUt",,/
I>
p · ... ee ."
""tMc A.
'* n\Gp!NAf
9 : X _ F thol aleruU I OM ;,
"*""'>t
P JIOO, : IbM CUldiUoa {al ~, and let 9 00 an ('k~n 0( I \(I X ,
U8
1"'=""
,..,I'J('
Ci""" thf: condil~ d. P'ptTe ... ie ~ kt · w".... F . 71It:Jo tII(TI'l X _F. Mnitot:rMll, moo •• Wt, #1IdI t1IU mill> ,,-4Imat ~w,~.
Olhtnv;"e. Then g halo ,he dellrtd propeny.
o
19.3 Sums of Radon Measures Let X be a locally oompact H............ fI .poooe. For aU pooill ... Radon m pz "","",
X And •.1.11: X - (0 ,+00), I"1!eaIllw /" 1diJ.'" .up,,/" 1 · I"dl' • ... ben! K ""'" \.h~ U~ ciMo 01 axnl*=' .w-u 01 X . Loet 1'1 , 1" be two P'l'iti.., R odQn lDI\aIIu,"", 01> X , And put I' .. 1'1 + 1',. Then/" 1tJ;. ,. /" /df'1 + /" /d1'2 &nd /" Idl' '' /" ldiJ.l + /" /diJ." So< /:
I'
on
X _ 10. +001. A mappi", I from X 1n\0 a ~ tpoIC"e io I'"measurabk if and only IfI~ II hoI.b 1'1" ill m urable and 1':t""!MMurabIt. Now I.t 1'1. 1'2 be two
~
.., N
..,N ..
Finally, "ze••~oo.;ng aU f"l'SlriclIobo 00
I'. (J)
We Qbee . ... that dee I ).
-•
--
,,"VI It tIOI.
_
-e.", v...(f»~ A\m",,.hle for all / E H - (X ). The mould", Radon " Uf' l ~ / ... ~( Il (f E 1'({ X , ell ill tBIled tbe AUlD of lhe p..". and \0 _,;IWI ~ .. p..".
!:...s ..
. ..
SUPl_ that (P .. }... A Io! ... rummable r....lly 01 j)Oiltl~ Radon '0(, Wf$I and ~ TMn " ol/) .. L..t: .. p.; (f) for &II x(0.+0:01; ... II frotll X; nlO'" IOjXilogiOtoap',. and m".
"wi)' disjoint v-u ou.ch lhat. N .. X illlouJly wnqlitihlc (Tboooe,u 19.2.1). Do:fino ... II F ' ure PG (XI X by I'o.l/l .. f J . I .... "" for I E HeX , e l. The .... ppon. 01 "" IJ i!lJCh.lded in K... and the",lOre ill, v-t.. II n:maIDI to be ""-n tM' L'"'! .. "" f f) .. j.l/) kor tad> , E 1f+fX l· Wrl'e S for the 1Uppr1 01 J and ,oJ' b- ,he (""'U1IAb1eo) ..,beet of A """'itAI,.. or " ' - Q E A JUdt IhM K" illltl_U S. Since !V n s ill ,,·Mg!!gIbioe.
I'fn .. / , , 19
0/ ~ru on T . Thm " "oaotorl~ uunl""lw 1J-,nCbk if and onl, if il io-lOIIl.orl) ~""" p., . ,~ /"" IllI '" E A "nd if u.. lam. IIJ (J .l.,dp., (I))~~ A of m ............. "'''' .....IIk. / .. u. .. .-., J .l.,dp (l ) ~AJ .I.,
lor
C'\'Cry
peelU""
meNu", "
~11on
ofit'D makeo it \>'Mible 10 d>ecl< tbM .. ,h_
mappilql io jl-adequate.
.\, k a """,,ri, _tWig ....."....., '""" T i...., M +{X ), .. Ad plOt .... f .\, IIp(l). Pro~;doll 1 ~. 4.3
C.d A ; 1 _
r..) fl A u ~I, amlin"""', ~n I _ ettety "' .. ~, """""'1;"..,.., ~Iion
J' (6) Ij A
I (z)da«z)
~
~; (n
u ~ ",mi(onIin......., /...-
1 ;:: 0
"""
r r
u ~ p.~,
IIp!l)
".'n~
X.
and
I (z ) d.\, (z)
(2).
it it " .• deqoo.ok.
P IIOOF: La I ; X _ 10. +001 N ~ ~DtinU(\\I&. &nd let P be tbe _ro:kIi....,.od oct of fun(tiono g E 11" +( X ) ,och thu II S I . r .... /1 E F • .!e.Jcn.te by II. tba function I .... .\,(g) on T. Similarly. put 11 / (1) - A; (I) = ""PoE.' 11.(1). W~ inln>o!. lbe IoIklwl"t; h,rpOtt-io., " ' 4< thaD lhal iD (a); GOlly thu tbe ....criaion of A 1.0 S _ ~ly oontinuout, 5 bel..., & ~k7-;1 Ahocc 01 T a>nUJnm, the ""wort of p . ror g E F, define by Ii, lbe function ... bid> Ifl'- .ilh II. 01> Sand ,"""" tbe val"" +00 00 5o
(-)r
I (%) J.,(:I)
r
dp (t ) J' I (:r}d.l.,(:r);
(.) ",.. " ........ /r K, Ie mnt.in......... t:leoo\e by
5 lbe _ of Ibtn I~ ulzl! • coml*'t lei K such \""1 I ONIis1>t& 00 J(C aM
r
such lbal tbe Iftilrktloo 01/1 K iI finite and """'tin....,... I.el G be ...... inl"C'.t>ie.open lei ronu.lnl"fl K , ~ .. mp" ' ''' which domin.alell I . h. lhoe k>woe.- .elIUolo.t iDUOIIII funCtion (T heon:m 111.4.1 ). and the ... lore .. hen I iI rea)·vaIu.d. Let H(X . R ) 8 F be the ... ~ of CJ,fI') a>DAiati"l of lhe Ii....... c0mbinations, witll cotlf">cienl.l In F, of fuT>Clions in X IX , R ). By li.-rity, 1M _~t io valid when I beIonp to X IX . R )0 F. N_. for e>"U)' I E C~(v). lhere exists " """""""" (J~ )~ ~, in H(X. R ) 8 F with the followin.s propert~, (e) (J~ )~U",," in CJ,(>., ). Let AI be the _ or t"'-e z E X l uch lhat (J. (r)). ~ , "'"'" con"""", to I (r ). Aa AI is .... neslijpble, the let N, . of I"",," l E T for ... hkh !of io "'" >"-'''-'&i~bIe, ill locaIl,. I'""neslijpble (""'poeti....!,., 1'""""lIli&ible), by P ropOoition II .U . S~ that t dooM not helon.s 10 N, U N • . Then the ",",,'IHQ' (J. )A2: ' con.u",," in eJ, (>.,) and It con.tl",," >.,·almo:wt evffyWhen: 10 I. Hence I beIoop to eJ,( >., ) and f Id)., - 11..... _ _ f lAd>., ( Pf'IlI'06ition 3.111). So the _ H Ii!I oonW, furn>ul& (5) 0
19.5 I.l-Adapted Pairs Let X and T be locally oompact H"uadorff If- ... I " ",."pins from T im X , and g" fUlx:tion from T low 10 , +«>1. Fo. ad> t E T . write t . ( I ) lor the Radon """,,"ure on X delined by tbe IJl.UII 1 at ,, (t). and >., for " t}t' I' )' Finally, let I' he. posit;"" Radon rneM= on T .
n..:
Definition UI.lI.l pair ( ... I) .....1"« conditions bold;
ijI ... id
to "" /'"wpted ... he ..ner t be !o!•
(a) .. 1lIId ,.,., ,,·_u••bk (b) For all 1 € H +(X ), tIM! function t _
1 ( .. (1))111) it _mmiaIIJ ~
inlegr&bie.
Prop .. ,,-den..: d . . 1) In S ; If K E "D. the _trkUon 011_ ( If ri!»'\" to K io ~Iy coRdn"",.. which 1mI'll.. !hal. " / 1r it oonl iOOO\l!.. T'herdon:. 1rI!l is )...neMur&~ 0
,,·_a.•
OlIld the ~lowl"4" ....suIt.
caD im~
1M .-.lta 01
""i"...,....
Lemma 19.1I. 1 AU~"'$ tIuI4 r II .. ,,11Joer oomicxnUnuous on K. FGr -..y I € T , u (l ) 2: 1(..1); iD- O. Write or(..-). fO< a11..- € X , lOr lbe Infimum 01 {utI) : I € K n ..-' ({..-I) }. The lW>Ction .. dominotee I by tbe pr"'-"'CIio«, 1\ illoom" ~u.ous on X by kmma. 19~. l (appt;ed 10 " I K ), and ..., h.t.~ .,(.-' )g( I) :os ~I) + t lOr "'""'l" I E K . Applyio« 10 v lOnnuL. (I ) 01 Section I!U. we obtain
r
1 (. )dv(. ) :OS
r
v{. ) dv(. ) -
r
v{ .. t )g(I) d;t(I)
:os
I;
(h (l ) + t) dIJ(l) -
r
hdIJ + t ,,(J).
Sinoe" iA bounded and r ill arbitrary, iDlqu..!>lity (2) """,It.&. R N(fOf """' pM! to tbe~....,..,u~. Tho>ee compooet _ K in T lOr which tbe _,"",ions of .. and , to K ""' COflIinOOYll! ror", a /,-d.onoo! .t-, "P, ia T . By "Theorem 19.3.1, " iltbe rum of a 81,mmab&e familJ- (p,. )~u of m mre8 carried by- P-IICI& As (.-,, ) is ",,+adapted. .... can oomidef ". _ f g(1 )e-. (.) dj.,, (I). By part A, I(z) "'v~ (z) ... 1(.-t)g(I) ..." . {I). FinallJ-, """"" tbe _"""'" v.. are .. ,mmobloo, with $um .. ( Pn>poIil;o., 19.~. 2), n
r
r
Jfz) dv(z) -
r
r ... L
... L
-r •
I (:z)"'v,,!:z)
r
1(.-I}g(t )"'"" II)
1(,,1), (1)11',,(1).
•
o P ropooillOQ 111.5.2 S.~ tJu.l". io ml;......... alld' )Ii ",",. , 9 it amtin .. • ...... , ond,- '(o) .. " .j,"~. Theft C . ,,1 ill'. adapUJ and I( ~) dv(;r) f (..t )g(t) dp(t) Nr eutr)l /' X - 10,+=1(1lIht:rt: .. - g(t )t . (. )d,,(t )}.
r
J
r
1(+(X ), "" ... 11&/1
(3).
1';'; • ,,·!nlqrab!e open """ U rool&initlg ,-'(0). Given ~ ;,. 0, put h' _ " +€. Itt_ As h'/ g io ~r ~nuoo.tt, tho:: runt:'lioll 7 dtAned on X t". 7 (", ) ., Int ("'(tl/gll) , t E " -'(z}} is ~ ...",icontin""... (L...""8 19.!:..I). Moo."""",, 11., fullowl'lll p'OpCHiee hold: (a) 7(",) ~ I e",) for all:r E X .
( bl 7 (1rI)g(1) ::; " '(I) lor all f E T.
I'f'OI)rnpooa ~boc\ H 4f K . tbere exlstl .. pUlition..t ..( H ) COIlIiisting of .. ~-II
. -1
Now H n ", - 1(1'0') iI p·ntd~bIe (Tt..on:m !9.S. I ). tile !let. H n (Co ) an: COOI..,:t, and tile ratriction of 1 ., to eao:h of lhem • OOIIIillUO\lll. Thus I'" mmpM:t!let. H in K for whid> / 0 "'I H ill conlin"""" !arm a p-dmoe cLu. in K (Propo.;lion !9.U ), .,hid> pnM'lII lhal / 0 " 11 • /O-_ .."rahIe (P",p"ailion 19.2.2). Con~!y, JUppoee that 1071, s is l-'· meMurable. Mne C. .. I'" oo!Iection of mm..,:t """ L in X for which / I L • conlin"",-,,- II • efIOUII:b 10 obow lhat C. • in X . Let N ... a ~ut.e!. of X luch that N n Lis ... not&l~bIe for all L E We ha,,, to I>fOV" that N il locally ... not&l~bIe or, equivalently, lhat ,,- I(H ) n S ill locally p.""SI~bIe ("Thoe 2) and f"j/ (a-HoW(z ) _ f"j/("I )I p(I)dp(l) < +00 (Theo""" 19.!>.1): tbue/" ......tlaUy ... I~ and , In r..tt .... I~ble.
B. NUl, we <X>n8ia-a from tlot 11m pan and lilian 19.5.2.
P~
0
Eurciu! fur Chapter 19 1
[,g 0 ~ .bo.loca1ly _ptoet"""", o:cnoidenld in &.ere ... 3 of Cb&pter 7. Fo< .,.. inleCfl" " ~ I. let p,. boo tboo RodotI _ uro 0 Po-,.....
2
Let. n &O>d" be .. in Exe c... I . I.
Let A boo .... claoo of ~nl'" _ In a Fl>o-..-...y " E A. dot,,~ e .". f~ 1M iod_ fua 398
~
I'
x-
[O,+co>[, .be. t",""
f' 1d11'... I- f' /,~I
(wbn'e
r
is the
function which I&"..eI wilh I on X ..00 VIUlIshtA out4ide X ). A mawina p from X into &lOpQlog;caI &po.ce is I' .... """",mable if..oo only if il is ,,,,"e..,urable on X (T~m 19.$.2).
&.....:/1 , po«. I ...... H..n..llto I'x. om!!.' We iJ ..,..I onIw iJ ill {CtJ.nonierJj =tewn!' .. U«IItWly " . "" j • • Irk; in tlti' - . f Id" x. - / !' 04ematica.
f'
sf'/'
T~m
20. 1.2 Ld T "' .. /.ocaIlr ...... po.ct Ho...un-JJ IJIG« And (U~ )..u. ... .".... toWI "111 '" T. F~ ado , kt I' .. 0 RJ.Un rrw»IUr\' "" U~ .1Och tIt-', for <JAdI,.ur e/i"# ,.. o. (J IM""Nt;J. U", n UI' "", 1M _~ ~ to,. 1'", ..M 1" "" U.nUI' _ Tl>ctioo 1 E H (T.CJ can be ..Tit..,., in the bm I _ Eli'S . I, wbere, for e.d> lDdu i, tbtno uiou; (I. E A..ucb thal. I, E H(T.CJ and rupp(h) C U"". For this purpOlle . ....., ot.~ .bat, 01 T CON.ain!na tbt . uppOrt 01 I , l ben \~ txln if K is .. c:om~ finitely maQJ" lo>U(K>J mappi~ It. : T I ) aud> .bat OUPP(Ito) is almjlfid &lid is <W!Iained in U", far 1 < j S n , N'ld ~ud> tbat EI S 'S~ lI,{t) - 1 For aU I E K . n.en the (UDCtionl! f • .. fll, oatOsfy tho> roquimd condil ione. Thloo! .oI~ pV ..... lhe un;q.+1 '" '" b , by dMinilion. 15_ 01 funct ....... In H(T. C ) eud> tbat 1lUpp{g.) c U., for I < i f '" and suw(h;) C U~, ki. 1 S j S n, ....:l ....,h thai. :C, S' S_ !I;{t ) ... 2 '5.>5_ h, (I ) .. 1 for all r E JUpp(f), the-n
L
1'... (19;)-
L
I'~, (lh.).
•
Similarly
Since 1lUw(/ f;IIJ) ill OO'try I E H'"(T).
J l/lll dll'l" !.~1J I'hd;i\ - :~ (T1>$)Jtlb
Th~n
7.U).
Now
f 1191dlpi - f I 01(""
f Ihd(g,, ) ~ J IdllIl'l '"'), aDd 10
19,,1 - 1, 1. 11'1. a
T hM>rem 20.2. 1 A H..me w.t 9 l' _ C it IGmlly l"rnt"1fl'bk. ~q 1 "191'/ - IlfIdlpl /or 0011 J : T - jO .+=1. M~. if 9 it .....U"""' ... lM , - '(OJ ;, I'.ml,¥,,~. 1
1 ""'" T inlo 2 &naGh 'po« .. UH>111411w 9J, - in~ JII" ......"tioIl, " .inlqildk;;n lIlil .... ~. f Id(g,,) "
.mI'"
if .... f Jlld;i· P ROaP':
.u..rtioo Ca'
n-
f>I>d tt.. flr"8t part of _
rtioo Ib) Mult from rf:IDfl UI.~.2 &nd IU.3. Now f 1l1~ ,,·int'l/l"llWe. 1'hDI v. ' T ~ C ;, 1oooJ1, " " .int.,. W. if Gnd <mI, if I,V. ;, Ioa>/I, I' .• nt~ In !II;' ....., v.ClJI') - (g,v. )" . P ~f'
o
Obvious,
'The_ ~!I') oIloxally J"lntegr .. bIe funct;"'" from T into C it. complex _ lpooee, and II>< wappina: 9 - 9" from ~u. ) loto M (T ,C ) it li.-r. gIl' - 9>1' if and only if I" - v.1V" - O. Ihat is, If II' - 9>1dV" _ I d( tg, - " IV,,) - O. Equivaltnlly, ,," _ "1' if and only if 9l Iooolly ".,b,,""I .....-rywt..re.
r 'I _
r
Oeflnltion 20.2.2 A Radon I_ute" OIl T it Mid to I>< .. n>eMun: wilh ' - I ', Oft abilolulf'ly continuous with mopoet to " , wht:neYo!r tt..re it a Iooolly J"lrrtegT&bIe function 9 from T into C .""" lhat .. _ ,,,. ",.,., , .... hicb 10 dt:fined Ul> to a locally J"rwgligible ""'-. it called .. denl!i/y of" r-elati,,, to 1'.
Hencefooth. IU _ that" it positi.",. If g: T _ C it locally J"j~ .. bIe and,,, it posili."" tbm ,,, -Igil'. and Oi) 9 - 1, l loeally J"alrno:>Aeoccywt..re. In lit!>.' words, a '6 '., &nd .ufficioent condillon lhat,,, be positive it lhat ,be positive locally J"almoot .....-rywt..re. F in&IIy, Dotice that
for aU locally J"inlqrable functions 9 .. V. from T into R.. A family (g.. ).~A of "."""""arable functinol from T into [O,+ceo[ it Mid to be locally a:ouDtabie .. henever tbe family (,;'()O. +«O[)). u it loc:aIly OOUMabie ( o.6 nition 19.2.2).
Propao;ition 20.2.3 u! (g. )GEA lie a 1oooJ1~ """,,/4bk Jamu, G/'-II, ,,_ in! J " 61, ",,,,,ticru from T into [0. +«0[ . Th< /oIlea.., """"i""", ""' "",oi>-
""'"
(oj 1'11= eN,., , , T - [0 . +oxo[, Ioa>/Ir " .~, n'litJo ~ "';!II E.~ A 110 '-Itr "."'mGot ~.
l'J Th< "'mar (gG " )..-EA 0/ .........""
;'...".motmdt"d m.... ~,.., "" T , and leI (I /I( a """,pkz _ on T .o.ch u...I IP! '" Mo . ..... _ AI .. 0 po.II"... o:>OUt4nl. TIle .. tMrt nilil an o ·~bk """,~.....s~t"d ",nen"" " 8",,11 u...I P _ UO . P ROO~, LeI. ,
e £bIT. oJ: eiooe 9 if P.'neaw....bIc anti
r
Igl~ dlPI '" .II
"' finit
f ''''!'
!to
r
1I I'do
ill C,b( T.P): ""'....... r,
< (/ "'.,,)'
< (/ ' • • ) (/ ",' ,. ,)
, "'(/,,,,)(/" ,'''') N.,..· the mtlppill& I - J ,~", a oontinOO\llt 11'-' Ionn on 4 {T ,o ). Sin(tion ~ a,oeh 1.... ( {J .. 00 , we "",y _ II(
I~
20. 3 TIw; Itodom-Nillodym n.-.m ou"",,*, tll&t .. ill reaI-.....JUI!d, uoivn'OalJy meM",able (Pros-ition 19.2.S) , and that it .......... ""wide L. Si""", If .. (! € T : .. (I ) < oj ill ... -...able and """"ined In L, v(H) ill the supremum of the numbers v(E ), _ben E " o r ~ the 01_ 01 all ooum .....wn. valid for complex ~." ' . • .,.... I" En:r cl. I; 2. _ si,-e ... OI;~ proof oflh .. romlt. Theorem 20.3.2 ~I,. De a pOIiti~ &U/"." m.... ~'" (>II T and /d" De A mil RadIJ1I "'...._"." T. TIle J"'~ «>n O. ,1>0: ... n \8t.s 6 > 0 8IIdr tlw the rdaliord II E H --(T) . 0 5 II 5 f. and f li d,. 5 6 imply f lid" 5 t . Th.. CODdI...... (a) imp/iN ..,...jit Mlr:r (h). Next. ~mo tlw ..,...jitlon (b) holds. and ~ K be a "'~i&ibJecomp6Ct ooel. Cb -- J E H " (T)...dr thai J '" 1 on K . 0 1....." > 0 ..... 6 > 0 he ... • 1. .... We QJ1 lind II E Ji" (T) such t hal II 5 J. ,. _ 1 on and S 6. Then v(K ) S /lib; 5 f. This ........ t hal K if "'~i&ible.
,..,td
K.
f
f "d,.
N"", (e) imp/iN (d), b, I"" HtwIon-N ibodym l l>eorem . F'lnally. (d) ;mp!k8 (a). by the .am.. ~n. .. t .. t hat of Tbeoreln 10. 3.2.
o T _ Ra.d 2 . P' ","ELJi"3 wilb tbe COOItmaio oc.ep-br...up, _ (6" ).",,. tbat hM tbe deoired pooptrtieo.
by
Now. Ior...,;t, t E T, let t, be tbe Radon
obc.lU.D tbe
_ueneon:nt l.2.1). Hence &II MOmIc Radon ....... ure and .. dilf_ Radon ............ an! difjoilll. Nexl , let " be .. n.don " ' Mure OIl T , and let "" be tbe n.don ..-ure defined b,< l be ..-,,({t) .t~ t E T. T hen (P - ",,)({t}) _ 0 for every t E T. 110" - "" ill dilf""". We amclude thal" C&II be written "" +I'.I,"bcTe "" if atomic and "" dilf_; nlOlWLtr, tbif decompa.ilion if unique. Finally, POlice thal resul~ e1milat to P~tionll 10.4.1, 10.4.2. 10.4.3, and n....uu 10.4.1 exiH for Radon.-ure, and ....... In " $ ; ' T to ....... 1ICe.
r...a.
20.4
Duality of lJ' Spaces
n.
LeI. n be .. Ioc&lly comPKI H. UIIdorff _ and " .. Radon meuure on Gl-en p E {I , +001, let q be iu mnjupte exptAknt. For all, E L~(P), ... rite 9. "" tbe ronlinllOlll linear form I - fill dp on L~ (I')' Thon, 9 : g _ '. ill an *"""try from L~(P) onto ...ubip!OOt of ( L~(I'»)'.
Le. iA be 1M maio proIoclption of 1'. When 1 < P < +«>, &ppIyI", T'heo<em 10.5.1,
LeW) ""I(>
mo."" L1::{ji)( L~u.»' ,., (L~[j!looring
l.g
......ionoc
,.)
bo,,1- III '1,,1·
I' 'ellrl'l .. I' ' "IeII,.1for all f
(0) ,al. ,,. ODd 9" ho ... . be _ _""I ~io ... _"""'" iollow ..... lint . h_
rn.e.
n.
~.
2
.s:
-
p~,
.s:
_
'''1'POrl
f
f
••!la• .-enion (d) 10 ' """
Let,. ODd ~ be ....., complox Radom _""'" 011 " IooaIIy -oqIi5iblo ... II Ioc&ll:r
q""""'" of _
...
2.
n..,
~.
Let 5 be ,1>0 _ring ;., n co",....i", "'f .... K n L' ( X, L compKI .... boe. N,-" , be i ............""'.1 _ [(Ju) oiJ>. of ".D,abIo
"
r
fJ1w-- r Gill: X -
[0. +oc);
(,J (I ...."pillf / /n>m X j'lLr Gil
n_u~
OIl X
Theorem 2 1.1.2 W I /Ie c ....pping /n>m X if\/.o 0 Banac4 ,pace F . II I " " .. 0UCtlti4l/J ,,·in~. til ... I ... eumlWIt o:v.).intqrUl< .nd
J I &,("v.» - J I " o:dj..
PROOF: T he Ii...,.,. ""'wi"W' g J gd{o:v.)) aDd g - J g" o:d" from q:(-("'Il) InlO P are oomlnuouo and they ...... on "h:{X. R ) I1j>P. 1O they are ident\cal. 0
Notice tbat , II/ iI intqral*, 10:0 ",utoa!.
~.m.i.ollJ
",v.rintqo::rable.
I " '" to """
_tWly ,...
Rllfulu limila. to P ropoolitloo18 H. I. l 1OU. " (I pouit; ... .........", on r , '" G ,. . ......... nsI:ok m4pJ1ing from r inlo 1", ,.. .. fflGwVle .. _II .. Illi ""bote! ",-I ( K 'l. Thm '" il l"propel , wei _ ..... "'" Il llion (a) to oonclude. 0
Decomposition of a Measure in Slicest
21.2
Let X bon iom~ 1I."".1....if ~, '" a mlt.ppi~ from X IRio a 1ouIlJ' _I*l lla~ $pooiI T , I' a po6it i.., Rtdon .....,.."re 0>11 T , " : i .... ~ a from T ioto M~ (X ), ..-!any _otwly I'-inlep-ab/e and vasueiy Jt-~ Put .. ,. IIw II S I: footrvuy II E H , .... ~f ,..{i) .. ~ (II) foo- aU 1 e T , d liv*'" by " . &pp/og I .... ~~ , and drootc by ... the!ilU6U.-e J11.1., dl'(l) OIl X . P lIOOf';
N~
r'. . . . r r 0.
to(.lU.,. It boondcd wd ., Ito .,..P«'Il sup !. .1.,(1) a'p(t). "
AIoo,; lhe 61\e1 of _;ons of }{ , tbe _ppi", t .... (1I.l., )(1) _ A.(1h) ,,01'1>('ld \0 t _ A.(f) unlfumly on every K E 'D (by Oini'. lheorem). Hence, b aU K E'D,
!.
A.(1) dp (t ) - ou p
"
.... H
!.
(hA. }( 1) dp{, ).
"
We conclude IMI
n.adore, " _ IUP.... H "" (Lemma 1.3.1 ). If N e T 10 locally I'"nt&lwble, ,hen. £or -::ry 10 E }{, 1\ is locally 1M'" nedicilM, and ,, - ' (N ) is IoeaJIJ" ...... nqli«ible ...1>...... ~ dedoo: thool. ,,- ' (N ) Ioloeally .... """Ii«ible (Propooit>on 111.3.1). SimiLarIJ", if / is "'P>eIIOUfablo, then / 0 " is "" •..,...,urabI<e. at / 0 " ill
.... meYu~.
21.3
C
Product of Radon Measures
In Ihlo _>on, X and Y "'" 110'0 IoeaJly comP'ct Ib.uodorff"fW"l FOr at\Y 110'0 oomP'ct oubeeu K, L of X , Y . define ali ....... ioIotDdry 101 from "H (X" Y, K" L;C ) onto "H( X.K ;"H(Y, L;C)) by (1oIf)(Z){W ) - I (z,w).
,,,,JIGOI!
Lemma 21 .:U ~ doted VI!d.o1" o/"H{ X" Y. K" L; C) ~Icd ., {, e 10 : g E "/-I( X , K ;C ), h E "H (Y, Li C )} ill "/-I(X " Y, K " L; C ) IIMI/.
PROOf"' Ler. I< E "/-I{ X, K ; FJ. where F _ "/-I(Y. L ;C ). P~lioa 1, l.le ....... I","" &i>"tO t > 0, lbe", nisi. fl .... , ... in H (X, K ; C ) and h" •.. , h" in F oud! IMI III«z) - L, :s,:s~g, (:r;)h; II:::: t for all z E X . The......:lt 1'oI1owo. c
Lemma 21.3.2 wI p "'" 4 """"""" R4dMi ..........., .... X . For ......., / in "/-I (X " y , K " L; C), the w_ JI (z , , ) a'p (z) /.0 H(Y, L; C ).
"'rIC"""
"lang.
PROOf': Put II'" wtll- Then j ~ t.:). iI the uniq"" Radon _ n oox:b that >.(g . , written
01. ,. and v. Thea."", 21.3.2 I"
uu. ow:>ttJl;"'" I" ® v i ... 1,,1® ivl.
PROOF: let I EO N+ (X " Y ), W~
"®", ii IOIld to be the product
and let ~ E N(X " Y, C l be !IUd> tnat IPI S I .
ha,..,
IAMI"
/41',,(:enl1'ore, I>'Uf) S (IPI@lvl)(f)and,finally, 1>'I:so 11-1· N_1ct. II E 1'('" (X ) and v E li" tY) - Given! >- 0, there exist and VI EO N{y,C).udt lhat Iu,1 S II, Itotl S D, and
2: 1,,1(11) -~, 1.-
~
(J"I{II) -
t)(I"' ( u) -
413
t) .
0,
u) l be .. 8OWl £or all " ~ I. Domne
N_
{:z En - U K.: I (r ) ;i z}.
There is .. II . negli&iblc .... boIe. II' or fY ouch that N (r' ,') is aU r' f 11'. Then, for all ~ f II', I(~ , ' ) is ""'fIlt6S\Inblt.
"H . n,,&li&ible lOr a
Wb", .... hA,... done r.... tbe prodl>Cl. of t"", II"ItaIIUre!I atmdo easily to fini te produet of Radon ~u"""
n b. .. """'_, H 'l'e "" """'a1ninr; K .udt ,hat ,,(K) < ,.(W) , aDd ~ .. real fIUIIIber o-*;,.(yiq ,.(K) < ~ " ,,(WI . "'" .....,. ~ > 0, _ lho. ."1tO;""",, d ~"""rablo "flO'" ~ U, V Olldt , hal
.""b
K c U e U e V e V c W and 6 - . ::; /'IU ) " 6 " ,,(V) ,.;: (+. ("""'".-uet V § ...... and nort Ul· l. Let K . w . 6 boo .. in pv\ l. S - l hat I""'" ."..... lMIu.od,obIo open V b ..-Md. K cUe 17c W on
""*
iDductively • family jU(.I).." or " .quadrablo opeD (a) V (O) _ . &de 1M< 1 10 (IK · "I."'·_~'obk Le< IK. I.u be .. ioOIop. and let fl' boo , boo iLl.........
to, 11 eqwiwed wnh l be d"""",,, lopOiosY.
...:I let ,/ be lbe
~~~
LeI" be Lvi , ... u U ",," on n, that ,,'((r)) .. 1 Ior..u z' E fY .
on rr _
"e,,',oq1i;Ible. 2. Let 6 ~ ((r . .. ) : :0: E n I be l be ~ of n, ODd wrl", /Ior Iu i""itaao< rupuo .1,;,. -..It _itb 1'I:ooo..em ' .2.5,
22 Operations on Regular Measures
u.t.,. _ _ on oemil'iDp, we P"'"""'" o.ppI;< •• iono or _"'" ~ ... "'. .,...q_;.... ' be d;.&;i."'iation or funct ...... _ .he SormuJa fo< +'DI!"
'1 .... I I
or _A'" ... ...........10k, _
. '"
,..or". 1~ tho .,G...., ,. of ftadooI
For .klo, ,ho - - , _ "'" oboc.roo:'c" .... _ "" be com>oIdoI. 22 . 1.1 Ld fllol! ~ Ioooll~ compo•.:! H~.."u,rg tJMI«, $ 0 oem>n,., in fl , I' ~ 'M;that .. - ' (B ) it......uiall) " .in' S. dl. fer all B E 5" . Theft I., 5") it " .,..;t«. MOT"'''' , if 1M comJllld ",""II 0/ fI' are .. (I"I ) ·~, IMn ""'1 it ~ """ J"i..u .... 10 u.. &Joft ............ (.1.).
... ..-..,* '"......
Fnc. _me thal II it ;>oIIitioe. Write 1) 10< , 1>0 (~of t'- compooct acta K C fI ouch thal it cootlnooos. For 011 K E 1), \.here nit'- .. !Jeq~na: (lI.. )."" of S"«u IUCb lhat K is induo.!ed in u..~ , ,,-1( 8.. ). Since 1) is.l..oen. In-It eW> romptoCt 8U~ PIIOOP':
.., of
fI",
n is .. u";on or J\o.r and 1'1- n. to .-(.101. FlnaJlJ". powe to . he ",..... oJ ~. in which " if c:oonpie.. W~ ....>"11 1.-",)1 ~ or{bo l), ....(;.) if resuIar. Eoocb 8 e S' is e>l\lentwly .. (.Io)-lmq;rable. aod IIBd( ..{).» .. ).("-'{B)) .. ..(p)( 8). The Ijn~"donno 1 _ f 141.-(;.) &Dd I - I Ib(A) "" Q;"( or{ll'l» .. Q;( " (IAIl) ""' conl illUOllf anti t bey o.c_ 01> S I(S'. C ). So '''''r are ldmIiAble spoooe Ie "u.}- ~ .. bene·...,. I 0 " .. A mappI", I from Il' into a 6ano.:h ' I*'" Ie _ntioJly " u. Hntovabie .. oo>e.~r 1 0 "If it _ioJly 1'.fn"'Wab!e; then f I n(;.) .. I I 0 "' Iiot a-rinf ~t.aI b, C ..."toitu
et'er)'
II""mIkI rJo~
B.rin:
oa.
P ROO'" LM.A. be the c!-. 01. aD 6n1te iDte' .. - (0 ) oocb that 0 S I -S I , J,K .. I , and I .. 0 on 0 - U. Ooli."" V .. (:c EO : I {z) > 1/ 2) and L .. ( ze n : I tz) O!: 1/1). Then. IillOO V - lI- (% , I{:r ) 2: 1/ 2 + lIn}. It .. llot uiiloo> 01. MW!_ 01 PItOOl', notre e>'*'
COIIlpKI
C_.
I
0
If U ill Open and :r e u, lakins K .. Iz) in PI""";tion 22.U. _ lind an Open &Ire we V sud> that % E V on:! V C U . Illollow-oo lhaltlot eta. 01' Open BaI .... _;" a boosie ,.". llot ~ 01 n.
P ropollllioll :n.2.~ Ld!Y an
n-
P AQOP: If F: and E" AnI compact BoJ.., IOtU In rr and rr. "",peaively, lben E' " E" ill. com~ C" roo thai. A' " A~ c .A. Now the c' - of all _ '" the form V' " V", w),e.., V' IU>d V~ all! open Hai.., oeu, ill a t.siII for \.be ~. By P1 j1 0, Ih {l are ".measu. able (Pn>J>
;..u
f
a,
J
V>.' (K)-
inf
'.~.
J1d.V>.w
'~'M
,.Inf... Jldl' l' I~' ~
and V)" ( K ) 2: V,,' ( K ) for all compact K . Howt!ver, by definltlOll of
r IdY,.,
bo aU fllllCtiono! I fr'OOl
r f)
IdYl 'S.
r
IdY,.
Im.1) [0.+001. lienee VA"(K ) 'S. Vp"(K ).
[J
Theorem 22.3.1 Th
,.
V~ ( E _ K. )
-kIO, E -
VA(K.)
U. K" lulso 1'· De&lCible. On tbe «bet
rinoe An K . io ~."'"Ili«ible for all n ~ I . it .. &lao 1'.~lW!. ThiI pnJII'I!I7 that An E ill'·De&lCibie. and finally that A .. locally ... ·nqlCible, Now , \eI. I E and \eI. (f.). ~ , be "'''qt>iCllOO In H (ll , C) thai. ""'''t" .. w I locally ~ ...t . Then (f. ). ~ , coo.tlp to I locally p-1U. and each I• ...... ..-urabIo; thUi I io 1I·"""",urable. We """ II>< now.ion of Section n .3, and PfOI"" II>< followl", ... ult. P EopotIltlon 22.4. 1 SZlppm IM.t G{,,+) And G(b-) etilt in it. Lei J _ (G(,,+) •G(~ » .. An illl."...j incn..ttd ill G( / ), Mill mdpointl G(,,+) and G{"' ). 1/ I : G (I ) - F .. • uch IMt (f " G)g iI ~.~, IN:n IIJ .. ".iIIlqro/lk. 011/1 f(f " G)gd~;, "'l""llo fflJtW or 10 - ff!JIW'"
G ( G(b-J; t' '"' +I ifG(a+ J < G (z:o) II.tId c' _ - I ifGla+) > G(",,); ..... +1 if G{",,) < G(6-).oo c" ~ - I If G(: G(6-). Then
r,
d (c (-+) 'mPflC1IP""p. ~ .... Inttod.... ph , " ' ,
01 , . n. " Ir ......J:yMo
wit~
-.
_I'
:tU !Jo ,b. -'ion ... dofino jrrwula»~ relati-elr I~~ and quu;""'''*'''m 'E 7"_ ..... ""'*"r -..ptId ......1'. 212 0.."..,.,. h "ly OIIaIpacl .!>ere exiou .. Ieft . _~ " ' re. Thill A "n;o UAiq .... Up 10 .. "",ltiplio:atl ... _ _ (1 Zl.2.1). 23.3 We drIi... 11>0 lDOdu .... 'uDCtion 011 a localJ;y. pro F that
tl ) U "'" lor olI:r: E X ; (b) (.1):1: _ . (LI:) for aU . , 1 E G Bod a11 :t E X .
l'hw G It tDl to "'" Qn X from tho loft. The ""lion .. oaid to be transit;"" w~, for all :1:. l' EX . there ~xi$t, , E G such that"., ..,. D.Do>le by T(O) tho mapping:l: "" f% from X jOin X . We ha"" 1(,t) _ n1(1) , In puticuI&r, for every • E C , T(' ) .. a bo~hism from X on\O X. Fo< map pi"ll I from X Into " oct Y . .... pm 1(.)1 - 1 01('-')' 90 tha, b(.lI)h hl.,) _ 1 (") k>r all., E X . At.>, If 1' '' '' Radon measure on X , ... ohaJ] dcnt!to: by 1(' )1' tbe image measUIe of /-I under 1(' ) , "" that
n.)
"'''''Y
lor all 1 E "N( X. C)
f
/(", ) 4(n ' )I')-
f
1("' ) ';/-1(" )
(1).
1nHe&d of d b t.)I') .... lIhall """"",,,,,,,, wTI~ dll('- ' .,), then (J ) .-.,..ds
f IC")d;.«.- ',, ) - f
l (tz )dII top(j Oll .. ). 0
"u
We retain tbe . 1KM! notation. Pr~don 23.1.2
1«»1"
A _ tIw Gil" """'pod "'0 .... Let" ". 0 100:. nlnfl'todr inNri.nt ' " ', .... on X . Tht:Io it. '"1Ilti,&r Xii" .....tin_ ",notion on G.
P ROOt": Let II; 1t( X, C), S ~ the oupport of I , "" • poItll of G, .nil V • m X ~ G [nto X , with r(1I )" (:U)I for all. , I I; G, z E X , and zt .. :r for &II :r E X . Denou br 6(.) the homeomorphism z _ from X onto X . For ~~I)" mappl", I fmm X into a IIt\ Y, ~ PIlI 6{.}/ _ I 06(.- ' ), 110 that (6(.)f)(6(,~) .. 1(" ) br all., E X . Alto, if " II. RwIon --.u", on X , ..... shall denote br ~(.)" I~ i"",«", ~~ 01" under 6(. ), 110 thal g>'OUl'
:u-,
10< o.U I / 1(:r) d(6(.),,)(.,) .. / 10
E 1'(X ,C)
I("'- ') ~")
(3).
,",d01 d(I(. )... ), ... .ru.u $Otuctjmes writ.! "Vi.ant, ~lativdy Invariant,.nII quasi-invariaJll ~ (wider G) on X are t\tfillt im-arianl. _,....." ' ut'fOl reIatl .... y \.........,. with t'fOIped to left (or ri«ht) t"MI·'Iooa... (~., ""'" SortJ,on 23.4).
I from G i!lLo & .eel Y , '"' de6"" I t". I {z) .. I (z - ' ). Abo, k>r ~ ftodoa _ ... I' on G, P is the lma&e ~ ... 0I/, ullder the ""'-"'_0 - Z .... z-'. Hence PUl ., /'(/) br I E "H(G, C). In ",her For ... ef) mapplnf;
_0»,
f
If "" agree
1(1
f(z )df I t... ~uppOrt of 1 + /" and Ii,,,,n .. compooCt nci,sborhood V of ~ ouch lhal., I'or """' Y
r ;> 0, ther" ezi$Io 9 E 1t:;. {V ),
(I, ,)+(f: g) Propert;e. (.0.), (b), and (c)
~
s: ((I + f
clear. vt
L
I :s:
): g) + r(h : 9)·
1 E 1(+
N1d
9 E 11'ence (d). For (~), Let 1 E H ... , 9 E 1!:;' , h E 1!:j. ; if
I :s:
L c,-n •• )g
gS:
's's_
L
's,s..
d;"I'('J)h
(with Co 2: 0, d J 2: 0, I" Ii in OJ, then
I :s: L
c,d,.,.{.,I;)h.
'0
(I: h ) :s: (I : g)(s! : hI· If "'" .. ppIJo (e) to 10, I, 9 and wI, 10. 9, we obt.o.ln (f ). fi nally, let I, /" h boa in H ... with h{.) 2: 1 on t he IUpport of I + /" and let . ;> 0 t... Ii""". P m F _ I+/' + (1/2) 0, there o:;.to .. " " _, ~borbood V of ~ NC:h tbat 11"1') - 'j>(l li :s: II a.od I.... C.) - '1>'( /11:s: ., for . - '1 E V . Then Iol, 9 be NI "",ment of H';. tV); for ...u 0 E C , we ba"" '" . "I'(')g :s: ('0'(. ) + IIh(.)~. Indm , 01 poinUI ..here 1C')9 '"lWisheI, this is clear; t b"" Ihis inequality bolds oul8iQe .V; in IV. '" 1"1.)+ II. Similarly, .... . "1'(.), ('1>'(. ) + ")"1'(1)9 . Now, if _ let c" ... ,c" k pceitiYf: nom""'" .. nd ., .... , ... boo eiemento of G """h thllt F :s: L,s..s_ C(")'(I, )9, "'" _ lhat
s:
I _..,F S
s:
L
'1'1'"1 (0. ), S
L
"' ("'(If) + 1I),{•• )g
ry'
ate
and similarly lor ' 0 ht'nce
L
(J : , ) +(J':,)s
C, ("'(8;) +",,'(8, ) +2'1) S (1+2'1)
L
Co .
+.,.
sillCe V' S I. Applying tbe definit ion 01' F and. !leXt. (b), (e), and (e), "'" collclude thAt (J :,) + (I : ,) S
(I + Z'1){F:,)
<
0, If II is K, : rttT than I on t""aupport ol f + I. It '=>ilow"l tlw 1(J + /') .. 1(J) + /(J'). By Propo.itioo 1.3.3, I can be c:UDded t.o. ~'-' form on 'It(C , R ); t his li,-, form is a P'l'Itl." _aero Radon !77U~ on C, and. by (a), is Idt. iTMUirur& . This is l he deaired left H...,. ~ Applying lbe f",~ng 10 lbe OJ>P(IIsite group. "'"
1 {, )g(II)<w(I)
..
/ dv(l) / I(.)g(c.)dl'(')
.,
/ <w(l) / J(r ' , ), (,) dp(, )
.. / 'C .)(/ J(rl.).tv(j»)
01'1'(. )
.. ,.(g,.C/lD,) , aod 011)" (D,I')', "'11 PI'-erywhere. beca .... D f aod DI' are 0'>:I , If J e 1( + II soch 1M, I'(f) ,. O. lhen J .. 0 ~he!- O. Tho. 1'10_ that " and (; "'"' ~rtion&l. 0
The"""".
Corolla ry Ever, left ..........n.I ( ......Prnot>mompod gro~" and lei,. /Ie G kfl HGlJT _1U't M' a rigAt Ho.ar ..........., .... G. G ;, """Cf'l:k if and onl~ if I'«( e}) > 0,
G ;, com"",,! if .",,1 nIr if 1" (0 ) < +00. P ROOP': The O. V io. finite ..... ,.(V) < +00; .. G it 11 ' "',,,,, OIl G. If.6.a " 1, ,he poup G io aaid to he unimodulat.
t"'"
Notice that" io rdatl.o:ly in..n..nt under ri&ht t,...",I"Ioo>I, with multiplier .6.a. ThUl.6.a io" COfltiDIJDUI ~ from G Into ~ . In particular, .6.a(~. - II- I ) .. .6.a (~) .. I for all., lEG; lherem. if lhe oubgroup G' 01 G luwutfld lit the mmmuUt.ton Is de .... in G, tl>ra G io LLIIimodular; thil Ihows lhat aqy COfl~ oemisimple Lie J1'OUp ill unhnodulat. If 'I' io an ieo" ... phisrn from G onto " Ioc.aIly """'pod poop G', ~ " 'I' .. .6.a. In partic:ular. ~ 'I' - .6.a if 'I' Is an "uton... phisrn 01 G. H~r.....M Ih, we 8e1. 6 .. Slr.oe
t:..::.
6(. )( 6 - '1') .. (6(. )6 - ' ) (A(.h.) .. (6 (. )-'6- ' ) (6 (. ),,) .. 6 - 1I' for aU. € G, 6 - 11'
_ II' Is" ri&ht H..... _ure. BUI
1(')'" - ("1(. )6 - ' )1' .. 6 (. )(6 - 11') _ 6 (.),,' . lienee, lor .-.y ri&ht H..... m ' I mre II, 1(.)11 .. 6 (. )11. Sir.oe (J io" ri&ht H..... m ure, i< _ ...t:.-' I' forllO O. W" ded..,., that I' .. a(A- 'I')' -Ai< " her.oe .. .. I , and li...Jly i< .. 6 - 1". Similarly, II .. 6", In Ihort , we "".., the l>Ilowiq ....WI.
""j
PItOQf' : Let I' be .. left. II"",
ole ' pee /)n
,,(V) _ I'(, - 'V. )
G. For ewry l eG, g
o.(a)I'(V),
.. be,""' I. foIlows.hal; A(. ) _ I.
a
As a ~. if G is O)mrnutali"", or discrete. or comPKt, It is u .... modular. If G is diocmt. the mo eu"" on G lor ...·hlch ...... poi,O\ I... ~ I is plainly ... Haat m E' I" .... Jy " '" ill 1 ~ form "XI' , ~"E C- aM X ;,,, omtin....... hom""""""'ilmJn>m C in/g/LJ. tn.noIaI;."...
~,,ntained in A" A- ' B. lIe~,
PlIOOf':
".,oi
r..c.
/ d..{,l /
IA(:r)ls(:trld;4:r) - /
oi"o=}I ~(>:) / I ,,(>:~) oi"v(r)
(I).
F\Qt. _ .... tlut.t v(B) ... O. By bypoothooi:is, v(.,- ' B ) ... 0 for """I")' >: E C . and 10 the rlf;btd $idf: 01 ( I) is 0, and he,.."., the fi&ht-4nd side is ahto O. AI a """""'I~ t!J.e«o niWl a local", J...~I~bIe "'" AI wch that f 1.e1:r,Idv(,) _ 0 lor :r M . Since" 'I O. _ conclude ,hat pM.. tha; Xl ll is Halllldonf. [)
of,
nc _ e.
e
A...""", Uta.! X ;, /«all], o»mprnpo.ct ".,.bborhood nl" I". (>flO>,hod), .• nci&hborhood 0( . ) in X. SiDCe e n {V " W) Itt l iiI)I)(tnpt)', W n Sat(V) 10 -.empey. ~ b ib ... Sat{ V). Nei&bb«hood V of o. So mpcIAen I-'(K ) ill """'JIft"t. h'? Fd, .. hen r'(K ) ". I, every u1tr.filu:r on r ' (K ) I..a • limit point. H ~th , ..., Io.t /I l>e • locally (Onlpea group operating COJItiDuoosIy (""" !l>e "«ht OD • k>caIly """,pea _"" X . and ...., ",,"unw: that 11 acto propo!fly on X , that ill, that the mapping /I : (;t, e) .... (;t, .re) from X ~ 11 itl1Q X" X is proper . Thm the i ...... of X" 11 under /l is c~ in X" X . In other words, if we C"OnSins '235. 1 and 23.5.3, XI H .. XI R illloc&Uy compect . We shall ~ by .. In. canonical projection from X onl(l XI H. romp.::!. ""'-'- of X I II ill the lm&ge u nder .. oC • """'PftC\ out.et oC X . If K , L ..... t'M) cnmpllet IlUbeo:to of X , tloen P(K, L ) '"' (e E /I , Ke n L 'i- 8) II! compllet in C oi~ , sinoIJ. /I is p"""",", /1- '(K " L ) is t. G """tin........ jun: e X , ~ ~ticn { ~ 1(:«) /)0/0"91 1 "H( H ,C ).
IH
~} ~ ftm~ri"'" 1": : % I (:r{)x({j - I d;ll({) i.
PROO~:
For all z e K , .... ha"" "' (1'» 0. H~".,. Inf •• "'H,,' (r ) _ Inf20EK " '(I') il lt rictly posit;"". The functlod h, eqUlllO ,1.. ' OD KH &lid ..hIc:b ....isbel OP X - KH , lie! in 1t~ (X, C). MOi ..... ~r, (tJa)~ _ ,,' h _ II by pan, (e) of 'Tben!m 23.~.1.
TbUl f .... f' If" &lid" then, for each I
[J
mt.~
1t(X , C ) onto 1t'(X , C ), &lid W (X ) onto 1(~ (X, C). to 1( ~ (X) &lid if II e 1t"( X ,C) "'Iis&es 1 , 1< If + If . 2 • .., define Pi u tbe functiou eqUll to
be"""
Jl(r) g(z). m r ) + mr)
(I: +If )(z ) > O. &lid O~"'bm!. Then II. ~ F I "'~
al puinlol z e X . uch lhal to 1(~ ( X, CJ . 1 1Io1:S &lid
1.'.
11 - ', + 9:1 .
If I ii " li....... 1onn of fini te van.tion over 'W (X ,C ) (Section 1.4). tbon 1" : I .... J(J' 1 10 • linur form o f finite van.tlon """. "I(X , C j (bo:o
.,hldo
ptO¥MOl.".., ~ "" XI H , rile/l 6 (Q~I
- ~(r ) from :H(X , C ) .. t..H (()~' for all (E H .
()) Ccnoerxlr, 1ft I' kG ................. X nodi th4l 6({)1' '' t..N «( 11' /.,.. Ail (E H . TIIen tAere ezi.to a l£nique ........ "'" ~ .... XI H nodi /JW 1' '' ~' .
o ~I
10 re.ol ( ...pectlve!y, l'O'itlve) if and only if ~ Is. ~(r) can be "";"en
( f( z )d)..'(z) " ( 2i,.. ... a ... "" XI H . ~ lamilJ (~J..., ;, !Immdtd ..... in M ( XI H ,R ) if.nntr if IN. fAmil, (~)"'I ;, k>vnded" .".,.,. in M (X , R ). TIIen $011'>(Al ) .. (aul'> ~ )I.
PJ
( ~).."
F"or...,., mol............,
~
on X / H , (~+)I .. P ' )+ an
(~ - )I
,. (~I)-.
(e) F.,.. ......... """'pJu ............ ~ '''' X I H , I~I ' -I~ I I· PIIOOF: S"wooe that the [.".;Iy (A;).u 10 bounereexisu a rMI m" m ... ,.' OIJ. X ! H sucb that " .. 11'. From,.11 :!: Al for .11 I, .... de
23.6
Integration with Respect to ,\:
Let H be • locally compo.cl. group ... hid> fOCU from Ii... r40;ht. coDtin........Jy and ",Ope' b". (lD • locally apII 71(X , C) onlO 'H{ XI H, C ). we .... thai 9 .. locally .I.-intqrable and that (g.l.)' .. (g Q Coovenody. jf , illocally ,\·intqrable, thm. by Propd 1",'\ _ 0, 110 N It loetJIy .\.."",liglbIe. It 10 It .\..mcuurabIe. lheo h Olf iI.V·meuurabloe by Propooilion 2 1.2.1 . ~raely, .... ume th&\ 10 o . iI ,\1·meaeun.bIe . and Ioe\ I(' boo! • compooa out.cl. of X / H . Cbooooo: I € "H. (X ) 110 tlw. 1 DO K', &I>d Ioe\ K ., aupp(f). Since outside :r(K ). K ' iI included in :r ( K ). There exiN
r "",,,,,",,
r_
• pN1l1ion of K ~11lI of. A' ·negli&ible let M and • llequtotf: ( K . )..ll oC o 23.11.2 1.'1.1 A 4114 A' 100 _ CI01IIJIlt;z ......,~ru """ XI H. :n...... ill nIer Uwtl A' 100 .. p - . . . .,w, baM A. il ;, nc« .....,. 4114 ~I t1I4I A~ 100 a ....... _.,w, ~A' . Pro~ltlon
PIIIXlr: T"IUt IoIIoor1 ftoat (.) and (b) of Proposition 23.6.1.
0
Propw.ltion 23.6.3 UI A.,. 0 .. 4 """,pia"eo .. e """ XI H . .. ;, A' · p'"",,' 1/ an4 tmIJ if H ;, ......"..,t.. In IA;, CGK...(A' ) . 8( /1 ).\.
,ha, .\ .. ",*,1_ Si""" " : u -IJ. Is l,.odequatt. for every
Plloor. Suj)pIIeO ,EO 1f·(X / H ) we ha""
r
(go .. ) '}8 1 From \Qp(>Io()', """' kI>oooo thal. locally cornl"'Ct Ipea! io po.:acom»Kt
if and
.... Iy II it can be partitioned jnto....., ","lIIiltg ,-n>J><mu.' Pro~ition
,.....ti...
(.) f' ;,
....e >d.mIitoU)' u:ro ....
,.) F.". • ...." ...... poId I~Nct K " - ' (K ) " ~
Gil)'
~I
0).
"""""Ill!:
Sil>Ot .. is open. thOIl1CU1 .. (n. ) form an ~n of X I R. "There iOOsI. an opeII _illl!: (V,) ... " loealIy 6niu, whlch reS".,. the 00'(, ill,(!: (:or(n.))., and • anillOOUl ~Itlon of unily (g,)"".,.. X I H wbonllnalft 10 doe OO"'illJl (V. )...,. For n.cb i E I , we ( hooeo: .. poinl ", E XI R 00 thal Vi C ..(n .. ). The function F, _ fJr< 0 "l/., helo"9 to 1-{"'( X) and Iw IiIppon. II included in ,,-'(V.). no.. ou.pporW 01 tbe F, tlwo Iorm .. \oaIl1y linite Wnily, and f' _ E .... F. II cootinll(lUS. For ~VCfy" E X/ lt. tbcs-e ",,;u,o an lnda i IOtlo thai. JiI; ("j '" 0, and " IM:il in V,. Then then: II .. point :r: E lot .. hIcb o.-(z ) - u ..... f •. (z) :> 0 and 9;( .... ):> 0, we F;(z ) :> 0, and , ev~lly, F (z) '" O. "T'hif poO..,. lhal F ..,isfoc,s (a). finally, let K be a a>mpolC\ .... t-I. 01 X I R. TMre .... ,,.. a finite ou~ J 01 l lUdo lhat, for i E 1 - J , Vi n K Ie empty. Th... ,, -'( K)n oupp(F.). ' Iot..u i E I - J . Sin{ * ",-) ;. ),I .int j,rihh. P ROOI': Let / E 1f(XI H, C ). Then h · (f 0 ... ) belongl ..
(
1.'1/1
- 1.>,.
t()
H (X . C ) and
d.l.{t)!{i: ) ( h(z{) dP({ )
il
" .
tMJ u-e.........,;tioru /ioU, 1' . T'Mn. for mil. E G , if 8. iI" ~ 01 "1O/ N(' »). utA _ I to A, _ h4'" 8. (,,1') " 11I. - 'r)/l1Iz) /otloil, ".ot/mMt .4_
ew" .....ure. P ROOF: (e) impliflo (b), by P'O!XMilion :23.S.1. If condition (b) hoido, the c_ 01 all locally bnec!~bIe subeeu of GI N is In''' riant under G, and 10 A .. q ...... iovariaot. Now ....""'" that A is qUMi-invariaot. For e,~'1 • E G , ""re(. »).1 and A' ~ equl.-.lent (Pf'O\)OIlt ion :23.6.2 and formul. ( I». TIIOC, dooe A' is equi..!tut 10 " (J>tw
M
"'"'-'U'O.
(e)
oM
'"tor in.........t.
n..... "",,u .. .....n......... t-nt:fil>nx}rom ex (GI ll ) into JO,+ooI.tod> I/olli x(, ... (:r» .. P(~)/p(':r) for all'. :r on C; ",o.toIoI>II6u~ 011 G_ Propoe.lt lon 23.9. 1 U! A 1: 0 '" .. m.e ~_I"" in"";.,. untltt SO(n-+ I, R ). Theorem 23.9.3 Ld C "" .. "-'I, ..,,,,pod ,...,..,.. G' .. cloMl normoJ fbb.. '""'" of G. G" 1M ,.,..,. GIG' . .. : % .... :i the OInonico>l pi uja:1in /rom G ~14 G", .,.., lei o . 0' .IUI 0" "" J4t HOIIlr "'....."""• .,.. G. G'. 0"",, G", !""tIptIdilO!l, .
(oj Up'" Io»rLl' OI>nIIont mtJlipIe of & .
L
I (%)~ . .. H ')J. is tbe idmtity. We identify A .. n l~ '~_ A • .. i,h n t 0/ A. MD,eo"" .
u'w:".....,..
[,..-' 0... 0,(H'f I(91'0. I») .. (p,(r, .... ,r,). 0..... 0) 1M 10111
:S k:S n.
yr
ate
PI\OOt': Lee. -.I .. ("jJ , SlS os~ I:IfI an tleml:nt of A , ~'" (r (IoC ill ~
+
I. R ) 0
Sy P ,O!>O'i.ion 23.111.3, tbe Euler ~ of g are deli""; lOr alm(llll &Il, in
SO(,, + I. R).
Noor, b SO(3, R ), 1I'fI c:i~ tbe. ~ric moMinl of Euler aIIlIB.
LeI. t be .. unitary W!CWor in R', and let lbe plane Ii ortboJsooal to i: be ~ with tbe oriewion for .. hi
r{r(e,), ,,) - (r(4, el 0 >ie. , 'I'll 0 r (e" ,,) 0 [r (a, el c r(4 , "'-'- witb .... IW'IIO . ,
(bl (g{", I, ·.· . g{ ..... ,I) io an orti"'W"'1&ll:uis of E /(Ir I :S j:S: n and .(r< ...... ,I) __ I. lei U- . U _ J•• ,. (d)
U · U - _ /~
...... l hat
.,(g(,,;) _
I
.. ,.
i:laI&nalc ~' 0(. ) 1M ullitary liOUP relati.-e 10 ... . r ...- every , lo 0("'), det('U · At - UI _ det(M ), and .. det{U ) io I Or - I. Si""",
a,•..... , _ ha....,
10. .. ,.-... 1 ~
From tM
+ .. , + a ....... ' - a ,... , .• ..., - - •.
l.
reiatioCl U· . U _ I ... , . "'" ded""" that
det(U )· 'U·
A
-D) B
_
wtJds to for all 1 '$ j '$ n + I . Thoen. lor II:Vt'f)' , E 0(. ). the matrill of , in the buill (Il<wiDc ...,u1t.
Lem ma
.mc
I + X~
.. .
X , X,
·• 1 + Xl.. ,
X. X ,
X. X ,
·•
•
1+ X!_.
...
X.XJ _1
X . X ._.
X. X j
XJX. df:lM~ _,(X •. "_.X, _., X j ., .. _.. X. _.) = Xj X._ the ooIo.ctoc ol Xj X~ in tbe matri~ M(X , ..... X.) iI - X;X._ Expo.ndin( deUf{X ." " . X.) aloG.!: the last column of M {X , .. ... X.). _ _ ,ha, The,~lore .
delM{ X, ,, __ . X.)_ (i + X!)dc!.M{ X ,, ... , X~_,J-
L
XJX! .
,S.S"- ·
T.kIna; b- A the IOeId C ol CIl, (I, ... ..9. ) to ($Inhll. )p,,_I(I , •.... I~ _ I).COIh(I. ). AIIG. let I, be tho "metloa' .... (sinh'.""""" ) from R 0010 H,. For ~ n 'a: I and. "'" every R ' , write r(9) be lbe matrb: of (DM'~' . · . · . DI.('~•. I.(f » iD. tboo ~k:.1 I..- of R~·" . If .. 2: 2. write KO (, ) lor t be n "n ..... trix wbooe Ii... row "' obWlII!d Il)' multJplyllIfI the lirst row 01. r (f:. , ... , 9~ ) by OW 9, . and. "''''''''' ith row, b &1.1 2 S j S n . ill "",. 0( r (f:. .. ... '~). Similarly. ...nte L"(I) !Or the n" n malrix whole lim row isobWlII!d by multiplylD,g the lim. row Ql J" (... .... B.) by $Ia9" and. .. boee ith row. for all 2 .s; < n, I..,qual to the ill> row 01 r (... . . .. B,,). Now . r t.l1 .. E R, defi ... h., (a ) 118 tho! byperbol;" rotation 01 R - '" whole mau;" with ....P"CC lO (~" ... ,"... ,) ill
'... 0 0 ) o ( o
58 B ..
ODIh{a ) oInb( ..) $loh(,,) (:(JIIh(a )
.
n({.... "'SlS. s . ) B: , ..here at .. (0,2101. 8~ .. fII .") loT 2 ~ j:S
lnf(~ . n - I ). and B: .. ,.10 ) 0 ••• ., hll) , ... be",
]0. +0[. roo- tech '" .. (e: )'SJSO£" in B , put r (w) _
hIO) - QI(r, j···g.Cr.J
forI S k Sn-i
g._,
,,(. ) - 91(r,) , .. (~_ , )h,, (9;; ) A'IUi"l n acilotmly .,.....In,,,,,,,
2. Suw:-.Ioa!. 1'" (11) 10 linite and 8 10 1"_~,abIo. Let ( A.. },,2' boo. 1 , Ii", ooq . _ of I' . in~ _ oonl-'nlq A."""" IIoa!. I'"(A) .. itof I'(A,, ). Gi""B C ;> 0, ...... e N be oudo tJw I'I A~) 1"(A )+c/3. S...... ,10M (I So l'(oA~ &11'" CBl iaf~~ , 1'18~) . S...... ,Iw .."(o A ('\ 8 ) _ inl I'/.A ('\ 8,, ). Thou, I""", _ 3 and 4. 10 .... f
e..
r
,,' (A- ' l _ +00).
7.
""'beI'
Su_.1oa!. A 10 1'"...... uraI;oIo, and .hal A "" B 10 k> G and ",4 " 1"(A - ' lI' "{O). Moc80 .... C A -' oJ_ Oed ..... Ihlt 8A -, _ "-II. I.nIft>o< poi"" IIo ,hao SA and AB -t. 100.", ...
67"'"
co..
Ia\.orioo p T . Leo f, and" be ,_ ~ ~u.mben""" thal the 0Q1>"-li000 "" , +..", _ p bao DO ...11« "''''ioa n" p ) i .. ,Iw>. io, 0, 0)_ Dec "'.. t., B lbe "'bpcup { up (lio1" ," +".,.) : n, E 72, n, E z }
C""
of T , t., C ,be ...
* ....,B of T . and poo. A _ 8 uC. ~
z'
I.
"* doli... "" T .. equn.u...o.
O. Denot~ Ilr H I A) .~ ... o f . _ 0" G b wbid> ~I AI " ".(A n .....). i.
Show.hat H IAI 10 •• _
l.
Le. 8 boo.
....
........ P fO (_
compact ........ of
b . E HI A). 8 -.1.8 ....
Ii
A
0...,.
pOn
1 of e:....,d08l) .
,.( 8 1 > (lt2)I'fA). Show .hat. boo d"loint. C-roo. ,hAl H I A) io..-.
_~.hat
Let C boo. ,....."... " ...... k>eaIIy _ : .~, writ_ oddl.if A, ".(1,01., _ ol n B.) . B_ _ U.~ A. and n..l'" B •• --"W:V. .... • hat 1'fA.) + I'fB.. ) .. JOtA ) + ..cB). FlnaUy, _ tbat ".«,01. _ _ . Jn B.. ) .. 11(8,.) too- - , . 0 E A...
Not. do!fini . A. _
s.._ that
11(8.0) > O. B1 - " 1 &hd ~ 01 &..!,d•• 2, tho l u _ I : . - ".(C ...... - .) n 0..) .. ,,(A... n(o + B.. I) "' -.lln""'" "" G....:I 1$ .. ".(- 8".)II( A.. ) .. ".(A.. ),.(8.. ). M~. I \aI;to tho val ... ".(8,.) II A•. P.- 1Il00. I .aboo Illy .boo -.1_ 0 _ ".( 8 ..), Mel I ..... C .. /" ( ,,(Boo) 10. d.",en ... _ 01 0 , ~ that ".(C) .. ..cA_l Mel that C io .boo 01 A ... 4, Su_ that ..cB.. ) ::. O. W$ d 01 0 10
_~bIe.
So D '" ... pp( l , .. " j n 8 ... """'" tho • ..cD) .. ",8.. ), If 0 E A. ....:I I E I), ___ ...... (. + VJ n "- 'I • Jot -.y -..""". ~hood V 011. Co"d ude IbM " - + I) C G ....:I . 1101 0 + 0 c: C . Conoidet HIC) " (. E G, ,,({C _ .lnC) ~ ".(C)}, For -=1>. E HIC) •
.,..,..,.hal
Ellen '
~.
b
Cb&pter 23
tbe """,p' .... 01. (C - . )n C in C io empt1 aDd that ... ,k , ..... E G tieo in H{C) if ODd 00LI,r if C t , io iDcluded in C. llenoo D io Included 1ft H(C) aDd C t H(C) '"' e. Dod""" from pw\ 7 01. Ewoe" 2 ,hat H(C) 11M _"'''''" 1 _. Coo>cludo tho., H (e) io botb open ODd :)ta, pty. W. do _ ~ulr< lhat B ' . toi"", 0. 0..:..-1 E B . Then ",( A t B ) .. "' ( A + (B -I)) aDd /'l A} + ,.(B) '" /'l A ) t,..(B _ I). Dod""" from pula 4, 5, and 6 .hat
(. ) ei.her ",, ( A + B ) 2 ,.(A ) t P{B ) 0< (b) .bono _ ... opeD compooIce 1M ru.....1010 Z _ A( " + E) n E) io comin"""" 011 R t." Ea . . .. 2, it .... _ ....iIl> Ide",k&lly .... (Q - (0»,.. Pro.oo IILN tiLl. """"iM" • comrod_ .nIb
,1>o_oI,_ ..
our
n.u..
0I"iii.... _ump0 _ 1'! , ... , f\ , L ~ ~ oocIilible, but f\ io _ ~ _urabiIo. Put Pi, D - f\ " , ODd _ lho.! + D io _ ~-....bIo, .....n I~ C ODd D ~ lrnea\iliblo.
• . Dod""" from
pw\
e
e-
yr
ate
23. MM' M tt", tt
4M
T IA G bf, t . _lIipko,tl ..
~ of _ricoo
(:
~ )""""
that • ;> 41
0:I>d , E R. W• ...., ...... i(r G wi. b tho """" ... ;"i..., of .ho
~"P
H 3 " _ClZ+~
Nd GIl - be .. I, Since SL(2. R )
,.
(OID1IUlw..n
(> E C : Im(»
of Gt,(2, R ). it It
:> o} ,
,11&. SL (2, H) "P'""". 0 _ - I), 0:I>d liI&t 5L{2. R ) KU;
'1'.,.. ( I/ "'IId. * d~),,. ....
1', ria ( :
: ) 1.
SL(2, R io aad Itt .. I:>< "'" _ •• LtC > - (G> +b)f(a + E P _ that A It \1>"Wiant
under ... 3.
H_ _ u"' "" 50(2. R ). Fb< _ry I in 1f: (SL(2. R). C),~ ... • '~""Ion p by ~ by " • • ,",""
of
t
r ""
Oo.d""" from Pt OjX4i.;..., '13,0,1 .hat A' : 1_
~n
io a H.... a
.""" ow
SL{~. R).
r-.
-..pee.
I. i l K .. _7 .......-i.. _ _ . IocaIIJ' &old. it io P*iblo ",dMcrll:>< lI_ 11".. _ SL{ ... K)( N._-'fUl,. ' r' ,~Cbpt .. 7),
'I'
ate
24 Convolution of Measures
<JI _~_ "" • Ioeally __ pert ",",p G io 01 flnt Impan...... ia ...,.jyoIo.. l.' {G) beioIc. B·n..... ",.. bea.. ..... QIl owIy..,.,...oI u..",o_ portai .... 1",10 Ilwe oIS '''' ''· B"I - , ;. .. ""'l' inten.ti"fl to 1100 I0:Il ftJUlar '"Pi ''OCioAclC""L' (C). Conwol~lioa
detno,,_
"""'.Ion of _ .t,' .... Two " ol**Yt..,... ....... -m_' , "1 oad """"It;""" 10.....,. _ _ _ W'1!,. 01> G"""", i>r.-,.!\metlon ,110 U {GJ ( .. it~
IS, S "-). ,. &ad 1_ ........,t.abIe (~_ 7U.1 """ 24,0).
u..s Focl :Spr"",,,,,",,,,hk, for
notry ; in J. GM Ih.at 1",1),(1 .. ~. Q),.,.' ,,/U 1'11ctioa (:r;)o£l _U) -
(f 0.")((,,,)... )
from
Xi -
n x,
into C.
"" - I!)
fu.l E Xi' Now. fur"""')' t :> 0, dim: erl8t1a ~ ... beet K 01 X j .uo:h tb.t lvl(X; - K) s ("/( t D/ U) (wt..re" ,. ® "'HII I'lM (and bouJ>ded ). We
.r
Io(n)d{JJ. ~ ,,}( •. z )
.r r dl'{.}
F\U"\hcrmore, I
_
I>(,.,) ct&-(z).
r lI(n)""'{z) .. lower oemIco!1l.in........
Now let. K be. 0)IIIp&
r
gd(jJ. . ,,) ..
r
(g - lK) tI(;. · ..) -
.....
Sinoe, -
r r r r ""(1)
g{uj dv(z),
4(.)
(g - IK)C", )dv(Z ).
f g{u)dv(z ).ad • - f Cg - IK)(-':) " (r) jIKd(JJ. o,,) ..
NO! jJ..ln~bie,_
jgd(jJ. o" j- jl.9 - I K)d(p o ,,)
., / .,.,C..)/
1,,{ttZ) dv(z)
..
1",l{-l{,j,,).
/ djJ.('" /
_un:
But . fllreac!>. E G . ... (.) ... 101. wltlt ~ fJ. H~. if K iII~ne&llciblt. it ;. aIoo 1' . "'Deglipble. n.;. Ili a'eo tIw " . ... if &" : 5$1·.-e ",ith bwo 8 . 0
o..HnlLIon 24.2.1 Let p be. "",&sun: orr G and let. f: X _ C be locally .6-~.t>ie. p and laM oald to be cortvolYl!.bie (with...spoed 10 8) wt:mcv.:r
/J and I~ aM conYOl ...hIe. In this ca9C, ""Y ~1III;ly of 1"
(J~)
w;lh I 1* aeling ctmli.."""", /rim> t.V IqI .... 10[ .wJo thaI 1"x (. )fJ .. :d'-" ·18 1M ."" ,h • E C. 111 ...., 1M "'" mm /J .,.. G, ® ~J io tk i~ ......... re 01 I' ® p 14 ~i.mI ('. zl"" (., . - I Z) from G " X "",to G ~ X .
xv.
"I.
P ROOF':
I
Let F Iii H !. ..... I(.-I:O)X- (.) io ~Ir /J-"'tqnobk ~ , ~ fur " koo:aIlr " . ...,."t~ 1Ot 1' 0 tI is tbe ima&t measure cI. (x-' Ii) 0" under tbe l>ooiofiClll>Ol"phjom
r.,.,).... (', . -1.,),
If
h(u )/ {z)I K{.)otp{. )dD(.,) ..
If
h{.,)/ (, - J.,)IK(·)X- I(·l d,,(.) " jj{.,j
(I).
N_ (.,z) _ lI(z)/ (.-' z) I K(' )X- '(') i!I "iii p.mea&unble. wit h compact support.. ~,~ ria:bt-Iw>d" of (I);" equt.! to
r
"11(z )II(z )
r
I{.- ' z )h "(')x-'(.) d,,(. j .
r r
• nd -'0 .,.. than
1111
d/f{z ) Is(z )
/(. - ' z)X-'(·'''''(· )
o
(where S '"' wPP(II)). Tht proof:' Qlmple\e.
C ond Id I : X - C k I«GIIr ~I ~ 1"" 0"'Mf ",,,diliQ,u MlIb.:
Proporoltlon 24 .. 2 .. 3 UI" k . _ _ 8 .. ;nlCf' ,6'e
(I)
I "
S..,.,.,K
1.\0;1 _
~N
(ONl in ........
(~) G GCU J"op< "~ II X oM f t'Gn~. ~tIII""" G "m:" .. oj """"~r ""''''
_JMd
KU.
(c) " .... .........,
...
~
•
~_
oj :>llAlUO, .... n' _,...:' flu .
.. euolllHllr " .int;"W. aupf for. 1«:01/, 8 .. n.c:giigolok Id of oahmpKt --". bmc:e Next . "
II 141 1.oXe ~ ,.. (.) t be ....tinuous endomor-
.
phlsm I -
~
"~(')I cl
L'c(fJ) . Then ..... ,.. {I) "' .. ~ulion of G on
L~(lJ). ~"e,
24.40 ooIu'ioIooll'EM{G.Cj aDd/ EZi{,8)
~13
Theorem 3~.~.1 Ld I' "" .. "'OOPM ... "" G ndL thCIl X-I'• .. ,, · in.t~.
end kll (e)"
E
171ft).
n-
0"'" I-~;
(') ,,t!.olmqJI 0(lfJ) .w .. 4enmr " • I E (O(p) ...110 oj QI ~ ~/rmr>Wt.
ruped to
p~tion
fl, ,n- 1«:01",
U.I.3;
(e) N~(" • I) ~ N~(f)( f X-I' . 011,,1). PROOF: X- I, o", and., ", iI arriod by a counl.able union S 01 com~ II!U. Let 11 E H (X ,C). Sin.ce ,, @/1is the lmqe rroe.~ of (x-',,)@ /1l1nder (. ,:r) _ (•. :r j, 'ho function (,, :rJ _ 1(,-' z )X-'(') ill ,, @t'-..-urabIe. Then (.,:r) _ I1(:r)/(,- ' :r)X- '{') ill ,,@lJ.rroe.urabIe, and
,_I
r '"' r
1"(:r)/(·-' :rlix- '(.)1,(.) d(l" I@t!){ •. z)
-r
011,,1(,)15(')
r
1I11:r)/(·- ':r)l x- ' (.) dDI:r).
r .r
1"(:r)f(,-':r)lx- 'I.)dD(:r) ..
1"{:r)I(·-':r)l d(7(.)fl)(:r)
1I1('~)/(~)ldp(lI)
ill Ie. , .....
r
"'.II)N.(7. -,(I1»),. "'.(f )X-'/.(.)N.(I1). Therdore,
1"{:r)/ (' - ' zlix - ' I') d(l"I@fJ)(" z) S N.(I) ( / X-I', dI"l) N,{I1),
and
(. ,:rJ _ II I:rJf(.-' :r)X- '(') ilL .-ntially ,,@t'-im~.
~umtly,
(',,) .... I1(.~)f( ~) .. _otially I' @ t'-lo~ \0 short, " and It! _ con¥Olvabie. Next. , tho mappiILS • - "r~ (.)i r..... ' G into L~1ft) iI cootinUOUl and N.(7. I.)i ) COl.of Tiwo>rem 244 .1.,. &nd / aN: coovo/vabW. P ropo> llitiol> 24.2.3 tbc>w1 thal 9 : r _ j{.-' .:a:)X- ' (. ) space. Finally. t". Section 24.1 . tbe "01....oh.tion in M~(G) io 7'
-i-' ive,
Clearly, tbe _un! t o. doIi"fd • of G, io . " " it [" M '(G) .
0
t". pIAci"l tbe unit . - &I tbe unit elenoeut
Now let G, X , tJ, x be loa In Section 24.1. e.nd ~ p, f (wil b ! :s p !f +0::» be t...o COI\iup$e 0.»01 .....1&. Fr...:ll ,.. in ." _ M"-''' (Gl. _ ~ l>f "r.v.) the O)lIlim"'" ~"m
j .... p-:""! of L~ (tJ) (Theorem 2U .!).
p ROOf': The _ un foIlo."s from !.he floCl thaI (.\. . p) . (ltJ) all .\. £ A., ,.. E A., e.nd 1 E VIP).
~.
1$1 . / P) for 0
N.,.- ~ lhoI..-e in ..·hid> X _ G , IM ... tl.oo ol G on X 10 (1, 1) - ~I, and tJ II l left U.... _ "' on G. lben X .. I The mawilll "r" (which DOl depeod on the "h 0 JJ.iolqrable. ~ro:£o"" (. ,z ) _ , (z)f(n)ol>(, - l) is (x") 0 .8-inl~. and (. ,z)_ g(. - ':r)J{ZjV(. - l) ill [email protected]·""d>le. ~Io .......... , ~,
1 ",
II
g(.-l r)!(r )w{, -l) djl(. )dP{:r).
From FUbini'. theorm>, ..., njtlgctc _!1D1lJ (1 :S p :S +00) . ALN, kt 1' ... . ............. 0 1..... U ' ''", w l3 fJ 01,. p undtr (I. zl .... ( I .•-'.,). and
eo
r
I/{.z)g(, - ' %)11,(.) 4< s 81(',%)
'" r
1/ (.... )g(z)l ls(.) "lUl'l 8 81(•. "j.
1Wf ~ ,he liM : .: .ilion 01 ,he ou,temenl. f ilially,
l~ppQII:
r
tbat
I/("IA'- ' zllls(')d!I' ® tII( • •,,)
"' finite. tlw ii, (1,:1:) .... ! ("'lft. - ' z ) l $(') II ,. 8 tJ.i"t~. For I'.a/mc:JoIt all , in S, .: _ / (.:jg(,- ' .:) it 8- i~bk. Thuo, fur locally I'.al~ all • tn G,..,'" / (Sb{.- IS) ill P.!~ MOlen..,. , the functio"
X/ · t,.-
f
J {.:)g(.- '.:) d8(:r)
;,; -eo1i•.I1r ".~ a.nd
/ (xl .
j ) dp - / /«.)g{. -',,) l s(.) 4(p @.8X.,:r)·
On th8 other Iw>d, (~, ,,) - l(u)g(z} IS(')X(~) ;,; ,,0,8-intqn.bie. Honoe, for b;allr.8-aJ"",* all z E G, ..... l (n W(z )ls (' )X(' ) 101 ".lntq;rBbiI:, and ..... I(unc,,)x(~) ~ !7m'iaIly ".~. Now .... I (u )x(,) 101 .-:nI.iaU)' ~ln\.eVlo~ lor locally ~,8-al",,* aU • E G. and . ... f l (u)x(')J,J(.) 100 _ially ,,"I~ FurthermII. a locally """'I*" group G. We .. rite X (.... pealYd)·, x'J for tbe left multi· plier (","""",Ivd)', tho, "Iht multiplier) of 8. Recall llur.l tbe ptOPffly, for a
2~.
-180
Co,.."ohtloa of M r . •
oompltx·.ct.lUoN! function on C, to be kM:alIy 6-ln~.bIe doftI not depet>d "" the O. thon and
•
•
•
I · ~ .. (/{J) . g .. I · (~Q). If ODe ollbo con\'OlU\ion produo:u ol f and g", conlillUOUi. it it Wliqucly ~. &tid It II called tboo convru" amtill_.,.. ...m.Ilu ov.,; .. " vnimeIion I . , II Ii...... '-Ilr fj·Ill"",.1 ~ 6y
(f . g)(:r) -
/ g(.-' :r)/(')x-'(,)d,6(.)
_ / 1(:r.-')g(.)r'(')dP('j
{I}.
o
PItOOP: This &:.IJcrA.s from Proposition 24.2.l.
.s
The-re m 2 .... 6. 1 ~'" GIld q lie 1_ amjtlgGu ~nU (I p .s +coJ. If Ix-' /' E er(C) ..ow: , E O(G), IN:n I GIld, ..... ..,.,.....m.w.., I • , .. ,;..... /oca.!lr (J·lllmQM. noellUJ (1 S P +00). Deli"" F ... filter OIl an index OIet I . Finally, let (Pj)~' be a family of """"""'" 011 G witb tbe foUo...iug JIf'OI>eI'lies:
s:
(Il) )C '/~ ill ",~;nt.egrable, for all i E / .
fbl
~,. ISlIP""
I x - llo "1",1111 finite.
fel / X- I' Odp.; ~ to 1 along F . Cd)
-,
For~' C'Ompect neighborbood V of ••
Iv. X-I ,. oil,..1OOD~ to 0
PR()OF : Fb: , E CO(e). Gi...., • > 0, let V be ,. rompect noighborhood of e IIUCh tbal. h!
'W
r r
N,ue exiou. oompoo:t nci.ghl>oU. ~ erilu G ..,m»Od .;1l :5 t for ; E A. Tbt;a
J
a-
Iv.
g(.,j - ().. • gl(z) - g(zl
+f
lv
TIl"" £ot ol1 ,
(1- Iv d(x- lII,))
(g{:r) _ g{.- ' ;oj) d(x- ' ",1(. ) - (
lv.
g(,-' z) d(x - ' ",1(') ,
e A and foe all '" E K ,
Ig(%) - (1.; . g)(:1:5 lr.up Ig{.)1+t il + t .~K
sup 19{:)I·
'fS- ' 1
c&Ilr
Ig{.-' r)1
.,
Iltr ) - (I• • g)(r)l ::;: 2€~» Ig(:)1+ ,,, +tN... (g) .
o Pro_lt;,;,n 2".7.6 S _. til,,1 J, E 1'/(G. C) for 011 i € I . MOOWOCi, AI.um. """'" "':'ta, for I!CCh """"""'" ntigJll>arllrml~ "n K . PROOf': OJ,,,,,,. > G, lei V be u in PrupositiQn 24.7.3. C , - A E:;1IO that V (JQlIlaI ... supp(f,) OJ>
o /;ow , ..... 4"" PI" "''' the usual .... ulu mati"" 10 tbe ~izuioD. of ,,- • ••..,... (III • Iox:alIy compACt. Next, ...;: «i"" an application aI re:n 24.7. 1.
"""p.
n-.
Tt-o ....m 24 .7.2 1d G be: G iooIJly """'Jl"Ct ,...,..,., aM id 8 (""""" fJ :> 0) /10. II re/lI.lmel) in...rieI", ......... ~ "" G. Efvi, L ' (/1) ooith 1M m..uip&&li.... (/ ...) -
~
I ' g. Tho< .. doNd _lor '~qIIa: W of L'(P) N II kft ""'-l oJIM. ~
BaMCh · '501", L ' (fJ) IV .
,E
if GM nIr if n')'
~ IQ W
I'" all •
E G n
H(C I K ) n H(K\C) .. H {C) n C(K\C! K). Equip 0 wit b ... Left II...,. _u'" "'G. H {G) is .... u~ (with ftIIp«:\. to 0DI\>'(I/L111on) of tbe ~ ~(C) (b«aLIM I ~ ouwon of "'G is t IM: whole '" G)- II foIlowt from (1) and (2) thai H(OI K ) ;. a lefl ide&! and 1i(K\C ) • ... ticf>t kIeal in HiG). Tho: 1nl('ll'(t;oo H i fl.'\GI K ) .. a ... ~a of H(G) (and aIoo of LhiC) . LeI. "'K be t he _maliud U.... rmMu~ 011 K. If _ put
IIi') _
j'i(
I {III') dmK (I ) ,j"'K{t' )
K •K
I'
ate
lot all functiono I E CIC ), ,he mapplq: I_I' is 8 prOjoocUon 01 d~ _ _ _ C(C ) om .. tho: _ ....- I ~ C{K\CI K): , hIo IoIIooft dimody from lhe Itft. and ~ iDvariaDoe..r mK . Furwmore, ,"" projediolll - P majlII 1I:(G) on\. 1((K\GI K ). Definition 24 .8 .1 Ir ' '''' uIled.,. Gd&nd pair.
alKd>ra 1I:{K\G/ K )
is COOlmuUti>"e, (G ,K) "
PTopa.;Uon 2".8.1 S _ 141 (G, K j .. a Gel/aM ""ir. 171m C ......i· .........T.
pROOf': Let.o. be tho: modular fUDCI.ion on G. A• .o.{K ) it ,. t"OmptlCt IlUtwoup ot ~ , .o.(K ) _ (II , and.o. _ I on K . For eood! I E 1I:(G),
11'dm(; - 111
- 11 '
(tot'j d"'er-oo ......... E G oudt l bat :r' _ u and ,; .. . ~. Proper-oo exi'lt.l I e K IlUCh that .-' ~Iortp 10 IAK. Th"" I _ I belongt 10 K .K. &Od ... may &pply Proposition 24.8.2. 0 Now..., Ill¥'
('T)1't)/oi nh('y) to (, - CGdo('y>:O)/oit>hh). SlDoo. _ g(",), tbe ... t lon 01 G on H~ III; doubly trantlti.." and (e, K) ..... Gelfand p&ir. When H~ if! equipp«i with the Rktt"""
Suppc.o
:I.
to
IJII.
I " ~ " -+01>. . be
01 ~(fJ} io eq..al
4. ,...... b a.be c,-clic pOUp Z/3Z. l.d .. be . _U ... OD. G. -.nth "'wort lhat; _ "('1 /11'1(0 doe! ""' ...... 00"11&", val_ S'- that 1110 """" '" ..... """,.....;.." ,.u.) • 1 - .. . I of ~(fJ) if 1Ir1c11y ... I.b.o.>
G.""""
W. bl < , (I, ......u ..... O < f $. Ill....,.. ..... H(..):5 qo lor 011 .. II' (0.6)Cboooo " 00.11&1 ' /" :5 , . Show .br. lu . ~._ , )«) - /ml lo .....han G~.
'1' , , /.'n, r, ') , (') '2 (....
"H -
I
+- H
H(,):;;q+2n
+ 270
,,~r
£»r:r
Levi'. t'-"em . 42
BertllUin poIynawial , 161 bi_niallaw, 130 Birkbolf'. crpIic .tot... """ 237
II<m>J ",.algebra.. 30 BoTeI ', normal numbo:< . 1>0" hm,
'"
Iknl-Canltlli ~mtnM. 31$, 123 I>ound1', th,,(H(w, 171
_"'C"""LICe in I..,., 3Z7
Boo",o oJ~&. 20 Bon!l IOta, 30
QOn~
in o>eMure. 105
0>0\1' "'3''''''' In probt.bility, 312
..
....
,
~nce
in tbe!MM. C.
produd, , 490
dow~......d-nIinuill , m.flH equioalmt fuoct ....... , ~ equi~ maoumo, MY e~ ID&pJIiq, 239
Apace, 2l
!13.
Fatou'. dlOOlcm. 49 Fq.. leer",,] , 496 F~tu"'" ~
""ier' . formul&, 2J.O 611 ... of lleCtiono, 18. finiU: variatioo, 1.6 F",,~ 1m ...1on lMoo,,,, . .m Fourier transform, 296 Fubini'l lheorem, lS3 funcbon of bounded "",I,,""' , 2Il6 G'pIIoCklng, G~, G~l&Iioo,
directed STOOP, 5 Dirkftlet
d~
:u.s
G."", . 419 p.mma function, ~ 2tifI Cebeoo.l>tt po/yDomiaI, 2J..8 (;elf. "" !*it, 48!1 Gliwnko-Cant.elli tbeorem, 3.\l Gooclw'ov'. thamm, 3.Y Gram'. dd~rminMl, i l l
Gram '. onlooconaliution, ill Haar m 'F ' U"'. 429
flal""", ...u, 30 Hamburfler'. """"",01. probiom,
'"
Hamel b&$io, 110 Hnuodorlf'. I ..,.,..,01 pro~ l.62 Hermite fundior>, 2J.6. Hermite poIynomial, :llIl HOlder'. inequality, ~ by~boIi(llmal law , J06 _mal ~! 1If>I>':f:. l3oI. nonnaln..! H..... meaaue. Ufo order dual, H Otdet 'deol, II (Ir't\er of .. u ' 29-1 order stalOn .... 3&S order ....mmabIe family, 18 order·boo'nded lioear Ionn. 14 «deled direct tum, to orde.ed V'O'Ip, !l Oidued _ 1I*>e, 9 wdlI.t.UoD. l1iIl
.rt.
,, -
~ ,booow~ j)on
....u hnur form, III 1ft!
p&tt
U&Ula:r
IIlCOI8U""
poW _dina ...... l50II
l'oIya'. o;riler>on. 3OB. poifItI~ neu~nt.
II.
poo.Itlve l i _ form, ltI politi.... _ ...,,"t. 34 poeitM pm , I prob&blUty "f*"", ill prod_ " IP ' ''''", 1& i l l prodllCt proboobilitr. 113 produt\~ m
prod"", I'eft'Ilril\f;. U8 produin~ ,!>leo fuDOtioa.!l6 Mandan! deviation, m 1IIandan! DOI'1Il.&! ......""' . 29.'> StioMtp. _un:, no St~I)eo.' mOlDent problem, 1.IW StoDe ~oc.ation t~, 20 Sto~' 1 conditlon, .ill Blrlctly regul&1' meuu"" 1tl6
m
at"""
law 0( Ia,. 1lIIlnbf,,., J2I)
strooc 8emirinc, 39
••
.ubdivilion OO . ill unibml), lntqn.blo ..... , lOS unimodillar ~P. ~
YooidA '"Pi Illation t""""'m, 23 YooidA- Hewill J.wmpoeitlon tbello..", ill
Copyrighted material
S)'!llbol Index
A"': com, ! metlt rL A (In"
&I"""
qM(O, C ): ..... 01 QUIIIt_1UtOJi U
~)
M (n, C): 'I'N'"' 01 f)e,,'el J
~- "
A - B : diBae..... A ll B"
QM(O, R ):
itA: ftIIIIkl .... 01 fu .....1on J to A (a ,.): Interval In R willi endpolou .. aDd 1[
"D
\0," : opftI ild.cn&l1II R :H{n, C ): _ _ 'pace town· me 01 oomp!ex· walued
- "
1I:(n. Ry. 1we d. '--.1""" fullC\ioQf III ?tIn,C ); 1.6
'H"' : "'" oI""';dve ru"cUo.. in :H(O, R ); 14
~of
ral quasi-
-"*" "
I,,: IryIk"," fund ..... of A (on"
" ..... "" OJ
_
M (O. R ): of - ' Donlell , n IlreoJ: U M +, _
01 po::.iti.... ~1 " :.-,..... 1.8
1': CCIIjupte of c.be
Oo,,!t!!
" ' U.IWp; U
LuI or
v~
abooiut\': ~. or ~kIo, 01. ,.; 16
,." and 1'-: poIitive.DC! ' .."'hoe pmU
0111>'1 mil "
....
II; U
. (C)
or C: " .rln& ..."aud "" C;
"
r hIed IT
r
st(S.F):
"'*'" d. s..lmpko fuatUona from (lInt !.he >'IdOl"
"*'"
()( (I . ,,): imfCr.t of IE £).{p): 4.S
F ; l4
St~ (S): aet of pCIIilioe
,
,,(n. ()( f I dp. ()( f l Iz) dp( z ),
S-.iwpko
/'I, (J)'
~...JuaI
funa ...... 36
QM (S .C }, n
"*'" of u" on lbe ...mrinc q......
r IJldV" brfuDclioa lhe to(~ I:
£' {p: R}
" "*'"
QM(S, R):
IDrMW'tII
of..eal qUM!.
f: 1 (1)1!IuaI rullC1icm1 1 auoh tlw.
1r,(f) < +00: 68 q:{P):
aiozzo; IOl
_pooa!
L'P'(P) : quo!.lcot"1*'!' 01 ';'(... );
'" C(A.,.; F):
IpKe
01. funak>M lhal
..... ,....-uabIe oa A;
'"
... pp(f): IUwon of tbe fullCUoa
[I" l~dVpl '"
for p E /O.+OO[and potIlfve/:
"
.
N,(n: tf" I/1'4V"p" 1or
the
_,..h.rl func\.loll / ,
.r;(p): "P"7.e&f.ol
aI
~
!: I'e' IUID oilhe IWOmabie ram..
ouppVo): oupport aI lbe R......, "
-..... "",., and,...., 17'9 " ' pure; ISS
..-e ol,.J. R..s..
-EF"US;
01 ,be u.trw::l
.\0, .l:rdilllmlioo&l
_urw; 138
M {O, R ):
prod!!CI
'"rei"" (2'." ' _ I (2'}g"'{:t"'); 181
nS,: Infinite product of tbe
~I
zml-aLPrM S; ;
m
.'
®,.,: Infinite prod!!CI of u.troct Dp(r ): deri_ive of the -. 7 mff: " &I.
r, 243
O(n + I, R ), sroup of orUqoarJ Ira
ha •• 1C'7 ID
R~" ' ;
'" e , ~ aw"'"""", S~ ;
'"
VIM, J) or V(J): variulca 01 M ~J: 2M
Var( X ): YVianceof X i :l.J.J
®P, : prOduct 01 ~ pooo."biliUei
lIN: Stldlja" 7'N,,'" ·"ed "IIilh M i 2:1D
-.:,
f .. /: Une In\teraI; ru
a(( X ,)",,); a·aJpa gme,.ud by l be n.Wi&bIe X i 312
E( X ): ""pected ..J\III! 01 X ; ill
D(X): ~ ma:.rb: 01 X ;
'"
JI;
40.1
..
.-(p): imact" 75'", WIder "I: 01 tbe Radon ""'''PI", P i 1'@lr.prOductoi R..s.., " . mrs; ill
r-gtlted ma8"ial
MO
Symbol I""""
,.(.): t.......u.tioo X ,. r _ .., (G -.:Ii,. bun tho! left. OIl X ): 4'26
8 \ G:
~lIp&a!of~
"'{. )/ : fl.metlao r - J(. -' r ); 476
SO{n
+ I, RJ: gou~ of rowiooo
u -'
"'*'" of G modulo H ;
'"
In R M
, 4'9
' .
6{.): ric~ tn "'letlao r (G actinc "..., lbe riChl OIl X) ; 42S
S H { .. + I, R ):
6(.J/: (nDeCIao r _/(u); 428
G L( .. , K ); ,roup of lbe .utomoor-
IfOUp of t.ypo,rbolic ..,. •• ;.... ill R· ·'; 455
ph ...... of
x: left ...uhiplltt
pvm
of •
orUtr ..
ore; 477
m
/: function I _ 1(.- ' ) OIl G; 429 jl: i=,1" T
"
u",
Uoa prod"'" of m
6:0: mduLar fuHCtIao OIl G : 433 cI X ( H
),' : _ ' " _ ••• ; '~tod
wllh~ ;
441
or: znll~
cI /J bf
fJ;
G / N : ,p&ee of Ief\
twtt.I
aoodulo H; 446
cI G
/ : """"",,ulion product of 11>1: - . m t P &nd lhe
f • p:
r Ct ): mean of 1_. ftt-; 441
.,
1"
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funcUOOI f : 46Q
act\Qc OIl X from t be rl&bI); 438
1'/. : qootitm
tho!
*.u 1'0 ....t 1', . "' . 1'00 : ..........01...
...
of I' .......,.
~
{m.ti~ 1.0
"~Id K ); .s4
. _ .- ' ; 4211
X / H : ",... iont
m- p1JUp of
SL(.. , K ): ..,...w
rtl&Il ody inv&ri&nt
K- ; 464
aavo/ulioo produet cI ~be function f and lhe meNu", /J: 479
/ • " cou>1)luu.... prod"'" cI lbe functlao / &nd tho! {unction
g; 480
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••
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This book s "ntefldad to serve as
I:lI
e book for a course In
measure theory at a gra.dua e level lin pore or apphed mathe ma1lcs. Th inclusion of some more developed matenal makes i also suitable for post-graduate stlldents. Apart from some ease 1n malhema ical reasor'lIng , 11 [s assumed only that the reader IS acquainted with the basic: results of topo ogy and func lonal analy,s ls.
The approach adopted here is based on the concept of com plex Daniell measure. This a lows the developmen or a 9 neraJ theory ha e compasses both Radon measures and (real· or comp ex-valued) abslrac ' measur~s on families of subsets of a g van se . While the setting chosen by the autnor is very gen-
eral. the material Included In he main theorems IS eaGY 10 undersland and 10 use. All the essential lools or applications of measure theory In analYSIS and probabil" y theory are fully develo ed. Numerous exercises help he reade to clarrly he heory.
111
•• ISB
~387- 94D'M-'6
9 760387 9 6L43 )
