THE LEBESGUE INTEGRAL BY
J. C. BURKILL
FellOfD of P�ttrhcnu�,
Cambridge
CAMBR IDGE AT THE UNIVERS ITY PRESS 1963
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THE LEBESGUE INTEGRAL BY
J. C. BURKILL
FellOfD of P�ttrhcnu�,
Cambridge
CAMBR IDGE AT THE UNIVERS ITY PRESS 1963
PUBLISHED BY
THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS Hl•ntlcr House. 200 Euston Roall, London. N'.W. 1
American .Uruneh: :12 Ea�t 57th Street, Xew York 22, N.Y.
West African Office: P.O. Box 33, Ibat.lan, Nigeria
Fir8t printed in Great Britain by John Wright th Sons Ltd., BriBtol
Rt.pri·ut,./ by ojJNttl·litl,ography by John Dickens� Co. Ltd., Nortluunpton
PREFACE My aim is to give an account of the theory of integration due
to Lebesgue in a form which may appeal to those who have no wish to plumb the depths of the theory of real functions. There
is no novelty of treatment in this tract; the presentation is essentially that of Lebesgue himself.
The groundwork in
analysis and calculus with which the reader is assumed to be
acquainted is, roughly, what is in Hardy's
A Course of Pure
Mathematics.
It has long been clear that anyone who uses the integral calculus in the course of his work, whether it be in pure or applied mathematics, should normally interpret integration in the Lebesgue sense. A few simple principles then govern the manipulation of expressions containing integrals. To appreciate this general remark, the reader is asked to turn to p.
42;
calculations such as are contained in Examples
4-8 might confront anyone having to carry through a mathe matical argument. result has the
Consider in more detail Exam pie
4;
the
look of being right rather than wrong, but the
limiting process involved is by no means simple, and the justi fication of it without an appeal to Lebesgue's principles would be tiresome. Anyone with a grasp of these principles wiH see that the easily proved fact, that
(1-tfn)n increases to its limit
e-t, ensures the validity of the passage to the limit. The attitude which the working mathematician may take towards the more general concepts of integration has been ex pressed by Hardy, Littlewood and Polya in
Inequalities. After
dealing with inequalities between finite sets of numbers and extending them to infinite series, they tum to inequalities between integrals and begin Chapter vr with these preliminary remarks on Lebesgue integrals: The integrals considered in this chapter are Lebesgue integrals, except in §§ 6•15--6·22, where we are concerned with Stieltjes integrals. It may be convenient that we should state here how much knowledge ot the theory we assume. This is tor the most part very little, und all that the reader usually
vi
PREFACE
needs to know is that there is some definition of an integral which possesses the properties specified below. There are naturally many of our theorems
which remain significant and true with the older definitions. but the subjerl becomes «JSier. as well as more comprehensive, if the definitions presupposed have the proper degree of generality.
. Since Lebesgue's original exposition a. number of different approaches to the theory have been discovered, some of then1 having attractions of simplicity or generality. It is possible to anive quickly at the integral without any stress on the idea. of measure. I believe, however, that there is an ultimate gain in knowing the outlines of the theory of measure, and I have "' developed this first in as intuitive a way as possible. During several years of lecturing on this topic I must have adopted ideas from so many of the books and papers on it that detailed acknowledgement would now be difficult. My greatest debts are to the classical books of de ]a Vallee Poussin, Cara thoodory and Saks, and the straightforward account (having a similar scope to this) given by Titchmarsh in his Theory of I also wish to record that one of my many debts to G. H. Hardy lay in his encouragement to write this tract.
Functions.
J. C. B.
September, 1949
Reprinting has allowed me to put some details into § 2· 2 which had been left to the reader. The first paragraph of§ 2·7 men tioning the role of an axiom of choice in the Lebesgue theory has been recast. I might have helped the reader more by discussing this axiom at its first appearance-on p. 3, in enumerating the qets E,,.. To do this now would disturb the type too much, and I can help him most by urging him to read an account of the foundations of the subject such as is given in the books specified on p. 87. There are other less important alterations. I thank Mr Ingham and Professor Besicovitch for constructive criticism. Jun�, 1958
J. C. B.
CONTENTS
Art.
Author' s Preface
Chapter I. SETS OF POINTS 1·1 1·2 1·3 1·4 1·5
The algebra of sets Infinite sets Sets of points. Descriptive properties Covering theorems Plane sets
Page v
I
3 4 6 7
Chapter II. MEASURE 2·1 2·2 2·3 2·4 2·5 2·6 2·7 2·8 2·9 2·10 2·11 2·12
Measure Measure of open sets Measure of closed sets Open and closed sets Outer and inner measure. Measurable sets The additive property of measure Non-measurable sets Further properties of measure Sequences of sets Plane measure Measurability in the sense of Borel Measurable functions
10 10 11 12 13 14 15 16 18 21 23 23
Chapter ]]]. THE LE BESGUE INTEGRAL 3·1 The Lebesgue integral 3·2 The Riemann integral 3·3 The scope of Lebesgue's definition 3·4 The integral as the limit of approximative sums 3·5 The integra,} of an unbo�nded function 3·6 The integral over an infinite range
26 27 28 30 31 33
.. Art. .
Vlll
3·7 3·8 3·9 3·10
C O NTENTS
Simple properties of the integral Sets of measure zero Sequences of integrals of positive functions Sequences of integrals (integration term by term)
Chapter JV.
4·1 4·2 4·3 4·4 4·5
4·6 4·7 4·8
Page
DIFFERENTIATI O N AND I N T EGRATI O N
Differentiation and integration as inverse processes The derivates of a function Vitali's covering theorem Differentiability of a monotonic function The integral of the derivative of an increasing func· tion Functions of bounded variation Differentiation of the indefinite integral Absolutely continuous functions
Chapter V.
0/w,pter VJ.
6·2 6·3 6·4 6·5
44 44 46 48
49 50 52 54
FURTHER P R O PERTIES O F THE I N TEGRAL
5·1 Integration by parts 5·2 Change of variable 5·3 Multiple integrals 5·4 Fubini's theorem 5·5 Differentiation of multiple integrals 5·6 The class LP 5·7 The metric space LP
6·1
34 37 38 40
58 58 61 63 65 65 61
THE LEBESGUE·ST IELTJES IN TEGRAL
Integration with respect to a. function The variation of an increasing function The Lebesgue·Stieltjes integral Integration by parts Change of variable. Second mean·value theorem
70 71 72 75 77
Solutions of some examples
80
C H A PTER I
SETS OF POINTS The refinements of the differential and integral calculus, which form the topic of this tract, largely depend on the properties of sets of points in one or more dimensions. This chapter con tains those properties that will be needed, in so far aa they are deacriptive and not metrical. The mles of algebra applied to setB hold whether the members of the sets are points or are objects or concepts of any kind. All that we require for a set E to be defined is that we can say of any given object x whether it is or is not a member of E. '
1·1. The algebra of sets. Let E be a set,t the members of which may be of any nature. The aum of two sets E1 , E1 is defined to be the set of objects which belong either to E1 or to E1 (or to both); the sum is written E1 + E1• By definition E1 + E1 is the same as E1 + E1, no question of order being involved. The
definition extends to any finite or infinite n umher of sets, E1 + E1 + . being the set of objects belonging to at least one E11• In the definition of an infinite sum there is no appeal to any limiting process. of any number (finite or infinite) of sets The product E1 E1 E1 , E1, is defined to be the set of objects belonging to every one of the sets En. The sets En may have no members common to all of them, and the product is then the null set-the set which has no members. . .
. • •
• • •
t Class and aggregate are synonymous with set; French ensemble, German Menge.
2
S E T S O F P O INTS
If every member of E1 is a member of E0 we say that E1 is contained in E0 and we write E1cE0 (or E0';lE1). The set of members of E0 which do not belong to E1, may be written E0-E1 or a.ltematively as OEb the cumplement of E1• It is easy to see that, the complements being taken with respect to a. fixed E0, O(E1+E2 + ... )= E1• E2 ..., and
O(E1E1
0 0
• • •
)= OE1 +OE2+ ...
.
is an infinite sequence of sets, the Limit seta. If E1, E2, upper limit, lim En, is defined to be the set of objects which belong to infinitely many of the E,.. The lower limit lim En is • • •
defined to be the set of objects, each of which belongs to all but a. finite number of the En. Clearly limE,.';l limE3• H the sets lim E., lim E,. are the same, we say that the sequence E1, E2, has a liinit, lim En. If a sequence of sets is increasing or decreasing, it has a limit: more precisely, • • •
(a) If Enc;En+l' then limEn= E1+E1+ ..., (b) If E3';lE+ n l' then limEn= E1E2 .... To prove (a), write E= E1+E2+ ... and observe that if xis a member of limEn then xis a member of E; hence limEncE. The result will now follow if we prove that EclimE,.. This is true because any member of Eis a member of En for some n and so (since the sets form an increasing sequence) for all greater n and therefore of lim En. A similar proof holds for (b), the product set possibly being null. More generally, the upper and lower limits of any sequence of sets, not necessarily monotonic (increasing or decreasing), ma.y be expressed in terms of sums and products. The formulae are limEn= (E1 +E1+E,+ ..,) (E1+E3+ ...) (E3+ .. ) ••. , limEn = E1E2 E3 ... +E2E3 + E3 + ...• .
• • •
The proof is left to the reader.
• • •
3
I N FI N I T JJ S E T S
1·2. Infinite sets. Two sets are called Bimilar if there is a one-one correspondence between the members of one and the members of the other. Thus two sets with finitely many mem bers are similar if and only if each has the same number of members. The idea of similarity is the foundation of any theory of infinite numbers. We shall give here only those outlines of this topic which are essential for later chapters. With infinite sets we have a phenomenon which cannot occur with finite sets, namely, that a set can be similar to a part of itself. For instance, the set of positive integers is similar to the set of even integers or to the set of perfect cubes. Any set which is similar to the set of all positive integers or to a finite sub-set of them is said to be enumerable. The one one correspondence may be displayed by using the positive integers as suffixes, so that the members of any enumerable set may be specified as x1,x1,x3, It is clear that any sub-set of an enumerable set is enumer able. The metnbers of an enumerable set of enumerable sets E1,Et, . . ••••
.
form an enumerable set. For let the members of Em be enumerated as x,,.1, xm1, xm3, ... The me•nbers of an the sets then form a double array:
Xu
Ztt
Zta
Xst
Xss
X23
Xat
Xat
Xas . . .
••
·
• ••
. ..
This array can be enumerated as a single sequence, for example, by taking terms along the successive diagonals in order X u, Xu, Xzt, X13' Z tt' Xalt •
•
••
As a particular case of this, the set of positive rational numbers is enumerable, for they are all included in the set
1 1 2 I 2 3
I' 2' 1' 3' 2' I'····
4
SETS O F P O INTS
Clearly, the set of all rational numbers (positive, negative, or zero) is enumerable.
A further application of the same argument proves that the
point8 of the plane of whick both co-ordinates are rational form an enumerable set. For if rltr1, ... are the rationals enumerated, the rational points of the plane can be displayed
as
(rt, rt)
(rt, rz)
(rt, ra)
(r1 ,r1)
(r1, r1 )
(r1,r3)
.. .
•
r•
• ••
and form an enumerable set of enumerable sets. The simplest example of a set which is not enumerable is the set of all points of an interval. Take the interval (0, I) and sup· pose, on the contrary, that aU the numbers between 0 and be enumerated as decimal
x1, x1, x3,
. • ••
Let each Xn be expressed
1 can as
a
the u's being numbers from 0 to 9. Write down a new number,
where Vn is determined from Unn by the rule that Vn
unn =I=
1 and vn = 2 if unn
is not the same
as
=
=1
if
1. Then ylies between 0 and 1 and
any Xn, for it differs from Xn in the nth
decimal place. This contradicts the hypothesis that the se· que nee
xux1,
• • •
included all the numbers of (0,
1 ).
1·3. Sets of points. Descripdve properties. that E is a set of points on a line.
E is bounded if all its points intervaL
!_point P,
0
_
c-.,/l;,.h
of abscissa
x,
if there is a neighbourhood which be!�
T to_�--�-�-
-
are
Suppose
included in some finite
is said to be an
interiO!J!_�nt
(x- 8, x + 8) of P_�
of E
every point of
DESCRIPTIVE PROPERTIES
A_ se� E is_.�ai� �_be _{)2,e1! if �v_ e_ty_pgi _ nt Qf!t is�-i�te�or
polnt. The simplest open set is an interval a < x ... without its en�points .-.. ApointP, of abscissa x (which may or may not be a point of E) is said to be a limit-point of E if any neighbourhoo� (x- 8, x + 8), however small 8, contains a point of E other than P. It follows that every neighbourhood of P contains infinitely many points of E. The set of all limit-points of E is called the derived set of E and -!s denoted by E' . Weierstrass proved that, if E is a bounded set having in� finitely many points, then E' contains at ]east one point. t A set E is said to be closed if E :> E' . (For example, consider the closed interval a� x� b.) It will be convenient to reserve the letters 0 and Q, with suffixes, for open and closed sets respectively. We shall prove first that these two ideas are complementary. If Q is cl08ed, then CQ is open (the complement natural1y mitst be taken with respect to an open interval). For let P be a point of CQ. Since Q contains all its limit points, Pis not a limit-point of Q . Therefore there is a neigh bourhood of P free of points of Q, and so CQ is open. �·-
-
-
--
----
----
-
If 0 is open, then CO (taken with respect is clo8ed.
----
to
a closed interval)
For no point of 0 is a limit-point of points of CO. It is to be observed that the set of all points of the line ( oo < x FA+l -
and Therefore
-
lim E.
= lim F•.
m(lim En) = lim mF•.
But, since Fn contains each of E", E•+h . . .
,
mFn �bound (mE., mE•+l' .. .), lim mF" � lim mE", and so giving the result. The necessity of the hypothesis that the E" are contained in a set of finite measure is illustrated by the example E. equal to the interval (n, n+ 1), for which the conclusion is false. (iv) Lim mE,.�m(liJ!! EnJ· The proof is oomplemental'J to that of (ill). Combining (iii) and (iv) we have (v) If the En are mea8urable and are all contained in a set of flniu meaB'Ure, and if they have a limit set E, then lim mEn eNt8
arul eq'IUll8 mE.
We extend these results to sequences of sets which are not assumed to be measura.ble. The following lemma will be useful. (vi) If E i6 any set, tMre are meaaurable sets F and G IJ'UCh
that
and
Fc:Ec:G mF =m.E,
m*E =mG.
MEASURE
20
F and G may be called respectively a mea8'Urable kernel and a measurable envelope of E. Let En be a decreasing sequence tending to zero. We can choose a sequence On of open sets all containing E The sets
such that
G be the product contains E. We have Let
set 0102•••• Then
G is
measurable and
mG�m*E mG�mOn
and for every
n.
Hence
mG =m*E. A complementary construction yields a set F with the
properties stated. (vii)
Let E1 cE2c ... , and let E = lim En. Then
EncE for every n, m*En�m*E and so lim m*En�m* E. It remains to prove the reverse inequality. Let Gn be a measurable envelope of En, so that mG, = m*En• Since
Hn = GnGn+J•••
Write then
E,�.cHncGn m*En = mHn =mGn.
and so Now the
Hn form
an increasing sequence of measurable sets,
and if His their limit,
EcH.
Therefore
m*E�mH = limmHn
(from (i))
=limm*En.
This proves (vii). The complementary result is
(viii) Let E1�E2=:J •••• Then m (lim En) =lim m* En, if the
right-hand Bide i8 finite.
.
PLANE MEASURE
21
2·10. Plane measure. The theory which has been given in detail for linear sets in §§ 2·1-2·9 can be extended to higher dimensions. It is sufficient to consider the plane. The founda tion stone is the open set, as it is for linear sets. The measure of a rectangle is its area, and as the (plane) measure of any linear set must be zero .(since it can be enclosed in rectangles of arbitrarily small area), it is indifferent whether the sides of a rectangle are regarded as belonging to it or not. The measure of an open set is then defined to be the sum of the measures of the meshes of a network of which it is com posed (§ 1•5). To justify this definition we must prove that the measure so defined is independent of the particular network taken. Take two networks, which may have different origins and axes in different directions. Let the open set 0 be composed 00
�
of the meshes � r" of the first network or of the meshes � &�c of 1
the second. Using the symbol 00
1
p,
for two-dimensional measure, let us
00
suppose that � p,rt > � fLBk · Choose n large enough to make 1
1
Inside each r�, we can place a closed rectangle R, and about each 8k we can circumscribe an open rectangle Sk, maintaining the inequality
But the Heine-Borel theorem (§ 1·4) shows that the last inn
equality is impossible. For every point of the closed set � R, n
1
is a point of 0 and is interior to some Sk. The set � R1 can 1
therefore be covered by a finite number of the Sk, and the sum n
of the areas of these Sk will be greater than � p.Ri. This 1
22
MEASUBE
contradicts the preceding inequality, and the definition of the measure of an open set is justified. The theory of measure of plane sets (or sets in n dimensions) can be developed as in §§ 2•1-2·9.
Ordi'fUJU set8. Let E be a linear set, which we shall suppose to lie on the �-axis. Let 0 be the set of ordinates of height k erected on E, i.e. the set of points (x, y) such that x is in E and 0�y�'"· We shall prove that 1'*0 = km*E
and
IL•O = km.E.
In particular, 0 is meaBUrable if and only if E ia, arul then fLO. = kmE. Enclose E in an open set 0 with mO <m*E + £. Let I be a typical interval of0, say a < x A ) the set of points of E0 at which f(x) > A , with a similar meaning for E(j� A), etc. The function f(x) is said to be mea&urable if, for all values of the constant A , the sets of one of the four families E(f> A ), E(j < A ), E(j� A ) , E(j � A )
are measurable.
24
1\ol E A S U B E
We prove that any one of these four conditions implies the other three. The sets of the first condition are complementary to those of the fourth, and similarly for the second and third conditions. To prove that the first condition implies the third, we observe that the set E(/ ?J:. A ) is the product of the sets E(/ > .A - *) for n I, 2, . . . , and so, being the product of measurable sets, is measurable. Similar arguments will complete the proof of the equivalence of the four conditions. The following sequence of results establishes the general principle that elementary operations performed on measurable functions yield measurable functions . (i) Iff i8 measurable and c is a constant, then J + c and cf are =
meaBU
rable.
This follows easily from the definition. (ii) If I and g are finite and mWJurable, the set E(f> g) is
mea&U
rable.
. If, for a particular x, f(x) > g(x), there is a rational r lying between them. Hence
E(l>g)
=
:E E(I > r) E(g < r) f'
summed over all rationals r, and is the sum of measurable sets. (iii) If I and g are finite and measurable, 80 are l+g and f-g. For E(l+ g > A) = E(f> A - g). The function A - g is measur able from (i) and the set E(l> A - g) is measurable from (ii). (iv) With the hypotkesia of (iii), fg is measu rable. If A > O, E(r > A) E(J> .jA) + E(J< - ,JA), which shows that the square of a measurable function is measurable. A pro' duct is reduced to squares by the identity =
4lg
=
(/+g)2 - (/-g) 2.
be a sequence of 'meaBU rable junctiona. PM:n M(x), the upper bound of the valu& at x of f1 , /2, i8 mea8'U rable. So iB the lower bound. (v) Let f1 ,j1,
• • •
• • •
M E A S U RABLE F ll N C T I O N S
25
E(M > A ) = ");E(fn > A),
For
n
which is the sum of measurable sets.
(vi) The limit of a
tio'IUI is mea8U rable.
mono
tonic sequence of measurable func
For suppose the sequence is increasing (/n �/n+1). Then the limit is the same as the upper bound, and (v) gives the result. (vii) If f1,j2, are measurable, 80 are the upper and lower • • •
limit functio'M of the Bequence.
Define Mn(x) to be the upper bound of fn(x), fn+l (x), . . . . Then, by (v), .lJfn (x) is measurable. Also Mn�lJtn+l· By (vi), limMn is measurable. But this is the same function as limfn · (viii) A continu01t8 Junction i8 measurable. If f is continuous, it is easy to see that the set E(f�A) con tains its limit-points, i.e. it is closed and therefore measurable. E X A M P L E S O N C H APTER II
( 1) Prove that Cantor's ternary set (Ch. I, Ex. 7) has
measure zero. (2) Let j(x, y) be a measurable function of x for each y, and continuous in y for each x. Prove that lim/(x, y) and limf(x, y) 11�a 11�a are measurable functions of x. (3) Let f(x) be a measurable function in (a, b). Prove that, given E, there is a continuous function �(x) such that J/(x) - �(x) I < E except in a set of measure less than E. (In general terms, 'any measurable function is nearly a continuous function'.) (4) Egoro!f'8 theorem. Let the sequence of measurable func . tions fn (x) tend to the finite limit f(x) in E. Prove that, given �' we can find a sub-set of E of measure greater than mE - 8 in which the convergence is uniform. A rough expression of this important theorem is that ' every convergent sequence of measurable functions is nearly uniformly convergent '. 3
CHAPTER III T H E L E B E S G U E INT E G R A L
3•1.
The Lebesgue integral.
The idea of the definite
integral which has come down through the centuries associates
J: (x ck ! )
with the
area
bounded by the curve
x-axis and the ordinates x
=
a, x = b.
y
=
/(z), the
Having developed in the
last chapter the concept of measure of a plane set of points we can, following Lebesgue, present the idea in a refined form. Let
E be a
set of points
x (which
may in a special case be an
inter .ral), and j(x) a function, supposed in the first instance to be positive.
Let fi be the plane set of points
(X, y)
such that
X takes
all
E and 0 � y � j(x). n can be described as the ordinate set of the function j(x) on E. If n is plane-measurable we shall say that j(x) IUUJ a Lebesgue integralt in E, written
values in
JJN.
and so there are points of R which belong to no J" for n Let x be such a point. Since x belongs to no In, it belongs to an /, of length l say, of � such that II, = 0 for n 1, But I has points in common with an I, for some n > for if not, l � kn < 2l.+1 for every n. Since lim l. = 0, this is impos sible. Let n0 be the smallest value of n for which I and I" have common points. l � k,._1• Then =
..., N.
N,
>N
But n0 and so, from the definition of x, x does not belong to Jn,• Since I contains both a point of In, and a point not belonging to Jn,, l � 2l._ > k,._l , and this contradicts the preceding inequality. Hence mR = 0 and Vitali's theorem is proved. We add two coronaries, the first of which embodies the form of the theorem most useful in applications. £,
CoRoLLARY I. Under the hypotke8es of Vitali's theorem, given we can find a, finite number of disjoint interval8 11 , In of � • • •
,
such that the ooter measure of the set of points of E not COtJered by them iB le&s titan E. CoROLLARY 2. The theorem (and the proof) hold in
more) dimen8iO'Tls if intertJal8 a,re interpreted
m.).
atJ
(or sqootvA (cubes, two
48
DIFFERENTIATION A N D INTEGRATION
4·4. Dift"erentiabllity of a monotonic: func:tion. The object of the next two theorems is to prove that a monotonic function has almost everywhere a finite derivative. We shall assume the function (x) to be increasing. The set of values of x for which one of the upper tlerivates 'Of (x) is +oo ha8 measure zero. Let E be the set in (a, b) at which D+4> + oo or D-q, + oo, and suppose mE k. Let K be any (large) number. With each point x of E can be associated a sequence of intervals for which 114> > Kax (where 11 is written for (x+h) - 4>(x)). By Vitali's covering theorem a finite number of these intervals can be selected, non·overlapping, and of measure greater than !k. Summing over these intervals we have "l:A > JKk, or, since q, is increasing, (b) - (a) > lKk. This is false for sufficiently large K unless k 0. The set of pcnnts at which an upper derivate of an increa8ing function i8 greater than a lower derivate ha8 mea&Ure zero. Consider for definiteness D+4> and D_ 4> and suppose that the set of values of x for which D+ > D- 4> has measure greater than zero. This set is the sum of sets E(u, v) in which ' D+q, > u > v > D_ q, =
=
=
=
and u, v are rational numbers. There is then a pair (u, v) for which E(u, v), E say, has measure greater than 0. Enclose E in an open set 0 of measure less than k + E. Any point x of E is the right·hand end·point of arbitrarily small intervals (x- h, x) for which fi 4>(x) - (b) and, since
f
J
a+h
a
1 a+A
(x)dx � luf>(a), we have lim k
4
(x)dx � �(a).
This proves the theorem. It is interesting to construct a continuous function (x) for which the sign < is w be taken in the theorem. Let E be Cantor's ternary set in (0, 1) (Ch. I, Ex. 7). In the intervals of OE (and at their end-points) define (x) as follows. For l � x � !, (x) l. For l � x � :, �(x) l, and for i � x � :, (x) = :. For /r � x � /.,, (x) = }; for t.77 � x � /.r, (x) f; and so on, repeatedly trisecting along Ox and bi secting along Oy. (A diagram will help the reader.) At a point x of E which is not an end-point of an interval of OE, �(x) can be uniquely defined as the limit of values of � taken in a sequence of intervals approaching x, and (x) is continuous at all points of ( 0, 1). Since '(x) = 0 at all points within an interval of OE and the sum of the lengths of these intervals is 1 , c/J'(x) = 0 p.p., =
=
=
and so
J:�'(x)fk
=
0. But �(1 ) - �(0) = 1.
4·6. Functions of bounded variation. Let j(x) be de fined for a � x �b. Take a set of points of division a
=
x0 < x1 < . . . < xk = b.
Let p be the sum ofthose differencesj(x,+1) -f(x,) for r = 0, 1, , k - 1, which are positive, and - n the sum of those which are negative. Then j(b) -/(a) = p -n 1 kand l: I /(x,+1) -j(x,) I = p + n = t, say. .•.
r-o
B O U N D E D VARIATION
51
Suppose that P, N, '1' are the upper bounds of p, n, t for all modes of subdivision of (a, b). It is plain that
P(or N) � '1' � P + N. Call the numbers P, N, '1' respectively the positive, negative and total variations ofj(x) in (a, b). Either these three numbers are all finite or '1' and at least one of P and N are infinite; if '1' is finite we say that f(x) ka8 variation in (a, b), and that '1' is its variation. variation, '1' = P + N. Whatever the If j(x) ka8 mode of subdivision, we have
total
bounded
bounded
p
=
n + j(b) -j(a), � N + j(b)-j(a).
This being true for all values of p, we deduce that
P � N +j(b) -j(a), P - N �f(b) -j(a).
or
A similar argument gives and so Then
'l' � p + n
=
N - P �j(a) - j(b), P - N = f(b) -f(a). p +p - {f(b ) - j(a)}
=
2p + N - P,
and since this is true for any choice of points of division, we may replace p by its upper bound P, giving
T � P + N. This combined with the obvious inequality
'l' � P + N
gives the result.
A junction of bounded variation is the difference between bounded i-ru;rea8ing junctions.
two
Let j(x) have bounded variation in (a, b) and a � x � b. Let P(x) and N(x) be the positive and negative variations of j(x) in the interval (a, x). The proof of the last theorem shows that
j(:x:)
=
{j(a) +P(x)} - N(x)
52
D I FFERENTIATION A N D IN TEGRATION
and the right-hand side is the difference between two bounded increasing functions of x. Conversely, if f(x) = cfo(x) - t/J(x), where cfo(x) and t/J (x) are bounded increasing functions, it follows from summing the inequalities
!/(x,.+I ) - f(x,.) I � {cfo (x,+l ) - tfo(x, )} + {ifJ(x,.+I ) - t/J(xr)} that f(x) has bounded variation. Examples (1) Prove that the sum and product of two functions of bounded variation have-bounded variation. (2) Prove that, if f(x) is of bounded variation and continu ous, the functions P(x), N(x) and T(x) are continuous. (3) Prove that the functions xsin (1/x), xlsin (1/zl) (defined to be 0 for x 0) are not of bounded variation in any interval containing x 0. Prove that xi sin (l/x8'1) has bounded varia tion. (4) Prove that a necessary and sufficient condition for the curve x == x(t), y y(t), a � t � b, to have finite length is that x(t) and y(t) have bounded variation. =
=
=
4·7. Differentiation of the indeftnite integral. If F(x)
=
F(a) +
J:!(t)dt, then F'(x)
=
f(x) almost everywhere.
F(x), being the integral of f+ - f_ , where /+ and f_ are the positive functions defined in § 3·1, is the difference between two increasing functions. Therefore F'(x) exists p.p., and it remains to show that F'(x) f(x) p.p. Suppose first that f(x) is bounded, say rJ(x) r � K. Let h take a sequence of values tending to 0. , =
Then and
! F(x+ htF(x) = u:+ltf(t)dt F(x+ h�- F(x) ' �F (x) p.p.
$!, K
DIFFERENTIATION O F THE INTEGRAL
53
By the theorem of bounded convergence, if c is any point of
(a, b), I.e }I.e{F(x+k)-F(x)}dx lim F'(x)dx = h--+0 � {![".li'(x)lk- !J:+".li'(x)lk} = F(c)-F(a), on account of the continuity of F. Hence J:{F'(x)-/(x)}lk 0 for all values of in (a, b). From § 3·8 (vii), F'(x) /(x) p.p. Now suppose that f(x) is unbounded. By the usual decom position of f into its positive and negative parts (/ f+ -f_ ), it is sufficient to give the proof for positive /(x). Let U<x)}*' be defined as in § 3·5. Then J:[J(t) -{/(t)},.] dt, being the integral of a positive function, increases with x. 1 , ,
(I
(I
=
=
c
=
=
Therefore its derivative exists p.p. and is never negative. By the result for bounded functions
d J.z{/(t)}.dt = fJ(x)}n p.p. dx F'(x) � {J(x)}*' p.p. 4
Hence
or, since n is arbitrarily large,
p.p. F'(x) f(x) � From this, f.li''(x)lk � fJ(x)lk = J'(b)-J'(a). •
54
DlFFEB E N TlATlON A N D lNTEGBA.TION
But the theorem of § 4·5 gives the reverse of this inequality, and so
J:{F'(x)-/(x)}dx
0.
=
Since the integrand is p.p. greater than or equal to zero, it is, by § 3·8 (ii), zero p.p., and the theorem is proved. AB the example given in § 4·1 shows, there is a sense in which this theorem is the best possible. The theorem contains as a special case a fundamental metric property of sets of points. If E is measurable we define the average density of E in an interval I to be m(EI)fml. The upper and lower right-hand densities of E at a point are the upper and lower limits of the average density in as k-+ 0. Similarly the left-hand densities are the limits of the average density in If all these four num bers are equal we speak simply of the den&ity of E at By applying the theorem of this section to the integral of the characteristic function of E, we have: The den&ity o a measurable set E i8 1 at almost all points o E, and is 0 at olmtJ8t aJ,l points o OE.
x
(x, x +k)
+
(x-k, x).
f
x.
f
f
f(x) (xn, x,. + kn)
4·8. Absolutely continuous functions. Let be de fined for a � are , Suppose that 1) , non-overlapping intervals in (a, H, given E, we can find 8 such that
x 'b.
(x1, x1 + k b). l: I f(x,. + k,.) - f(x,.) I for all choices of intervals with l:k,. 8, f(x) is said to be abso lutely contin1W'U8. interval (x, x +k), we see that an absolutely By taking • • .
<E
"
r-1
n, fn = 0 if I x I > n or I y I > n. Then /,. ill bounded urable function and ffj,. tkdy exists and is equal to either of the repeated integrals of fn · From the hypothesis, fJdy exists for almost all x. fJ,.dy inc with n and tends to fJdy. 88
a.
meas
reases
By § 3·10 (A), But the left-hand side ill
ffJ,.tkdy and its limit,
88 n-+oo,
Jfftkdy. Hence ff!tkdy = fd ffdy, and the rest follows. a: Ifj(x,y) i8 a mbk juncti&n of (x,y) and if ftkfi fl dy ea:iat8, then ftkf!d71 = fayfJtk. From the last result, ffil l tkdy exists, therefore so does ff!tkdy and the conclusion follows from Fubini's theorem.
ill
meJUU
THB
S·S.
CLASS V
65
Dift"erendation of multiple integrals. This is an
operation of theoretical interest rather than one which occurs in day-to-day analysis, and we only mention the main result (for double integrals). 8 is
If a Bq'U4re of side h, containing (x, y), then p.p. · • lim � J f f(x, y) dxdy f(x, y). A-+oh Js =
A proof can be based on Vitali's covering theorem § 4·3 (note Corollary 2).
f(x)
is in the Lebesgue 5·6. The class LP. \Ve say that class LP (where > 0) for a given set of values of if is
p x f(x) measurable and l f(x) I P is integrable in the set. For example: (1) x-l is in P, for p 2, in (0, 1). (2) In a finite interval (a, b), a function in liP is also in Lq for 0 � q p; a bounded function is in J;P for every p. in (0, co), but not (3) The function x-l(I + f log x f )-1 is in in LP in (0, oo) for any value of p other than 2. The most interesting case is p � 1 and we shall assume this.
0
(i) (ii) (iii)
if x =1= y;
d(x, y)
=
d(x, x) = 0.
d(y, x).
d(x, z) � d(x, y) + d(y, z)-the triangle inequality.
Functions of JiP are elements of a metric space if we take
d(f, g) = Np (f-g).
Two functions differing only in a set of measure zero are indistinguishable as elements of the metric space. With this convention, the properties (i), (ii), (iii) of the distance function are satisfied, (iii) being Minkowski's inequality. The reader will recognize in the following discussion an extension to the space of functions in JiP of ideas such as limit point in the theory of sets of points. and NP (fn - /)-+0 as n-KO, we say that If/, and f are in J,�f(LP) or alternatively that f. converge8 strongly to f (with index A nece88ary and sufficient C01Ulition that fn�f(LP) is that NP (f - /,.. �0 as m and n tend to infinity. Two such limit
LP,
p). m
functi0'1UJ f ran differ only in a set of measure zero. Moreover, there i& a sub-sequence n,. suck that /,r�f We first prove sufficiency. Given E, there is n0(E) such that for m � n0, n � n0, /,. I P lkt: < Ep+l ,
p.p.
film 1
and so the set in which /m -/, I > e has measure less than E. Replace E by e/2, . . . , ef2k, . . . successively and let n1, , nk, . . . be the indices corresponding to the n0 of the last paragraph. Then • • •
FURTHER PROPERTIES
68
except in a set of measure at most E/2"-1 . Since the measure of the exceptional set tends to 0 as r-+eo, it follows that the series
� {!.J:+,(x)- /n1(x)} ,
k-o
is absolutely convergent p.p., that is to say, there is a function / ) defined p. p. to which the suh·sequence J ) converges as
. (x
(x
,
r�.
We shall prove that /, converges strongly to J with index p. By Fatou's lemma (§ 3·9),
i.e.
Jl!-f,. i�'dx �!,.�{11.., -In i�' dxfO �t>+l
if n ;> no,
J,�J(LP).
To prove the uniqueness of the limit function (ignoring differences in sets of measure zero), suppose that /.,. also con· verged to g(LP). By Minkowski's inequality, Np(/-
g)� Np(/-/,) + Np(/, - g),
and the right·hand side tends to zero as 1H-OO The necessity part of the theorem also follows at once from Minkowski's inequality. For •
�
Np(/m - /.,.) Np(/m -
.
f) + Np(/-/,).
E X A M PL E S O N CHAPTER V
( 1) Investigate the question of existence and equality of the double and repeated integrals of the following functions over the square 0 � � 1 , 0 � y � 1 ,�a: ' xtt+-ytti ' (i) (ii) (x y ) ( 1 -xy (iii) /(x,y) (x-1 !)* for 0 (x). In the same wa.y
J/t¥
is defined to mean
JJ<xW}de
where I is the set of f corresponding to the set E of x. · If I takes both positive and negative values and I = f+ - 1 , as in § 3·1, then fdt/> is defined to be f+dt/> - f
J
�
� I!_ _, _ _, __ -.,.. __ .,... --r--. r-
J
J -dtl>·
-- - - - ---
t =-;(x) I I I I I I I I I
f(x)
0
a
b
X
Fig. 2.
Text4fig. 2 illustrates (in the first quadrant) an increasing function e = tf>(x) having a horizontal stretch and a discontin uity. The graph of it is 'projected' on to the g axis, and ordinates of a function f(x) (shown as positive and increasing) are set up on the projection. The shaded area represents
t¥· i f
•
74
THE L E B E S G U E· STIELTJES INTEGRAL
The altemative method of definition of the integral of a bounded is as the common limit of approximative sums
f
8
=
8 =
ft.
L l,.+1 e,., 1
ft.
L l,. e,.,
1
where e, is the variation of q, over the set E,. for which l,. �f(x) < l,.+1 (cf. § 3·4). Extension to unbounded is made as in § 3·5. If + is a function of bounded variation, then q, q,1- �� where 1 and 2 are increasing functions (they are taken to be the positive and negative variations of � ). We then define
f
=
Ifdl/> I11lfh - I1dll>l· =
If the integrals on the right exist, so does this is appropriately written
I11 dl/> 1.
Iftllh Ifd.h and +
It is easy to adapt the arguments of Chapter III to yield theorems about the LS integral. A set of x of measure zero is to be replaced by a set over which the total variation of is zero. A property which holds except possibly in a set over which � has zero variation is said to hold p.p. (�). We give two illustrations of useful results derived from those of Chapter III. THEOREM OF DOMINATED OONVBRGENCE.
If, for
aU
n,
I.p I dl/> I exi818, and /,.-+/ p.p. (4>), then IJdl/> limII diJ>. The theorem of bounded convergence is a special case of this. INTEGRATION OF SERIES. If either I I or :EI I u,. I I dl/> I la finite, then I :EI•,.dll>·
I /,.(z) I � ,P(x), wkt:te
=
..
=
I N T E GRATION BY PARTS
E:t:amplu
75
( 1) Establish the equivalence of the 'geometrical' and the 'approximative sum' definitions of the LB integral. ( 2) Prove that
Ifdd/
=
/(b) f>(b) -j(a) f>(a).
We shall investigate under what hypotheses on f and � this holds. If the integrals are defined in the original 8ert8e of BtWtjes (§ 6· 1 ( 1)), then, if either integral on the left·hand 8ide exist!, so doe8 the other and the farmula is true.
I
Suppose that fd exist-a. Let a = x0 � �1 � x1 � . . . � X3_1 � �. � x. = b be any dissection of (a, b). Define €0 = a, €a+I = b. Then if • � � f ) {/(xr ) -f(x,._1)}, '1' 1 ( ,. =
and
we have identically,
T+� = /(b) �(b) -f(a) �(a).
76
THE L E B E SG U E - S TIELTJES INTEGRAL
Observe that if either max (x,. - Xr-d Ol' m.a.x (€,. - �,_1) tends to 0, so does the other.
[!d4>
Since
exists, it is the limit of
T1
as max (z, - z,._1)
J
0 . Therefore T tends to a limit, and so df exists and relation (I) holds.
tends to
We now link this up with the LS integral.
bounded
If j,tfi have variation and there is no value of x fur which they are both diacontinluouB, the:n, if a, b are points of con tinuity off, cp, the formula (I) holiJB. (If a, b are discontinuities of f or q,, the right-hand side of ( 1 ) is to be replaced by
f(b - 0) t/J(b - 0)-j(a + 0) . =
We may suppose that + is increasing and that I and
g are
of constant sign (say both positive). Write �
==
tions be
Jf{xCe>l g{xCe>l de, :e: =
)
=
the integrals being taken between the appropriate limits.
78
THE LEB ESOUE·STIELTJES I N T E G RAL
The result will now follow from the theorem of change of variable for Lebesgue integrals (§ 5·2), if we satisfy ourselves that the integrals are unaffected by the many-valuedness of the inverse functions x and X. We find that may be un defined in an enumerable set of �, and f in an enumerable set of E, corresponding to intervals of constancy of q,(x); J may also be undefined in an interval in which (so that S: is constant); both the integrals vanish over such an interval. We now prove the second mean-value theorem (for the L integral). Iff
g
g=0
is mmwtonic and g integrable, then f:Jgdz =/(a)f�gdz +J(b)L6glk for some � satisfying a�;� b, where a, b are values for which J is co-ntinuous. (H a,b discontinuities ofj, J(a) and f(b) on the right-hand side are to be replaced by J(a + O) and j(b-0).) Let G(z) = J:glk. Then the L8 integral J!dG exists and we have by the last ...heorem, are
J:Jglk = J:!dG = [!GJ:-J:odj, by integration by parts, =/(b) G(b) -G(�){/(b)-/(a)}, by the first mean-value theorem (§ 6·3, Ex. 3), and this is equal to
\
\
EXAMPLES
79
The brief account that we have given of Stieltjes integrals should enable the reader to manipulate them with confidence. Differential properties (depending on the notion of differ· entiating with respect to a function tfo) do not often come into question. EXAMPLES O N CHAPTER VI
(I) If tfo"'�' state sufficient conditions for
£!# �J:fd.f>. ..
(2) Construct a Stieltjes-Fubini theorem (§ 5•5). (3) Prove that
� (n )J.1xm(I - x)"'-m�(x). J.l�X(x) n=m m 0
==
0
SOLUTIONS O F SOME EXAMPLES Hints a.re given for the solutions of all but the easiest examples, and more detailed solutions of those which are most important. CHAPTER
I
P. 9, Ex. 6. Let E11 be the set of x such that /(f) -,-
Jflo
x
2
f
1•
2 ll
A
- cos 1 dx > - . If
x
x
n.
I;Jf . 1.
I � cos t dx existed, it would be > X X 1 § 3·7, p. 37. ( 1 ) g bounded or (2) r, g integrable. If (1), then fg is measurable and, if I g I � K, integrability of fg follows 1 from that of K l f l · If (2), use 2 1fg l �r+g • Ex. 2. !, (x) = nxe-nz' in (0, I). Ex. 3. Let H be a subset, with mH > mE - 8, in which j,_,.j uniformly. Then there is n.0(8) such that I f-fn I < 8 in H for n > n.0•
P. 42, P. 42,
fE-H JH
f.it-f I = t + IB-H• ..
� {upper bound of I !-!, l } m(E - H) < 2m, � II mE for n > no.
Jl !-/,. l-+0 as �. P. 42, Ex. 4. (1 - :r < and-+ e..... Use theorem of domin Hence
e-, (x) M;n) in 8�,, for all r of � 1. Let (a1, 61), , (a", b*') ' . . . be the intervals of 0. Extend the interval (a1, b1) to the left by taking (�1, b1) such that • • •
b1 - �� = K (b1 - a1). Do the same to (a1, b1), giving (a1, b1). If (�1, b1), (a1, b1) have an interval in common, move this to the left so as to give an interval (�, p.) containing (a1, 61) and (cxt, b1) of length equal to (b1 - a1) + (b1- a1). We have thus either one or two intervals, say (A1 , B1), (At, B1). Carry out the same construction with (as, b3) giving (as, b3). If (a3, b3) has an interval in common with either (A1, B1) or (At, B1), carry it to the left ..as before. We then have either
one, two or three intervals. This construction leads to an enumerable set n of non overlapping intervals (�1, p.1 ) , , (�, of total measure KmO. The set such that, for any (�, p.) inside (�, p.,.), the measure of the set of points of ( �' not in 0 is at least (K- 1 ) times the measure of the set common to 0 and (�, p.). There fore, in an interval (�, of which the left-hand end-point does not belong to n, the average density of 0 is at at most 1/K. To prove the density theorem, we first show that an open set 0 has zero density p.p. in CO. Take p intervals of 0, say ()P, with miJP > mO - �. Given K, construct the set n of the lemma for the set 0- 0P. Then
0 is
• . •
p.)
p.)
p.*')'
. • .
�op + P> � mfJP + Km(O - Op )
� mO + (K - l)E. All the points of CO at which the right-hand density of 0 is
greater than 1/K lie in a set of measure at most (K - 1)E. Since E is arbitrary they form a set of measure zero. Let K--+«.>; the right-hand density of 0 zero p.p. in CO. Similarly so is the left-hand density. Let now E be any measurable set; we prove that its density is zero p.p. in CE. Enclose E in 0 with m(O - E) < E. The set
is
85
SOLUTIONS
of points of CO in which E has not zero density is contained in the set in which 0 has not zero density, i.e. has measure zero. Since £ is arbitrary, E has density zero p.p. in CE. Interchange the roles of E and CE. CE has density zero p.p. in E and so E has density 1 p.p. in E. P. 57, Ex. 4. Let I l l < K. Let E,,., = set where u � , � 11