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j \j £ I}, called a partition of unity subordinated to the covering {Gj | j £ I}, such that j < 1, and for j £ I we have supp j C Gj. Moreover, the family {supp <j>j \ j £ I} is locally finite and Yjj£i4>j{x) = 1 for all x £ G. In addition, for any e > 0 there exists a function ip £ Cg°(R n ) such that 0 < ip < 1, diam(supp ip) < (1 + e)nll2 and 2 n 2 (^DkeZ", V'jbfa) = i> (x — k), is a partition of unity, i.e. J2keZ tp (x — k) = 1 for all x £ R™. It has the property that for m £ No given, there exists a constant M = M(n,m,e) such that \daipk(x)\ < e~^M for all \a\ < m, and there exists a number N = N(e, n) such that the intersection of N + 1 of the supports of the functions ipk is empty. C. For any £ > 0 there exists 4>e £ C°°(R) such that -e < (f>E(x) < 1 + e for all x £ R, 4>e\[o,i] = 1 and whenever x < y it follows that 0 < (j)e{y) — 4>e(x) < y — x. A proof of part A and B is given in [237], Corollary 1.3.2 and Theorem 1.5.2, respectively. A proof of part C is indicated in [102], Problem 1.2.1. In some situations we have to work with lower semicontinuous or upper semicontinuous functions. A function f : G —>• (—00,00], G C R n open, is said to be lower semicontinuous at XQ £ G if f(a;o) < 00, then for every e > 0 there exists a neighbourhood U{XQ) C G such t h a t i(x) > / ( z o ) — £ for all x £ U{XQ), and if f(a;o) = 00, then for every M > 0 there exists a neighbourhood U{XQ) C G such t h a t i(x) > M for all x £ U(x0). If f is lower semicontinuous at all points x £ G, then it is called lower semicontinuous on G. T h e set of all lower semicontinuous functions on G is denoted by %i.s.c{G). We call a function g : G —>• [—00, 00) upper semicontinuous if —g : G —> (—00, 00] is lower semicontinuous. The set of all upper semicontinuous functions on G is denoted by rHu.s.c.{G). P r o p o s i t i o n 2 . 1 . 4 A. Leti,g £ 'Hi.s.c.{G) and a > 0. Thena-i, f+g andiAg are lower semicontinuous too. B. For any non-empty set % c 'Hi.s.c. {G) the function defined by u(x) := sup {f(a;) | / £ V.} is also lower semicontinuous. C. Let u : G —¥ [0, 00) be a non-negative lower semicontinuous function. Then we have u(x) := sup {i(x)
I f £ C 0 (R") andO < i(x) < u(x) } .
D. Let u £ T-li.s.c.{G). Then there exists a sequence ({V)V£H of functions f„ : G —> R such that f„ < f„+i for all u G N and VL{X) = lim iu{x) — supf„(a:).
continuous
2.1
19
Calculus Results
E. Let u £ V.i.s.c.{G) and K C G a compact set. such that u(x0) = inf u(x). xeK
Then there exists XQ £ K
A proof of Proposition 2.1.4 is given (partly) in [126], p.89, or in [8], § 3. D e f i n i t i o n 2 . 1 . 5 A family {ij : G -» [—00,00] \j £ 1} is called filtering increasing if for j \ , J2 £ I there exists J3 £ I such that ij1 < f,3 and f,-2 < f,-3. T h e o r e m 2.1.6 Let H C 'Hi.s.c.{G) be a filtering increasing family such that f(a;) ;= sup{g(a;) | g £ T-L} is upper semicontinuous. Then f is continuous and f is approximated uniformly on compact sets K C G by the family H, i.e. for every e > 0 and every compact set K C G there exists g£ £ % such that 0 < f(ir) - ge(a;) < s for all x £ K. A proof of Theorem 2.1.6, which is often called Dini's theorem, is given in [8], § 4. Let G C R n be an open set. We call u : G —> C Holder continuous exponent A £ (0,1] if sup K
g
) - y i
£ C X defined for s > 0 and p £ P by UPt£ := {x £ X \p(x) < £ } . Then the set Bo of all finite intersections of elements ofB'0 form a base ofO £ X which turns X into a locally convex topological (Hausdorff) space. Conversely, let X be a locally convex topological vector space and for a convex, balanced and absorbing set U C X define the Minkowski functional Pu(x) : = i n f { A > 0 \x€ AC/}. Then P\j is a seminorm and the family of all these seminorms generates the topology on X in the sense described in the first part of this proposition. A linear operator A : X —» Y mapping a topological vector space X into another topological vector space Y is continuous if and only if it is continuous in 0 £ X. Moreover, when the topologies in X and Y are generated by families Px and Py of separating seminorms, then A is continuous if and only if for any q £ Py there exist finitely many seminorms pj £ Px, 1 < j < I = l(q), such that q(Ax) < c max Pj(x)
(2-43)
holds. In particular, a linear functional u : X —¥ C is continuous if and only if there exist finitely many seminorms pj £ Px, 1 < j < I = 2(u), such that |u(a;)| < c max Pj(x)
(2.44)
holds. Let q be a seminorm on X. We call q continuous with respect to the family Px if (2.43) holds with A = id^- Suppose that Pi and P 2 are
22
Chapter 2 Essentials from Analysis
two separating families of seminomas each turning the vector space X into a locally convex topological vector space. These topologies on X are equivalent if and only if every seminorm in Pi is continuous with respect to Pi and every seminorm of P\ is continuous with respect to Pi. In this situation we call Pi equivalent to P2. For a locally convex topological vector space we denote by X* the space of all continuous linear functionals u : X —¥ C, i.e. X* is the dual space of X. Let X be a locally convex topological vector space. The weak topology on X is the weakest topology on X which makes every u. £ X* continuous. Note that every weak neighbourhood of 0 € X contains a neighbourhood of the type {x e X I \UJ(X)\ < £j for 1 < j < m}
(2.45)
where u, € X* and £j > 0. Next we want to introduce a suitable topology on X*. For this let x € X be given. We can define a linear functional lx on X* by lx(u) := u(x) for all u<EX*. Denote the set of all these linear functionals on X* by X. The weakest topology on X* which makes all elements of X continuous is called the weak-*-topology on X*. Every weak-*-neighbourhood of 0 S X* contains a neighbourhood of the form {u € X* I | u ( ^ ) | < £j, for 1 < j < m}
(2.46)
where Xj G X and £j > 0. For the weak-*-topology we have the following compactness result. Theorem 2.2.2 (Banach—Alaoglu) Let X be a topological vector space and U(0) C X a neighbourhood ofOeX. Then the set K:={ueX*
I [u(sc)| < I for allx £ U(0) }
(2.47)
is weak-*-compact. For a separable space X, i.e. there is a countable dense set in X, we have more results. Theorem 2.2.3 A. Let X be a separable topological vector space and let K C X* be a weak-*-compact set. Then there exists a metric on K turning K into
2.3
23
Measure Theory and Integration
a metric space with metric topology equivalent to the weak-*-topology (on K). B. Let U(0) C X be a neighbourhood ofO£X,X being a separable topological vector space. Further let (u„)„gN be a sequence in X* such that |u„(:r)| < 1 for all x £ U(0) and i / £ N . Then there exists a subsequence (u„fc)fc6N of (u„)i,gN and u £ X* such that for all x £ X we have lim ul/k(x) = u(x). k—>oo
R e m a r k 2 . 2 . 4 In general we call a topology metrizable, if there exists a metric generating an equivalent topology. A locally convex topological vector space which is metrizable such that the metric space is complete is called a Frechet space. W h e n working with mappings f : X —>• Y from a topological vector space into another, it is often sufficient to restrict the attention to sequentially continuous mappings, i.e. mappings with the property t h a t xv —>• x in X implies t h a t f(a;„) ->• f(a:) in Y.
2.3
Measure Theory and Integration
We assume the reader to be familiar with the basic notions and results of measure theory and the theory of integration including the notion of properties which do hold almost everywhere (a.e. as abbreviation), the completion of a afield, central theorems like the Caratheodory extension theorem, the dominated and monotone convergence theorems, the theorems of Tonelli and Fubini, the Radon-Nikodym theorem, etc. For these topics and for the following results which we state without proofs and without reference, our standard references are the book [18] of H. Bauer, the beautiful work [217] of P. Malliavin, and the monograph [256] of W. Rudin. Let Q be a set and S C V(£l). We call S a n-system in fl, whenever A,B € S implies A D B e S. A 7r-system is called a d-system in fi, if ft G S, A,B £ S and A C B then B\A £ S, and for any sequence (j4„)„ e N, Au £ S such t h a t A„ C Av+\ it follows t h a t (Ji/gN ^ 6 &. T h e o r e m 2.3.1 ( M o n o t o n e class t h e o r e m ) Suppose that S is Then it follows that d ( 5 ) = a(S).
air-system.
Here d(S) denotes the d-system generated by S and cr(S) is the cr-field generated by S.
24
Chapter 2 Essentials from Analysis
C o r o l l a r y 2.3.2 Let ft be a set and S a •n-system in Q. Further let rl be a vector space of real-valued functions f : fi —»• R such that 1 £ %, XA € H for all A € S, and for any sequence (fv)v€^, f„ G H, such that 0 < iv < f„+i and f = supf„ < oo it follows that f € T-L. Then "H contains all real-valued bounded functions
which are measurable with respect to o~(S).
For a locally compact space G its B orel-o-field is denoted by B(G), but for G = R™ we write B^n\ Note t h a t B^ is already generated by the bounded open sets in R™. T h e Lebesgue measure on R n is denoted by \(n\dx). By definition measures are non-negative, and measures on B(G), G being a locally compact space, are called Borel measures. Let fi be a Borel measure. We call H locally finite if fJ-(K) < oo for any compact set K C G. A Borel measure is inner regular if for any B € B{G) we have / i ( 5 ) = sup {n(K)
| K C B and K c o m p a c t } ,
and \i is outer regular if for any B € B{G) it follows t h a t H(B) = inf {fi(U)
| B C Uand Uopen} .
A Borel measure fj, is called a Radon measure if it is locally finite and inner regular. On fi(") Radon and Borel measures coincide. T h e support supp fi of a Radon measure /x is the complement of the largest open set G such t h a t /i(G) - 0. Let (fi, .4) be an arbitrary measurable space. T h e total mass of a measure H on (Q,,A) is denoted by ||/x|| := //(fi), and .MjJ~(fi) is the set of all bounded measures on (fi,-4), i.e. fi € A4jj"(fi) implies ||/x|| < oo. By Ml(fl) we denote the set of all probability measures, i.e. measures fi with total mass ||/z|| = 1. A sub-probability measure is an element of .Mjj"(fi) such t h a t ||/x|| < 1. For a G 0 the Z>irac measure at a is denoted by e a , but in case fi = R n and a = 0 we often write £ instead of £Q. Now let G be a compact space. A signed measure on (G, B(G)) is a mapping /i : B(G) —> R such t h a t we can write y,{A) = Hi(A) — fJ>2(A) for two measures /xi,/X2 G A^j"(G). Whenever p is a signed measure there exist two measures Hi and y% in M£(G) such t h a t fi = (1° — (i% where /i° and /i° are mutually singular, i.e. there exists a Borel set A0 C G such t h a t
A(A°) = /i?(G) and /^(^°) = 0.
(2.48)
2.3
25
Measure Theory and Integration
Let (fi, A, //) be a measure space. We call AG A a fi-atom if fi(A) > 0 and for any B C A, B e A, either //(£?) — 0 or //(A \ 5 ) = 0 holds. In particular we have / / ( J B ) = 0 or n(B)
=
p(A).
T h e total mass of // is given by ||//|| := //°(G) + //°(G). T h e set of all signed measures is denoted by M.(G). Let G be a locally compact space. A signed Radon measure // is given by two mutually singular Borel measures //J and / J ° . W i t h |//| := /i° + ^2 w e consider the completion of 23(G) with respect to |//| and denote this cr-field by B^G). For A G B^{G) such t h a t |//|(A) < oo we define //(A) = //?(A) — n\(A). T h e bounded signed Radon measures on G are denoted by M.b(G). It follows t h a t (Mb, ||-||) is a Banach space. T h e o r e m 2.3.3 The Banach space (Mb(Rn), ||.||) is the dual space of n Coo(]R ;]R). For /J, G ^Vl6(]Rn) a linear continuous functional on Coo(]R™;R) is defined by UH
/" u(a:)/x(da;) = /" u(a:) ^°(da;) 7R" 7R"
/
u(x) (4(dx).
(2.49)
JU"
This theorem is also related to the various variants of Riesz' representation theorem. A linear functional I on C ( G ; R ) or any of its subspaces, G being a locally compact space, is called a positive functional if l(u) > 0 for all u > 0, u € C(G; R) (or in case where I is defined only on a subspace of C(G; R), u > 0 should be taken only from this subspace). T h e o r e m 2 . 3 . 4 A. Let G be a metrizable locally compact space which is countable at infinity. Then I is a positive linear functional on Co (G; R) if and only if there exists a unique locally finite Borel measure fj, such that Z(u) = / u{x) //(da;)
(2.50)
JG
for all u € Co(G;R). B. In case that G is compact, I is a positive linear functional on C ( G ; R ) if and only if (2.50) holds with a unique measure // £ Ai^(G) and all u e C(G; R ) . C. A linear functional I on Coo(R n ; R) is positive if and only if there exists a unique Borel measure // on R™ such that (2.50) holds for all u G Coo(R n ;R). Moreover, for all u G C 0 0 ( R " ; R ) it follows that JRn u(x) //(da;) < oo. We may also extend the considerations to complex-valued measures. This is possible in two equivalent ways. Either we define a complex-valued measure
26
Chapter 2 Essentials from Analysis
with the help of two pairs (M?, M°) and (^°, fi°) of mutually singular measures by (i(A) = fi{A) - &{A) + i(£(A) -
fi(A))
(2.51)
whenever (2.51) makes sense, or we introduce complex-valued measures as the dual space of Co(G), now we consider of course complex-valued functions. T h e total mass of M given by (2.51) is now defined by ||/i|| := Y^=i M?(^)> a n d IMI is the measure $^,- =1 M?- We denote the space of all complex-valued measures by MC(G); M.f(G) is the set of all bounded complex-valued measures. Let ( f i i , - 4 i , / / i ) and (0,2, .4.2, M2) be two measure spaces. T h e product measure of MI and M2 is denoted by Mi ® M2- We introduce the mapping Ak : Rnk ->• R n , (xi,... ,Xk) i-+ xi + ... + Xk and give D e f i n i t i o n 2.3.5 Let Mj G At+(R™), 1 < j < k, be measures. The image of Mi ® • • • ® Mfc under Ak is called the convolution of these measures and is denoted by Mi * . ..*/zjb := Ak(fJ*i ...® Mfc)-
(2.52)
Obviously we have Mi * . . . *Mfc G A^j"(R n ) and ||MI * • • • *Mfc|| = ||MI|| • . . . • ||/ifc||. T h e convolution is associative and commutative, i.e. (Mi * fj,2) * M3 = Mi * (A*2 * M3) and Mi * M2 = M2 * Mi
(2.53)
for all Mj ^ A^j"(R"). For any non-negative measurable function on Rn we have /
i(x) (Mi *M2)(dz) =
JR"
/
/
f(a; + y)Mi(da;)M2(dy)
,/R" ./R"
= 1 1
i(x + y)^(dy)tx1(dx)
(2.54)
which yields in particular (Mi*M2)(5)=/
Mi(5-y)M2(d2/)= /
M2(S-z)Mi(dz)
(2.55)
n
for all i? G B( \ Using (2.51) we may extend the convolution to .Mf,(R n ) as well as to M%(Rn), and (2.53) remains valid for this extension. Moreover we have Mi * (M2+M3) = Mi * M 2 + M i *M3
(2.56)
2.3
27
Measure Theory and Integration
and «(Mi * M2) = {afJ-i) * M2 = Mi * ( 0, € R or € C depending on whether Hj € M^(Rn), respectively. In each case we have fj,*e
(2-57) Mb{Rn)
= £*H = H
or
Mf(Rn),
(2.58)
Thus it follows t h a t _Mb(R n ) and Mf(Rn) are algebras with respect to the vector space operations and the convolution as product. Moreover, £ is a unit in these algebras. For integrable functions f, g : R n ->• C or R, iX^ and gA^n) are elements in M%(M.n) or .Mi,(R n ), respectively. Thus we can consider the convolution iX^ * gA< n \ and we find f A ^ * gAo, Vt S M^(G), we may define ^lim-fo+,-77,) dt
S_K)
S
(2.63)
2.3
29
Measure Theory and Integration
in these topologies. For example for the vague convergence (2.63) does mean t h a t for a l l u G C 0 (G) lim - ( / u(x) rjt+s(dx) «->o s \JG
-
/ u(x) JG
r]t{dx)
exists and leads to a measure fi, defined by / u(x) fi(dx) = lim - ( / JG
u ( i ) Vt+s{dx) -
s->0 S \JG
[ u(x) r,t(dx)) Jo
(2.64) J
for all u G C 0 ( G ) . Let (Cl, A) be a measurable space and f, g : Cl —)• R be measurable functions. We write (f A g)(x) := min(i(x),g(x))
and (f V g)(x) := max(f(x),
g(x)).
(2.65)
Both functions f Ag and f Vg are measurable too. Thus we may define f+ := f VO as the positive part of f and f ~ := — (f A 0) as the negative part of f. We always have t h a t f+, f_ > 0 and f + (a:) • i~(x) = 0 for x € 0 . Moreover it follows t h a t f=f+-f"
and | f | = f + + r .
(2.66)
Note however t h a t f+ and f - do not have disjoint supports since they coincide on the set {x £ fi | f(a;) = 0 } . In case t h a t fl = G is a topological space and A = B{G) is the Borel cr-field, the set of all measurable functions f: G —>• C is denoted by B(G), and Bf,{G) is the set of all bounded functions f S B(G). When we want to emphasise t h a t we have to work with real-valued functions we write f G B(G;W) or f G Bb(G;R), respectively. For any measure space (CI, A,n) the spaces Lp(Cl,n), 1 < p < oo, are the usual Lebesgue spaces (of equivalence classes) of measurable functions f : £~i —>C with finite norm
Hf|| L P :=^JfW| P Md^))
P
,l L P (R") is a Ziraear operator. Moreover we have IIWHLP
< IHIL?
(2.79)
and lim || J e ( u ) - u|| L P = 0.
(2.80)
£->0
In case p = 2 we also have (JE(u),v)0 = (u,Js(v))0 for all u, v G L 2 ( R n ) , i.e. J £ is a bounded selfadjoint Moreover, for u G Co(R n ) we have lim/
jv(x - y)u(y) dy = u(x) =
*-e- (JTJ)TJ>O converges vaguely to EQ and (^(x
operator on L 2 ( R n ) .
u(y) ex(dy),
(2.81)
— .))v>o converges vaguely to ex.
Note t h a t the approximation result holds also true for u € Coo(R n ). Moreover, the special type of j e is not necessary if only the convergence is of interest. For any <j> € L 1 ( R n ) such t h a t / R „ <j>(x) dx = 1 let us define <j>e(x) = e~n ( | ) . Then it follows t h a t ||u * <j>E - u|| L P -> 0 as e ->• 0 for all u G L p ( R n ) , 1 < p < oo. Once again the result remains true for u G C 0 0 ( R n ) . A proof for this is given in [289]. Now let u, v G L P ( R " ) , 1 < p < oo, be functions such t h a t u > 0 a.e. and 0 < v < 1 a.e., respectively. It follows t h a t J £ (u) > 0 and 0 < J £ (v) < 1 hold. Furthermore, since |((0 V u{x)) A 1) - ((0 V v(x)) A 1)| < |u(x) - v(a;)| a.e.
(2.82)
holds for all u,v G L p ( R n ) , we find t h a t a sequence ( u „ ) „ e N , uv G Cg°(R n ), converging in L p ( R n ) to u G L P ( R " ) , has the property t h a t ((0 V u„) A 1 ) „ 6 N converges in L p ( R n ) to (0 V u) A 1 and ( 0 V u „ ) M £ C 0 ( R n ) . T h e next theorem characterises the conditionally compact subsets in L P (G). T h e o r e m 2.3.18 A subset K C L P ( G ) , G c l " a Borel set, 1 < p < oo, is conditionally compact if and only if the following three conditions hold
33
2.3 Measure Theory and Integration 1. sup ||u||LP < c; 2. lim sup ||u(. + h) - u ( . ) | | L P = 0; |h|->0u€if
3. lim sup
G\BR(0)U
= 0.
X
LP
iVoie rAai for a bounded Borel set G C R™ t/ie Zas£ condition is empty. Let g : G —• R be a strictly positive measurable function and consider the measure fx := gA(")|G on G. The space L P (G, /x) consists of all measurable functions u such that u • g G L P (G), or equivalently, for any v G L P (G) the function ^ • v belongs to L P (G, fi). Moreover, we have ||u|| LP , G •. = ||ug|| LP , G j. Thus a set K C L P (G, fj,) is conditionally compact if and only if the set Kg := {v = K(x, A') is for all A' € A' a measurable function and A' H-> K(X, A') is for all x G SI a measure. A kernel K is called a Markovian kernel ifK(x,Sl') — 1 for all x G SI, and it is called a sub-Markovian kernel ifK(x, SI') < 1 for all x G SI. Given a kernel K from (SI, A) to (SI', A'). We may define an operator K op on all non-negative measurable functions u : SI' —> K by (K op u)(x) := J u(x')K(x,dx').
(2.83)
It follows that K op u is a non-negative measurable function on SI. Suppose that Kop is an operator from all non-negative measurable functions u : ST.' —• R to the space of all measurable functions v : SI —> R having the representation (2.83). Then the right hand side in (2.83) is called the kernel representation of K op .
34
Chapter 2 Essentials from Analysis
T h e o r e m 2 . 3 . 2 0 Suppose thatK.op is an operator from all non-negative measurable functions u : Cl' —>• R to all measurable functions v : Cl —> R. Assume further that Kop has the property that u > 0 implies K o p u > 0, and that for every sequence (u„)j,gN of non-negative measurable functions \xv : f2' —> R such that 0 < u„ < u „ + i it follows that supKopU^ = K o p ( s u p u t / ) . Then there exists a kernel K /rom (fi,»4) io (Cl',A') Kopu(z) = /
such that Kop has the kernel
representation
u{x')K(x,dx')
for all measurable functions
(2.84)
u : $7' —>• R, u > 0.
Clearly, we may extend K o p in the usual way to suitable integrable functions. In particular, if all the measures (K(a;, .)) x en are bounded, K o p is well defined on the space of all bounded, real-valued measurable functions u : Q' —¥ R and K o p u is a bounded measurable function on fi. Suppose t h a t N : G" x B(G) -> [0,oo], G,G' C R n Borel sets, has the property t h a t for every x € G' the mapping N(x,.) : 13(G) —• [0, oo] is a measure. Still it is possible to define the operator N op u(a;) := /
u(y)N(x,dy).
JG
P r o p o s i t i o n 2 . 3 . 2 1 Suppose that for all u € C Q ° ( G ) the
mapping
a; M- / u(y) N(z, dy) is continuous. Then x \-t N(a;, A) is measurable for every A S B(G), N(:r, dy) is a kernel.
i.e.
This proposition is proved by approximating the functions x A , ^ € B(G), pointwise by suitable sequences of functions belonging to Co°(G). Suppose t h a t Ki is a kernel from (£li,Ai) to ( ^ 2 , ^ 2 ) and K2 is a kernel from (^2,^.2) to (^3,^.3). Suppose further t h a t the measures (Ki(a;i, . ) ) I i e n i as well as (K 2 (a;2, -))x 2 en2 a r e bounded. Then the operators K„ p , K„ p and K ^ o K ^ are defined by K^p := ( K i ) o p and K ^ := ( K 2 ) o p , respectively, acting on bounded measurable functions, and K^ p o K ^ is just the usual composition of operators. For a bounded, measurable function u : $^3 —>• R we find Kopu(x1):=(K2opoKfp)u(x1)=
[ [ Jn2 Jn3
u(x3)K2(x2,dx3)K1(x1,dx2).
2.3
35
Measure Theory and Integration
On the other hand, by Theorem 2.3.20 the operator Kop has a kernel representation with a kernel K3i(a:i, dx^) which yields K31(Xl,A3)=
f
K2(x2,A3)K1(x,dx2)
(2.85)
for all x\ S fli and A3 e .43. In particular, when a family (Kt)t>o of kernels from (R™, Bn) into itself is given, we may consider the kernels K,it{x,A)=
[
Ks(y,A)Kt(x,dy).
(2.86)
Often we have to use the following two results. L e m m a 2 . 3 . 2 2 Let E be a metric space and (£l,A,n) be a measure space. Further suppose that u : ExQ, —» R is a function with the following properties: u> i-t \i(x,u>) is for all x G E integrable with respect to fi, x i-> U(X,OJ) is continuous at XQ £ E for all ui £ ft, and there exists a function h £ L 1 (r2,/n) such that |f(a;,o;)| < h(u>) for all x £ E and w £ fi. Then the function x i->JQ{(x,u>) fi(dui) is continuous at XQ. L e m m a 2 . 3 . 2 3 Let I c R be a non-empty interval and (Cl,A,fJ,) a measure space. Assume that u : I x fi —>• R has the following properties: u(x,.) £ L 1 (fl,/i) for all x € 1, for all ui £ Q, the function x H> U(X, UI) is differentiate on I, and there exists a function h £ L 1 (f2,//) such that \-^u(x,u>)\ < h(w) for all x £ I and u> £ fi. Then the function x i-» Jn u(x,u>) /i(du>) is differentiate, the function to i-> -^n{x,ijj) is an element in L 1 (fi!,/i) for all x £ 1, and — I u{x,u)
fi{du) = I -^(x,u>)
Ai(dw).
(2.87)
We need various results on the integration of functions with values in vector o
spaces. Let I C R be a closed interval such t h a t 1 ^ 0 . Further let (X, \\.\\x) be a Banach space. We denote by C(I; X) the space of all continuous functions u : I —¥ X, and C 1 (I; X) is the space of all these functions which are continuously o
differentiable in I and have onesided derivatives at the endpoints of I. For any compact interval [a, b], a ^ b, we may define the Riemann integral f u(t) dt for u £ C([a, b]; X), and then, as usual, we may define the improper Riemann integrals /»oo
/ Ja
pb
u(t)dt,
/ J— 00
/-oo
u{t)dt
and
/ J — 00
u(t) dt
36
Chapter 2 Essentials from Analysis
for suitable functions u : I —• X, I = [a, oo), I = (—oo, 6] or (—oo, oo), respectively. The following result is taken from the book [88], p.9, of St.Ethier and Th. Kurtz. Lemma 2.3.24 A. Let u G C(l;X) such that Jj ||u(t)|| x dt < oo. Then the integral J. u(£) di exists and we have the estimate
fu(t)dt
< [\\u(t)\\xdt.
(2.88)
In particular, for every compact interval I, every u G C(I;X) is integrable. B. For u G C1([a, b]; X) we have
jT (j|U(i))di = u(6)-u(a).
(2.89)
C. Let (A, D(A)) be a closed operator on X and u G C(I;X) such that u(£) G D(A) for all t G I, Au G C(I; X) and u as well as Au are integrable (over I). It follows that J, u(t) dt G D(A) and we have k(f
u(t) dt j = f Au(t) dt.
(2.90)
Another way for denning an integral for functions u : I —> X is due to S. Bochner [37]. We follow the book [315] of K. Yosida. Let {£l,A,n) be a measure space and u : S l - ^ I a finitely valued function such that ' k
u(u,) = ^ x n »
(2.91)
for Xj G X and Clj C 0, //($!,•) < oo. We call functions of type (2.91) fi-Xelementary functions. The //-integral of u is now denned by
J
A;
u(w) /i(dw) := J2 xolJL<Sli) G X1
( 2 - 92 )
3= 1
Definition 2.3.25 A function u : Cl —• X is said to be Bochner-/z-integrable if there exists a sequence (u„)„gN of Q-X -elementary functions converging li-almost everywhere, such that lim f ||u„(w)-ii(w)|| x /i(da;) = 0.
(2.93)
2.4 Convexity
37
The Bochner integral has many properties analogous to the Lebesgue integral. For elements u G C ( I ; l ) , l c l a c o m p a c t interval, the Riemann integral coincides with the Bochner integral, thus in Lemma 2.3.24 we could have also worked with the Bochner integral. Finally, we mention how to define an integral for functions u : Q —> X where (Q, A, (J,) is a measure space and X is a topological vector space. For details we refer to W. Rudin's monograph [255]. We suppose that X* separates points in X, i.e. for x, y G X, x =£ y, there exists u G X* such that u(a;) ^ u(y). Now let f: 0 —• X be a function and for u G X* define the function u(f) : Q, —> C (or R) by u(f)(w) — u(f(w)). Suppose that u(f) is /Li-integrable. If there is some x G X such that u(x) = f u(f)(w) /z(du;)
(2.94)
Jn for all u G X*, we define the /x-integral of f by x=
I fd/i.
(2.95)
Jn A criterion for the existence of the /x-integral (2.95) is Theorem 2.3.26 Let X and X* be as above and for a compact Hausdorff space G let (G,B(G),fi) be a probability space, i.e. {G,B{G),JJL) is a measure space and fJ-(G) = 1. If i : G —>• X is continuous and if the convex hull of f(G) C X has a compact closure in X, then the ^-integral x = JQidfi exists.
2.4
Convexity
We need only a few notions and results from convexity theory, but also some concrete applications. Our standard references are the lecture notes [17] of H. Bauer, G. Choquet's lectures on analysis [57]-[59], once again W. Rudin's monograph [255], and for Choquet theory we refer to the book of R. Phelps [238]. Let X be a vector space and K C X. The set K is said to be convex if for all x,y G K and A G [0,1] we have Aa; + (1 - X)y £ K. The intersection of arbitrarily many convex sets is convex again, hence for any set H C X we may define its convex hull conv(H) as the smallest convex set containing H, i.e. conv(tf) = p | {K D H I K convex}.
(2.96)
38
Chapter 2 Essentials from Analysis
Let K C X be a non-empty convex set. A point XQ € K is called an extreme point oi K ii K \ {x0} is convex too. Equivalently, xo is an extreme point whenever xo = Xx + (1 — A)y for some x,y & K and A € (0,1) it follows t h a t XQ = x = y. T h e set of all extreme points of K is denoted by ext(K). A subset C c X i s called a cone with vertex at 0 G X , if for all A > 0 it follows t h a t AC C C. We call C a peafced cone if C n ( - C ) = {0}. Further, C is said to be a cone with base if there exists a hyperplane H in X such t h a t 0 g H and for every x G C \ {0} the intersection of {Az | A > 0 } and i7 is non-empty. T h e set B := C C\ H is called a 6ase of C. A cone is said to be convex if it is convex as a subset of X. It follows t h a t a cone C is convex if and only i f C + C c C . A central question is whether we may describe a convex set by using only its extreme points. An important first result is the theorem of M. Krein and P. Milman. T h e o r e m 2.4.1 (Krein—Milman) Every compact convex set K ^ 0 in a locally convex topological vector space is the closure of the convex hull of its extreme points, i.e. K = conv(ext(K)).
(2.97)
Suppose t h a t X is a locally convex topological vector space with dual space X*. Further, let K C X be a non-empty compact set and [i a probability measure on K, i.e. a Radon measure on the Borel sets of K such t h a t fx(K) = 1. We call x € X a barycentre of /U if u(x) = /
u(y) /i(dy)
(2.98)
JK
for all u e l ' . T h e o r e m 2.4.2 ( C h o q u e t ) Let K C X be a non-empty compact convex set and suppose that the induced topology on K is metrizable. Then for every x G K there exists a probability measure ^ on K with barycentre x such that s u p p / i C ext(K). Choquet's theorem may be used to obtain concrete representation results. Let i f be a compact Hausdorff space. Consider a linear subspace H C C(K; 1R) such t h a t 1 G H and denote by H* the dual of H (with the weak-*-topology).
39
2.4 Convexity The set A(H):={leH*
\l(l) = l=\\l\\}
(2.99)
is a compact convex set in H* and every functional lx(i) :— i(x), x G K and i £ H, belongs to A(H). We call H point separating if for all £1,2:2 G K, xi ^ X2, there exists f G H such that i(x\) ^ f(x2). The set dHK := {x G K I lx G ext{k{H)) }
(2.100)
is called the Choquet boundary of iJ. Theorem 2.4.3 Suppose that H is point separating. A point x belongs to djjK if and only if /x = ex is the only probability measure with i(x) = f f(y) M(dy)
(2.101)
JK
for all { € H. Theorem 2.4.4 Let H be a point separating subspace of C(K;W) such that 1 G H. For any I G H* exists a measure fi on K such that l{() = f t(x) n(dx)
(2.102)
JK
for all f G H. Moreover, fJ.(G) = 0 for any Baire set G such that GtldtfK
= 0.
Recall that the Baire sets form the smallest cr-field in K such that all elements f G C(K;W) are measurable. In presenting Theorem 2.4.2-Theorem 2.4.4 we used the monograph [276], but note that there are no proofs for these results in [276]. Proofs were given in [238]. Finally in this section we want to discuss shortly Hausdorff's moment problem. Let (ci/)„gN0 be a sequence of real numbers and put Jc„ := c„ + i — cv. We call the sequence (C 1/ )^ 6 N 0 completely monotone if and only if {-l)k6kcv
>0
for all k,v G N 0 .
(2.103)
40
Chapter 2 Essentials from Analysis
Theorem 2.4.5 (Hausdorff) A sequence {cv)v&io *s completely monotone if and only if there exists a measure on [0,1] such that cu = f tv n(dt), v e No, Jo
(2.104)
holds. A proof of Theorem 2.4.5 is given in [313], pp. 148-154.
2.5
Analytic Functions
The reader is of course assumed to have some knowledge in the theory of complex-valued functions of one complex variable. As a standard reference we mention the monograph [2] of L. Ahlfors which is to our opinion still one of the best books in the field. Furthermore, we would like to mention R. Burckel's book [51] which is ideal in both precision and historical scholarship. From the theory of complex-valued functions of one variable we quote only the theorem of G. Herglotz, see [122], pp.508-511, and Hadarmard's threeline-theorem, see [51], p.147. Theorem 2.5.1 A. (Herglotz) Denote the unit disk in C by D and let f : D —> C be a function. This function is analytic with Re f > 0 if and only if it has the representation
f(z) =ic+ T ^±^
M(dC),
z € D,
(2.105)
where /i is a finite measure on (—n, w] andc = lmf(0) € R. The representation is unique. B. (Hadamard) Let Cl :— {x + iy | 0 < x < 1, y G R} and Cl its closure. Further let f be a bounded continuous function on fi which is analytic in Cl. Then the function M 7 := sup {|f(7 + ij/)| | y e R } satisfies M1 < M^1'Ml for 0 < 7 < 1. Here we want to discuss shortly some basic results from the theory of complex-valued functions of several complex variables and the theory of functions of one variable with values in some topological vector spaces. As standard references for several complex variable theory we mention L. Hormander's book
2.5
Analytic Functions
41
[151] or the book of St. Krantz [189]. A good reference for functions of one complex variable with values in a topological vector space is W. Rudin's book [255], we refer also to the monograph [179] of T. Kato. Let G C C™ be an open set and f : G - > C a continuous function. We call f analytic (or holomorphic) in G if f is analytic in each of its variables, i.e. for any point a = (a\,..., an) £ G the functions Zj H> gj(zj)
:= f(ai,...,
a,j + zj:...,
an), l<j X be a weakly analytic function. Then the following assertions hold: 1. f is a strongly analytic
function;
2. if j is a closed path in G, such that ind 7 (w) — 0 for every w 0 G,
then
we have f[(z)dz
= 0,
(2.111)
and if z £ G and ind 7 (z) = 1
f(z) = J_ I l^L dw. v ;
2niJ7w-
z
(2.112)
2.6
43
Functions and Distributions
Obviously, formula (2.111) is Cauchy's theorem and formula (2.112) is Cauchy's integral formula in this new situation. From (2.111) it follows t h a t if 71 and 72 are two closed paths in G such t h a t for all w £ G we have i n d 7 l ( w ) = ind 7 2 (w), it follows t h a t f f(z) dz=
f
f(z) dz.
(2.113)
Moreover, the function f is strongly continuous.
2.6
Functions and Distributions
T h e purpose of this section is to collect various material for functions and distributions as it is needed in later chapters. In principle it should be possible to find these results in L.Hormander's monograph [152]. As further standard reference we mention F. Friedlander [97], W. Rudin [255], and C. Zuily [317]. In our presentation we often used the book [165], First let us introduce some spaces of functions and their topologies. T h e Schwartz space S ( R " ) consists of all functions u € C°°(R n ) such t h a t for all m i , m2 € No R » l i m j ( u ) := sup ((1 + \x\2)m^2 I 6 R
"
V
\dau(x)\)
< 00
(2.114)
\I
Moreover, for any function £ C°°(R n ) which together with all its partial derivatives is polynomially bounded, i.e. \da(f)(x)\ < pa(x) for all a £ NQ and with suitable polynomial pa, the function 4> • u belongs to <S(R"). For <S(R") we have the following density results.
44
Chapter 2 Essentials from Analysis
C o r o l l a r y 2 . 6 . 1 A. The space Cg°(R n ) is dense in <S(R n ). S(Rn) is dense in L p ( R n ) for p > 1 and in C 0 o ( R " ) .
B. The space
Moreover, using the Friedrichs mollifier we have L e m m a 2.6.2 For any u G L p ( R n ) , u > 0 a.e., there exists a (u„)t,gN, u„ G <S(Rn) and u„ > 0, such that lim ||u„ — u|| L P = 0.
sequence
V—XX)
n
n
Let T : R ->• R be a differentiable mapping and u G <S(R n ). In general u o T does not belong to <S(R"). It is sufficient to take T(a;) = CQ € R n for a fixed value XQ such t h a t 4>(CQ) ^ 0. Then u o T is a non-zero constant which does not belong to <S(R n ). However, if T is a diffeomorphism such t h a t all its derivatives are bounded, then u o T £ <S(R n ). Of course, u o T is the pullback of the function u G <S(R") under the mapping T and often it is denoted by T*u := u o T. In particular, for any bijective linear mapping T : R™ —>• R n we have u o T G <S(R") for all u G <S(R"). We introduced already the space Co°(R"), and more generally the space C Q ° ( G ) , G C K " open. Defining a topology on Co°(G) is more complicated. First we give a topology on C°°(G). For this let (Kj)j&^ be a sequence of compact sets Kj C G such t h a t Kj C Kj+\ and U j l i Kj = ^> i-e- (Kj)j€® is a compact exhaustion of G. On C°°(G) we obtain a separating family of seminorms turning C°°(G) into a Frechet space by P«,K-,(u) := sup l ^ u ^ ) ! ,
a G NJ.
(2.117)
xeKj
It is obvious t h a t a second compact exhaustion (K'j)jew will lead to an equivalent family of seminorms. Let K C R™ be a compact set and m G N 0 U {oo}. T h e space C^(K) is defined by Cf(K)
:= {u : R™ -> C | u G C m ( R " ) and u|*= = 0 } .
(2.118)
(Note t h a t this is a definition slightly different to t h a t given in Section 2.1 for the space CQ(E) and a general topological space, but this will not lead to any confusion later on.) On ^(K), m G No, we have a norm (hence a family of separating seminorms) given by Wc(u):=gm(u):=
£ i|a]<m \s
sup|cTu(z)lx€K
(2.119)
2.6
Functions and Distributions
Moreover, C^(K) ('(R") having a continuous extension to <S(R n ), i.e. 5 ' ( R n ) C 2?'(R n ). B. The topological dual space S'(G) of C°°(G) is called the space of distributions with compact support. It consists of all distributions u G T>'(G) having a continuous extension to C°°(G), i.e. £'(G) C V(G). In case of G = R n we also have £ ' ( R n ) C <S'(R n ). For a distribution u G W(G) we define its support supp u as follows. For G C G C R n , G, G open sets, we have always C Q ° ( G ) C C O ° ( G ) . Hence, we may consider u| C oo«3\
46
Chapter 2 Essentials from Analysis
which is an element in 2^'(G'). Two distributions u, v G 2?'(G) are said to coincide on G C G if their restrictions to C Q ° ( G ) coincide. T h e support of u G T>'(G) is by definition the complement of the largest open set G C G on which u coincides with 0 £ V(G). In the sense of this definition S'(G) is the space of all elements u € V'(G) having a compact support. Let us collect some examples. For u G Ljoc(G) a distribution is defined by u() := fG(x)u(x) dx, 4> G C Q ° ( G ) . Moreover, for a locally finite measure \i on B(G) we may define fi(<j>) := JG (j>(x) fi(dx) which gives also a distribution. In each of these cases we identify u or jj, with the corresponding distribution. Now we have V(G)
C L/ o c (G) C P ' ( G ) ,
(2.120)
and MC(G)
C V'{G).
(2.121)
Furthermore, it follows t h a t L P ( R " ) , 1 < p < oo, all polynomials and all measurable, polynomially bounded functions belong to <S'(R n ). In addition, if/x G Mc(Rn) such t h a t there exists N G N with the property t h a t / R „ (1 + \x\2)~N n \fj,\(dx) < oo, then fi belongs to S ' ( R ) . In particular, we have L e m m a 2 . 6 . 7 For u G L P ( R " ) , 1 < p < oo, and a polynomially measurable it follows that u G £ ' ( R n ) .
bounded
Clearly, X>'(G), S'{G) and S ' ( R n ) are vector spaces and we may consider t h e m as topological vector spaces when taking the weak-*-topology on these spaces. W i t h these topologies we have the following density results. L e m m a 2.6.8 A. The space C Q ° ( G ) is sequentially space <S(R") is sequentially dense in <S'(R™). There are some natural operations on V(G). and-0GCoo(G)
dense in T>'(G). B. The
We may define for u G T>'(G)
(dau)(<j>) := ( - l ) H u ( d a 0 )
(2.122)
(tf>u)(0:=u(^)
(2.123)
and
2.6
47
Functions and Distributions
for all <j> S Cg 0 (G r ). T h e convolution of functions with distributions can be denned in several situations. T h e o r e m 2 . 6 . 9 For a distribution fined by
u and a function
the convolution
{u*)(x):=u(<j>(x-.)) in each of the following
andGG^(G);
2. u e £ ' ( G )
and4>€C°°(G);
3. u G S'(R ) Moreover,
(2.124)
situations:
1. u e P ' ( G )
n
is de-
and <j> G «S(R n ).
in each case u * is a C°° -function
and we have
da(u*) = (dau)*(f> = u*(da), and for u G <S'(R"), <j> G <S(R n ), i/ie function growth.
(2.125) u * 0 /ias ai mosi
polynomial
Using the convolution we may extend the concept of the Friedrichs mollifier to V(G) by setting J £ (u) := u £ := u * j e . It turns out t h a t u £ converges in V{G) to u. Moreover, when u G £'{G) or S'(G), u£ belongs to these spaces too and t h e convergence takes place in the topology of these spaces. It is also possible to define the convolution for two distributions u i and U2 provided one of them has a compact support: (ui *u2)(c/>) : = u i *(u 2 * u(tp(., y)) belongs to C°°(G') and we have d£(uOK.,y))=u(c£V(-,2/))-
(2.130)
Suppose that T : G\ —>• G^ is a diffeomorphism, G\, G^ C R™ being open sets. Clearly, for <j> G Cg°(G,2) it follows that o T G Cg°(Gi), thus in the notation introduced before, the pullback T*0 belongs to Co°(Gi). For u G V'{G2) we define the pullback T*u G V(G{) by
^^:=U(^W(T_1)V)'
(2J31)
for all <j> G Co°(Gi). Obviously, det(dT) denotes the Jacobi determinant of the differential of T. Whenever T is such that T*4> G <S(R") for all
'(Rn \ {0}) be a homogeneous tension to V'{Rn). Then u belongs to <S'(Kn)
distribution
with ex-
We will use two results on the structure of distributions. T h e o r e m 2 . 6 . 1 3 A. Suppose that u G V'(Rn) there exists a number m G No such that
u=
and that supp u = {0}.
J2 c G Co°(G), > 0, we have u() > 0. Then there exists a measure \i G M.+ {G) such that u() = f 4>{x) n(dx)
(2.134)
JG
for allege
Cg°(G).
(Recall t h a t measures are always non-negative in our definition). In order to apply Theorem 2.6.13.B, let's say in case G = R n , it is sufficient to show t h a t u(|(£| 2 ) > 0 for all <j> G Cg°(R n ). T h e problem is of course t h a t in general the square root of a function <j> G Co°(R™), (j) > 0, need no longer belong to Co°(]R n ). But one can smooth it out by using the Friedrichs mollifier in an appropriate way. Let Gj C Rnj be two open sets and u^- G C(Gj), j — 1,2. On Gi x G 2 C R™1 x R" 2 we define the function ui u 2 by (ui ®u 2 )(a;i,a;2) : = u i ( a j i ) • u 2 ( x 2 ) , Xj G Gj. We call ui®U2 the tensor product of ui and U2- T h e space C 0 X > ( G I ) ( 2 ) C Q D ( G 2 ) is the set of all finite linear combinations Y^T ,- 2= i Uji ®Uj 2 where Ujk G Co°(Gfc). T h e tensor product generalises to distributions. T h e o r e m 2 . 6 . 1 4 Let Uj G V(Gj), j = 1,2. Then there exists a unique distribution u = u i ® u 2 G £>'(Gi x G 2 ) called the tensor product of u i with u 2 such that u(4>i ® fa) = ui(