Radon measures and the dual of C(K), the barycenter map, the Strong Krein-Milman Theorem, and Jensen's integral inequality Preliminary version 0.1. Radon measures. Recall that (S; Σ) is a measurable space if S is a nonempty set and Σ is a σ-algebra of subsets of S. By a measure on (S; Σ) we mean any µ:Σ ! [0; +1] such that µ(;) = 0 and µ is σ-additive, that is, µ(S A ) = P µ(A ) whenever the sets A 2 Σ n2N n n2N n n are pairwise disjoint. Similarly, given a Banach space X, a function µ:Σ ! X is called an X-valued measure whenever µ(;) = 0 and µ is σ-additive. Using this terminology, we see that any finite measure is just a nonnegative R-valued measure. Theorem 0.1.1 (Total variation). Given an X-valued measure µ, the formula ( n n ) X [ jµj(A) = sup kµ(Ai)k : n 2 N, Ai 2 Σ pairwise disjoint, A = Ai ; i=1 1 for A 2 Σ, defines a measure, called the (total) variation of µ, which can be equivalently defined by +1 +1 X [ jµj(A) = sup kµ(Ai)k : Ai 2 Σ pairwise disjoint, A = Ai : i=1 1 Moreover, for X = R the variation jµj is always finite. Given a set E 2 Σ, we can consider the corresponding \restricted" σ-algebra ΣE := fA \ E : A 2 Σg = fA 2 Σ: A ⊂ Eg: If µ is an X-valued measure, the restriction of µ to E is the X-valued measure µE := µjΣE on (E; ΣE). One can also view µE as a measure on (S; Σ) defined by µE(A) = µ(A \ E);A 2 Σ: We shall not make formal distinction between these two views of µE. Theorem 0.1.2 (Hahn and Jordan decompositions). Let µ be an R-valued measure on (S; Σ). Then there exist two disjoint sets P; N 2 Σ such that P [ N = S and + − µ := µP and µ := −µN 1 2 are finite (nonnegative) measures, called the positive variation and the negative variation of µ, respectively. Such a decomposition of S is called the Hahn decomposition. Moreover: (a) the Hahn decomposition (P; N) of S is unique in the following weak sense: 0 0 if (P ;N ) is another such decomposition then µP = µP 0 and µN = µN 0 ; (b) µ = µ+ − µ− and jµj = µ+ + µ−, and hence the variation jµj is a finite measure; + − (c) the measures µ ; µ are minimal in the following sense: if ν1; ν2 are (non- + − negative) measures such that µ = ν1 − ν2 then ν1 ≥ µ and ν2 ≥ µ ; this \minimal" decomposition µ = µ+ − µ− is clearly uniquely determined and it is called the Jordan decomposition of µ; (d) the variations µ+; µ− are given also by n n + X + [ µ (A) = sup µ(Ai) : n 2 N, Ai 2 Σ pairwise disjoint, A = Ai ; i=1 1 n n − X − [ µ (A) = sup µ(Ai) : n 2 N, Ai 2 Σ pairwise disjoint, A = Ai ; i=1 1 where t+ = maxft; 0g and t− = − minft; 0g are the positive and the nega- tive parts of a real number t. 0.2. Radon measures on compact Hausdorff spaces. Let (K; τ) be a compact Hausdorff topological space, and let Borel(K) denote the σ-algebra of Borel sets of K, i.e., the smallest σ-algebra containing all τ-open sets. Definition 0.2.1 (Radon measure). A Radon measure on K is a finite (non- negative) measure µ on (K; Borel(K)) which is regular, that is, for each Borel set B ⊂ K, µ(B) = supfµ(C): C ⊂ B is compactg: It is easy to see that if µ is a Radon measure on K then µ(B) = inffµ(G): G ⊃ B is openg ;B 2 Borel(K): Definition 0.2.2 (Radon X-valued measure). Given a Banach space X, a Radon X-valued measure is an X-valued measure µ on (K; Borel(K)) which is regular in the following sense: 8B 2 Borel(K), 8" > 0, 9C compact, C ⊂ B, 8A 2 Borel(B n C), kµ(A)k < ". 3 Proposition 0.2.3. Let µ be an X-valued measure on (K; Borel(K)). Then µ is a Radon X-valued measure if and only if its variation jµj is a Radon measure. Moreover, for X = R, µ is an R-valued Radon measure if and only if both µ+; µ− are Radon measures. Lemma 0.2.4. Let µ be a Radon (nonnegative) measure on K. Then the union of an arbitrary family of open sets of null µ-measure has null µ-measure as well. Proof. Let G be a family of open sets such that µ(G) for each G 2 G. Put H := S G (the union of all members of G) and assume that µ(H) > 0. By regularity of µ, there exists a compact C ⊂ H with µ(C) > 0. There exists a finite subfamily G ⊂ G that covers C, but then µ(C) ≤ P µ(G) = 0 is a 0 G2G0 contradiction. Corollary 0.2.5 (Existence of support). Every Radon measure µ has a sup- port, that is, the (unique) smallest closed set spt(µ) ⊂ K such that µ is concentrated on spt(µ): µ(K n spt(µ)) = 0. (Indeed, by the above lemma the union of all open sets of null µ-measure is the largest such open set, and spt(µ) is its complement.) Definition 0.2.6. For an X-valued Radon measure µ we define spt(µ) := spt(jµj): Notice that, in the particular case of X = R, if µ is an R-valued Radon measure then spt(µ) = spt(µ+) [ spt(µ−). 0.3. The space of Radon R-valued measures as the dual of C(K). Definition 0.3.1. Let M(K) denote the vector space of all Radon R-valued measures on K, M+(K) the set of all nonnegative Radon measures on K, and M+;1(K) the set of all probability Radon measures of K (i.e., R-valued measures µ 2 M(K) such that µ ≥ 0 and µ(K) = 1). Theorem 0.3.2 (The Banach space M(K)). The formula kµkM := jµj(K) defines a norm on M(K) in which M(K) is a Banach space (even a Banach lattice). 4 For µ 2 M(K) and f 2 L1(µ) := L1(jµj), the corresponding integral is defined as Z Z Z f dµ := f dµ+ − f dµ− : K K K Theorem 0.3.3 (Riesz Representation Theorem). For every compact Haus- dorff topological space K, C(K)∗ = M(K) in the following sense: there exists a bijective correspondence, which is a linear isometry, between the two above spaces, and this correspondence C(K)∗ 3 Φ 7! µ 2 M(K) is given by the representation formula Z Φ(u) = u dµ for each u 2 C(K). K Corollary 0.3.4. Notice that the above theorem implies that Z jµj(K) = sup u dµ : u 2 C(K); kuk1 ≤ 1 K for every µ 2 M(K). 0.4. Extreme points in M(K). Given a Banach space Y , let BY denote its closed unit ball. Keeping in mind the Riesz Representation Theorem, we can consider on M(K) = C(K)∗ also the corresponding w∗-topology σ(M(K);C(K)). Then the sets BM(K) and M+;1(K) ∗ ∗ are convex and w -compact. Indeed, BM(K) is w -compact by the Alaoglou Theorem, and M+;1(K) = BM(K) \ fµ : µ(1) = 1g where 1 2 C(K) is the constant function 1(t) = 1 (t 2 K). Notice that, since the hyperplane fµ : µ(1) = 1g is a supporting hyperplane to BM(K), this also implies that M+;1(K) is an extremal set for BM(K). Given a closed (hence compact) nonempty set F ⊂ K, we clearly can identify the set fµ 2 M(K) : spt(µ) ⊂ F g = fµ 2 M(K): jµj(K n F ) = 0g with M(F ). So we shall often consider M+;1(F ) as a subset of M+;1(K). The following simple lemma shows that this subset is w∗-closed. F Observation 0.4.1. Let F be a (nonempty) closed set in a compact Hausdorff ∗ space K. Then M+;1(F ) is convex and w -closed in M(K). 5 ∗ Proof. We already know (see above) that M+;1(F ) is convex and wC(F )∗ - ∗ ∗ compact (where wC(F )∗ is the w -topology σ(M(F );C(F )) ). Now it suffices to notice that Tietze's extension theorem and the definition of the w∗-topologies easily imply that ∗ ∗ (M+;1(F ); wC(K)∗ ) = (M+;1(F ); wC(F )∗ ): We are done. ∗ By the Krein-Milman Theorem, each of the sets BM(K), M+;1(K) is the w - closed convex hull of its extreme points. Let us determine the extreme points of the two sets. Given t 2 K, let δt denote the corresponding Dirac measure, that is ( 1 if t 2 A, δ (A) = t 0 if t2 = A. Though this measure is defined for all subsets of K, we can view δt (by restricting it to Borel(K)) as an element of M(K). We clearly have even δt 2 M+;1(K). Exercise 0.4.2. Consider the \Dirac map" δ : K !M(K) that assigns to each t 2 K its Dirac measure δt. (a) Show that in the norm of M(K) the image δ(K) is a discrete metric space whose any two distinct points have distance 2.