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Version of 26.8.13 Chapter 46 Pointwise Compact Sets of Measurable Functions This Chapter Collects Results Inspired by Problems in Functional Analysis

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Version of 26.8.13 Chapter 46 Pointwise Compact Sets of Measurable Functions This Chapter Collects Results Inspired by Problems in Functional Analysis Version of 26.8.13 Chapter 46 Pointwise compact sets of measurable functions This chapter collects results inspired by problems in functional analysis. 461 and 466 look directly at measures on linear topological spaces. The primary applications are of course§§ to Banach spaces, but as usual we quickly find ourselves considering weak topologies. In 461 I look at ‘barycenters’, or centres of mass, of probability measures, with the basic theorems on existence§ and location of barycenters of given measures and the construction of measures with given barycenters. In 466 I examine topological measures on linear spaces in terms of the classification developed in Chapter 41.§ A special class of normed spaces, those with ‘Kadec norms’, is particularly important, and in 467 I sketch the theory of the most interesting Kadec norms, the ‘locally uniformly rotund’ norms. § In the middle sections of the chapter, I give an account of the theory of pointwise compact sets of measurable functions, as developed by A.Bellow, M.Talagrand and myself. The first step is to examine pointwise compact sets of continuous functions ( 462); these have been extensively studied because they represent an effective tool for investigating weakly§ compact sets in Banach spaces, but here I give only results which are important in measure theory, with a little background material. In 463 I present results on the relationship between the two most important topologies on spaces of measurable§ functions, not identifying functions which are equal almost everywhere: the pointwise topology and the topology of convergence in measure. These topologies have very different natures but nevertheless interact in striking ways. In particular, we have important theorems giving conditions under which a pointwise compact set of measurable functions will be compact for the topology of convergence in measure (463G, 463L). The remaining two sections are devoted to some remarkable ideas due to Talagrand. The first, ‘Talagrand’s measure’ ( 464), is a special measure on I (or ℓ∞(I)), extending the usual measure of I in a canonical way. In 465§ I turn to the theory of ‘stable’P sets of measurable functions, showing howP a concept arising naturally§ in the theory of pointwise compact sets led to a characterization of Glivenko-Cantelli classes in the theory of empirical measures. Version of 9.7.08 461 Barycenters and Choquet’s theorem One of the themes of this chapter will be the theory of measures on linear spaces, and the first fundamental concept is that of ‘barycenter’ of a measure, its centre of mass (461Aa). The elementary theory (461B- 461E) uses non-trivial results from the theory of locally convex spaces ( 4A4), but is otherwise natural and straightforward. It is not always easy to be sure whether a measure has§ a barycenter in a given space, and I give a representative pair of results in this direction (461F, 461H). Deeper questions concern the existence and nature of measures on a given compact set with a given barycenter. The Riesz representation theorem is enough to tell us just which points can be barycenters of measures on compact sets (461I). A new idea (461K-461L) shows that the measures can be moved out towards the boundary of the compact set. We need a precise definition of ‘boundary’; the set of extreme points seems to be the appropriate concept (461M). In some important cases, such representing measures on boundaries are unique (461P). I append a result identifying the extreme points of a particular class of compact convex sets of measures (461Q-461R). 461A Definitions (a) Let X be a Hausdorff locally convex linear topological space, and µ a probability measure on a subset A of X. Then x∗ X is a barycenter or resultant of µ if gdµ is defined and ∈ A equal to g(x∗) for every g X∗. Because X∗ separates the points of X (4A4Ec), µRcan have at most one barycenter, so we may speak∈ of ‘the’ barycenter of µ. Extract from Measure Theory, by D.H.Fremlin, University of Essex, Colchester. This material is copyright. It is issued under the terms of the Design Science License as published in http://dsl.org/copyleft/dsl.txt. This is a de- velopment version and the source files are not permanently archived, but current versions are normally accessible through https://www1.essex.ac.uk/maths/people/fremlin/mt.htm. For further information contact [email protected]. c 2000 D. H. Fremlin 1 2 Pointwise compact sets of measurable functions 461Ab (b) Let X be any linear space over R, and C X a convex set (definition: 2A5E). Then a function f : C R is convex if f(tx + (1 t)y) tf(x)+(1⊆ t)f(y) for all x, y C and t [0, 1]. (Compare 233G,→ 233Xd.) − ≤ − ∈ ∈ (c) The following elementary remark is useful. Let X be a linear space over R, C X a convex set, and f : C R a function. Then f is convex iff the set (x,α) : x C, α f(x) is convex⊆ in X R (cf. 233Xd). → { ∈ ≥ } × 461B Proposition Let X and Y be Hausdorff locally convex linear topological spaces, and T : X Y a continuous linear operator. Suppose that A X, B Y are such that T [A] B, and let µ be a probability→ ⊆ ⊆ ⊆ 1 measure on A which has a barycenter x∗ in X. Then Tx∗ is the barycenter of the image measure µT − on B. proof All we have to observe is that if g Y ∗ then gT X∗ (4A4Bd), so that ∈ ∈ g(Tx )= g(Tx)µ(dx)= g(y)ν(dy) ∗ A B by 235G1. R R 461C Lemma Let X be a Hausdorff locally convex linear topological space, C a convex subset of X, and f : C R a lower semi-continuous convex function. If x C and γ<f(x), there is a g X∗ such that g(y)+→γ g(x) f(y) for every y C. ∈ ∈ − ≤ ∈ proof Let D be the convex set (x,α): x C, α f(x) in X R (461Ac). Then the closure D of D in { ∈ ≥ } × X R is also convex (2A5Eb). Now D is closed in C R (4A2B(d-i)), and (x,γ) / D, so (x,γ) / D. × × ∈ ∈ Consequently there is a continuous linear functional h : X R R such that h(x,γ) < infw D h(w) × → ∈ (4A4Eb). Now there are g0 X∗, β R such that h(y,α)= g0(y)+βα for every y X and α R (4A4Be). So we have ∈ ∈ ∈ ∈ g0(x)+ βγ = h(x,γ) <h(y,f(y)) = g0(y)+ βf(y) for every y C. In particular, g (x)+ βγ <g (x)+ βf(x), so β > 0. Setting g = 1 g , ∈ 0 0 − β 0 f(y) 1 g (x)+ γ 1 g(y)= g(y)+ γ g(x) ≥ β 0 − β − for every y C, as required. ∈ 461D Theorem Let X be a Hausdorff locally convex linear topological space, C X a convex set and ⊆ µ a probability measure on a subset A of C. Suppose that µ has a barycenter x∗ in X which belongs to C. Then f(x∗) f dµ for every lower semi-continuous convex function f : C R. A ≤ R → proof Take any γ<f(x∗). By 461C there is a g X∗ such that g(y)+ γ g(x∗) f(y) for every y C. Integrating with respect to µ, ∈ − ≤ ∈ fdµ γ g(x∗)+ gdµ = γ. A ≥ − A R R As γ is arbitrary, f(x∗) f dµ. A ≤ R Remark Of course f dµ might be infinite. A R 461E Theorem Let X be a Hausdorff locally convex linear topological space, and µ a probability measure on X such that (i) the domain of µ includes the cylindrical σ-algebra of X (ii) there is a compact convex set K X such that µ∗K = 1. Then µ has a barycenter in X, which belongs to K. ⊆ proof If g X∗, then γg = supx K g(x) is finite, and x : g(x) γg is a measurable set including K, ∈ ∈ | | { | | ≤ } so must be conegligible, and φ(g) = gdµ is defined and finite. Now φ : X∗ R is a linear functional → and φ(g) supx K g(x) for every g RX∗; because K is compact and convex, there is an x0 K such that ≤ ∈ ∈ ∈ φ(g)= g(x ) for every g X∗ (4A4Ef), so that x is the barycenter of µ in X. 0 ∈ 0 1Formerly 235I. Measure Theory 461G Barycenters and Choquet’s theorem 3 461F Theorem Let X be a complete locally convex linear topological space, and A X a bounded set. Let µ be a τ-additive topological probability measure on A. Then µ has a barycenter in⊆X. proof (a) If g X∗, then g↾A is continuous and bounded (3A5N(b-v)), therefore µ-integrable. For each neighbourhood ∈G of 0 in X, set F = y : y X, g(y) gdµ 2τ (g) for every g X∗ G { ∈ | − A |≤ G ∈ } R where τG(g) = supx G g(x) for g X∗. Then FG is non-empty. PPP Set H = int(G ( G)), so that H is an open neighbourhood∈ of 0. Because∈ A is bounded, there is an m 1 such that∩A − mH. The set ≥ ⊆ x+H : x A is an open cover of A, and µ is a τ-additive topological measure, so there are x0,... ,xn A { ∈ } 1 ∈ such that µ(A (x + H)) . Set \ i n i ≤ m S ≤ E = A (x + H) (x + H) i ∩ i \ j<i j n S for i n, and y = (µE )x ; set E = A (x + H).
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