A study on Choquet's Theorems and their applications

Teena, Thomas and Paul, Tanmoy (2019) A study on Choquet's Theorems and their applications. Masters thesis, Indian institute of technology Hyderabad.

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Abstract

This project is a literature survey of various theorems and their applications in Choquet theory. For a compact convex subset D of a locally convex topological vector space E, each point x P D is a barycentre of a maximal probability measure on D. This is, in fact, a generalized version of Minkowski's Theorem for finite dimensional spaces. This measure exists uniquely if the compact convex set is a simplex. If the compact convex set is metrizable then the above measure is supported by the ext(D) but very few information is available if the set is non-metrizable. Measures supported by the extreme points are the maximal measures. For a non-metrizable compact convex set the set of all extreme points may not be of Borel category, hence for such cases, the support of maximal measure can have a non-empty intersection with a Borel set disjoint from the extreme boundary. The Choquet-Bishop-De Leeuw Theorem, hence, states that - For an arbitrary locally convex topological vector space, each point of a compact convex subset is represented by a maximal probability measure which gives zero value to all Baire sets disjoint from the extreme points. Further, we study the analysis of function spaces, namely, CpKq in the context of Choquet boundary. If M is a uniform algebra of continuous functions over a compact Hausdorff space K then the state space of M is defined; it is a w*-compact convex subset of M*. The extreme points of the state space are precisely the point evaluation functionals. This motivates to define the Choquet boundary of M, as a subset of K. Choquet boundary is a boundary and its closure is the smallest closed boundary for M, called the Silov boundary. Here we also study the notion of peak point and the result that when K is metrizable then the set of all peak points is dense in the Choquet Boundary. As an application of this notion, we discuss the well- known result by Saskin, which states that for a Korovkin subspace of C(K) the Choquet boundary is the whole K and also vice versa.

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