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Baire classes of Banach spaces and strongly affine functions. (English) Zbl 1215.46009

This paper represents a further investigation into the theory of affine functions on compact convex subsets of locally convex Hausdorff topological vector spaces in the tradition of Gustave Choquet.
Let \(A(X)\) be the space of real-valued continuous affine functions on a compact convex subset \(X\) of a locally convex Hausdorff space, let \(A_1(X)\) be the space of pointwise limits of bounded sequences of elements of \(A(X)\), and let \(A_2(X)\) be the space of pointwise limits of bounded sequences of elements of \(A_1(X)\). Let \(B_1(X)\) be first Baire class of functions on \(X\) consisting of pointwise limits of bounded sequences of continuous functions and let \(B_2(X)\) be the second Baire class of functions consisting of pointwise limits of bounded sequences of elements of \(B_1(X)\). A real-valued affine function \(f\) on \(X\) lying in \(B_2(X)\) such that, for any Radon probability measure \(\mu\) on \(X\),
\[ \int_X f(x)\,d\mu(x)= f(r(\mu)), \]
where \(r(\mu)\) is the point in \(X\) that is the barycentre of \(\mu\), is said to be a strongly affine element of \(B_2(X)\), or, alternatively, to lie in the space \(B_2(X)\cap A_{\text{bf}}(X)\). Using a subtle inductive process, the author constructs a metrizable Choquet simplex \(X\) and an element \(f\) of \(B_2(X)\cap A_{\text{bf}}(X)\) which does not lie in \(A_2(X)\). A result of Talagrand shows that a compact convex set can be found for which this holds, but it is a new and interesting result that \(A_2(X)\) and \(B_2(X)\cap A_{\text{bf}}(X)\) can also differ when \(X\) is a metrizable simplex.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
26A21 Classification of real functions; Baire classification of sets and functions
Full Text: DOI

References:

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