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A new type of affine Borel function. (English) Zbl 0562.46005

A function f defined on a convex compact set K is called strongly affine if for each probability measure \(\mu\) on K, we have \(F(b_{\mu})=\int_{K}fd\mu\), where \(b_{\mu}\) is the barycenter of \(\mu\). We construct a separable Banach space E which has the Schur property and such that there is \(x\in E^{**}\setminus E\) which is Borel and strongly affine on \((E^*,weak^*)\). If K denotes the unit ball of \((E^*,weak^*)\), x is Borel, affine and strongly affine on K. However, x cannot be obtained from affine continuous functions on K by taking pointwise limits and repeating this operation any number of times.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
03E15 Descriptive set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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