## 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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