Preserving affine Baire classes by perfect affine maps

TL;DR

This paper proves that under certain conditions, the affine Baire class of a function on a compact convex set is preserved when composed with a perfect affine surjection, extending known results in affine Baire class theory.

Contribution

It establishes a general condition under which affine Baire classes are preserved by perfect affine maps, broadening the understanding of affine Baire class stability.

Findings

Affine Baire class preservation under perfect affine maps.

Extension of known results on affine Baire classes.

Application to strongly affine Baire mappings.

Abstract

Let $\varphi\colon X\to Y$ be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on $Y$ in the space of real-valued affine continuous functions on $X$ is complemented. We show that if $F$ is a topological vector space, then $f\colon Y\to F$ is of affine Baire class $\alpha$ whenever the composition $f\circ\varphi$ is of affine Baire class $\alpha$. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.