$F\sb \sigma$-additive families and the invariance of Borel classes

Abstract

We prove that any $F_\sigma$--additive family $\A$ of sets in an absolutely Souslin metric space has a $\sigma$--discrete refinement provided every partial selector set for $\A$ is $\sigma$--discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps $F_\sigma$--sets to $F_\sigma$--sets and has complete fibers, admits a section of the first class.

The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves $F_\sigma$-sets, is shown as an application of the previous result.