$F\sb \sigma$-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.

Keywords

• $F\sb
• \sigma$-additive
• families
• invariance
• Borel
• classes