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

We prove that any $F_\sigma$-additive family $\mathcal{A}$ of sets in an absolutely Souslin metric space has a $\sigma$-discrete refinement provided every partial selector set for $\mathcal{A}$ 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.