Borel 1 type mappings and the respective equi-families

TL;DR

This paper studies classes of functions related to Borel class 1, introduces equi-families, and explores their properties and closures, extending previous results to functions with sections that are equi-continuous or have generalized Lebesgue properties.

Contribution

It defines and analyzes equi-families of Borel class 1 functions, reducing their study to orbit maps and extending results to functions with specific section properties.

Findings

Equi-families can be characterized via orbit maps in product spaces.

Closure of equi-families under pointwise convergence is studied.

Generalization of Grande's result to functions with equi-continuous or Lebesgue sections.

Abstract

We investigate classes of functions from a topological space to a metric space that are related to those of Borel class 1. Following the idea defining an equi-Baire 1 family (due to Lecomte) we define the respective equi-families of functions from the considered classes. We observe that studying of equi-families can be reduced to the exploration of a single orbit map with values in a product space. We consider the closure of equi-families with respect to the topology of pointwise convergence. Finally, we investigate functions $f\colon X\times Y\to Z$, for metric spaces $X,Y,Z$, with sections that are equi-continuous, equi-Baire~1 or have equi-generalized Lebesgue property with respect to measurable sets of class $\alpha$. In particular, we generalize a result of Grande.