Borel regularity is equivalent to Lusin's theorem and the existence of Borel representatives

TL;DR

This paper characterizes Borel regularity in topological measure spaces, showing it is equivalent to Lusin's theorem and the existence of Borel representatives, especially in metric measure spaces.

Contribution

It establishes the equivalence between Borel regularity and key measure-theoretic properties, providing a new characterization in general topological measure spaces.

Findings

Borel regularity is necessary and sufficient for Lusin's theorem.

Borel regularity ensures the existence of Borel representatives.

The results apply specifically to metric measure spaces.

Abstract

In this article, we characterize both Lusin's theorem and the existence of Borel representatives via the regularity properties of the measure in general topological measure spaces. As a corollary, we prove that Borel regularity of the measure is both a necessary and sufficient condition for these results to hold true in metric measure spaces.