On Almost Uniform Continuity of Borel Functions on Polish Metric Spaces

TL;DR

This paper proves that on any Polish metric space with a finite Borel measure, every Borel function can be approximated almost uniformly by bounded, uniformly continuous functions, extending known results to more general spaces.

Contribution

It establishes that Borel functions on Polish metric spaces are almost uniformly continuous in measure, broadening the scope beyond locally compact spaces.

Findings

Every Borel function can be approximated in measure by bounded, uniformly continuous functions.

Bounded Borel functions can be approximated in L^p norm by uniformly continuous functions.

The results extend the known almost uniform continuity to more general measure spaces.

Abstract

We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$ there is some bounded, uniformly continuous function such that the set of points at which they would not agree has measure $< \varepsilon$. In particular, this result complements the known result of almost uniform continuity of Borel real-valued functions on a finite Radon measure space whose ambient space is a locally compact metric space. As direct applications in connection with some common modes of convergence, under our assumptions it holds that i) for every Borel real-valued function there is some sequence of bounded, uniformly continuous functions converging in measure to it, and ii) for every bounded, Borel real-valued function there is some sequence of bounded, uniformly continuous functions converging in $L^{p}$ to it.