A Note on Almost Uniform Continuity of Borel Functions on Polish Metric 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.