A note on almost uniform continuity of Borel functions on Polish metric spaces

With a simple short proof, this article improves a classical approximation result of Lusin's type; specifically, it is shown 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 less than $\varepsilon$. This result also 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.