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 ε > 0 there is some bounded, uniformly continuous function, such that the set of points at which they would not agree has measure less than ε. 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.