Geodesic
A note on almost uniform continuity of Borel functions on Polish metric spaces
Problemy analiza, Tome 11 (2022) no. 2, pp. 24-28.

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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.
Keywords: almost uniform continuity, Borel functions, extension theorems, finite Borel measures, Lusin's theorem, Polish metric spaces. 
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Y.-L. Chou. A note on almost uniform continuity of Borel functions on Polish metric spaces. Problemy analiza, Tome 11 (2022) no. 2, pp. 24-28. http://geodesic.mathdoc.fr/item/PA_2022_11_2_a1/

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