  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.
