Let $(R, B)$, where $B$ is the Borel $\sigma$-algebra, be our measurable space.
In this measure space how do you that the function $$f(x) = \lfloor x\rfloor$$ is measurable?
2026-03-25 03:01:46.1774407706
Is the function $f(x) = \lfloor x\rfloor$ is measurable?
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Yes: in particular, it's a countable sum of step functions, so is a uniform limit of step functions, so is regulated, and all regulated functions are measurable.