How to prove (by definition) that if $f:X\to R$ is measurable function, then [f] is also measurable? Here, I have to prove that for every $[c,\infty)$, $[f]^{-1}([c,\infty))$ is measurable.
Any help is welcome.
Thanks in advance.
2026-04-04 21:40:31.1775338831
Measurable function by definition
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If $c\in\mathbb{Z}$: $[f]^{-1}([c,\infty))=f^{-1}([c,\infty))$
Otherwise: $[f]^{-1}([c,\infty))=\\f^{-1}([c,\infty)) \cap (f^{-1}([c,[c]+1)))^c$
And these sets are measurable.