Function Borel measurable

Question

A non-decreasing function $f:\mathrm{R} \rightarrow \mathrm{R}$ is that function $\mathcal{B}(\mathrm{R})-measureable$ and is it per definition?

Answer 1

A non-decreasing function $f \colon R \to R$ is Borel-measurable but not per definitionem. A function is Borel-measurable if every preimage of a measurable set is measurable. One can show, that it suffices to prove this condition for generators of the Borel-sigma-algebra like $$\mathcal G =: \{ (a,b) \colon a,b \in R\}.$$

We want to show that for each $(a,b) \in \mathcal G$ the set $f^{-1}((a,b))$ is again measurable. But $f$ is non-decreasing and one can show that for non-decreasing (or non-increasing) functions the preimage of an interval is again an interval (try to prove this!). But the intervals are Borel-measurable which proves your claim.