Problem. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a monotone increasing function. Show that $f$ is measurable.
Solution. We know that the set of discontinuites of any monotone increasing function $f$ is measure zero (since it is at most countable). We define a continuous function $g$ such that $g(x)=f(x)$ except the discontinuous points of $g$. Then $g(x)=f(x)$ almost everywhere. Note that any continuous function $g: \mathbf{R}\rightarrow \mathbf{R}$ is measurable. Also note that if $g$ is measurable and $f=g$ almost everywhere, then $f$ is measurable. Hence we conclude that $f$ is measurable.
Is my solution correct? Thanks.