Added Lusin's theorem.

src/measure/measurable-maps/real-valued.tex

@@ -50,3 +50,12 @@
 
     Finally, if the above functions are $\real$-valued, then they are $(\cm, \cb_\real)$-measurable by \autoref{lemma:extended-real-measurable}.
 \end{proof}
+
+\begin{lemma}
+\label{lemma:monotone-measurable}
+    Let $f: \real \to \real$ be a non-decreasing or non-increasing function, then $f$ is Borel measurable.
+\end{lemma}
+\begin{proof}
+    By taking $-f$, assume without loss of generality that $f$ is non-decreasing. In which case, for any $a \in \real$, $x \in f^{-1}((-\infty, a])$, and $y \le x$, $f(y) \le f(x) \le a$, so $y \in f^{-1}((-\infty, a])$. Thus $f^{-1}((-\infty, a))$ is an interval and hence measurable. Since the open rays generate $\cb_\real$ (\autoref{proposition:borel-sigma-real-generators}), $f$ is Borel measurable.
+\end{proof}
+