Added the Legendre transform.

src/topology/main/semicontinuity.tex

@@ -22,7 +22,7 @@
 \end{proof}
 
 
-\begin{proposition}[{{\cite[Proposition 7.11]{Folland}}}]
+\begin{proposition}
 \label{proposition:semicontinuous-properties}
     Let $X$ be a topological space, then
     \begin{enumerate}
@@ -33,7 +33,7 @@
         \item For any $f: X \to (-\infty, \infty]$ lower semicontinuous, $f$ is Borel measurable. 
     \end{enumerate}
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Proposition 7.11]{Folland}}}. ]
     (1): For any $\alpha \in \real$, 
     \[
     \bracs{f > \alpha} = \begin{cases}