Updated the Lebesgue non-negative integral formula.

src/measure/lebesgue-integral/simple.tex

@@ -16,7 +16,7 @@
     is the \textbf{Lebesgue integral} of $f$.
 \end{definition}
 
-\begin{proposition}[{{\cite[Proposition 2.13]{Folland}}}]
+\begin{proposition}
 \label{proposition:lebesgue-simple-properties}
     Let $(X, \cm, \mu)$ be a measure space and $f, g \in \Sigma^*(X, \cm)$, then:
     \begin{enumerate}
@@ -26,7 +26,7 @@
         \item The mapping $A \mapsto \int \one_A \cdot f d\mu$ is a measure on $(X, \cm)$.
     \end{enumerate}
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Proposition 2.13]{Folland}}}. ]
     (1): If $\alpha = 0$, then $\int \alpha f d\mu = \int 0 d\mu = 0 = 0 \cdot \int f d\mu$. Otherwise, the mapping $y \mapsto \alpha y$ is a bijection. Hence
     \[
     \alpha f = \sum_{y \in f(X)} (\alpha y) \cdot \one_{\bracs{f = y}}