Added a variation bound for scalar measures.

src/measure/vector/variation.tex

@@ -90,3 +90,24 @@
     Let $(X, \cm)$ be a measurable space and $\mu$ be a signed/vector measure, then $\mu$ is \textbf{finite} if $|\mu|$ is finite. 
 \end{definition}
 
+\begin{lemma}
+\label{lemma:variation-bound}
+    Let $(X, \cm)$ be a measurable space, $K \in \RC$, $\mu: \cm \to K$ be a signed/complex measure, then:
+    \begin{enumerate}
+        \item If $K = \real$, then $|\mu|(X) \le 2\sup_{A \in \cm}\norm{\mu(A)}_E$.
+        \item If $K = \complex$, then $|\mu|(X) \le 4\sup_{A \in \cm}\norm{\mu(A)}_E$. 
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+    (1, scalar): Let $X = P \sqcup N$ with $P, N \in \cm$ be a \hyperref[Hahn decomposition]{theorem:hahn-decomposition} of $\mu$, then
+    \[
+    |\mu|(X) = |\mu(X \cap P)| + |\mu(X \cap N)| \le 2\sup_{A \in \cm}\norm{\mu(A)}_E
+    \]
+
+    (2, scalar): By (1), 
+    \[
+    |\mu|(X) \le |\text{Re}(\mu)|(X) + |\text{Im}(\mu)|(X) \le 4\sup_{A \in \cm}\norm{\mu(A)}_E
+    \]
+\end{proof}
+
+