Added the Radon-Nikodym theorem for finite measures.

src/measure/vector/vector.tex

@@ -42,5 +42,26 @@
 
 \end{proof}
 
+\begin{proposition}
+\label{proposition:vector-measure-properties}
+    Let $(X, \cm)$ be a measurable, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$ be a vector measurethen:
+    \begin{enumerate}
+        \item For any $\seq{A_n} \subset \cm$ with $A_n \subset A_{n+1}$ for all $n \in \nat$, $\mu\paren{\bigcup_{n \in \nat}A_n} = \limv{n}\mu(A_n)$.
+        \item For any $\seq{A_n} \subset \cm$ with $A_n \supset A_{n+1}$ for all $n \in \nat$, $\mu(\bigcap_{n \in \natp}A_n) = \limv{n}\mu(A_n)$.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1): Let $A_0 = \emptyset$. For each $n \in \natp$, let $B_n = A_n \setminus B_{n-1}$, then $\bigcup_{n \in \natp}A_n = \bigsqcup_{n \in \natp}B_n$ and
+    \[
+    \limv{n}\mu(A_n) = \sum_{n = 1}^\infty \mu(B_n) = \mu\paren{\bigcup_{n \in \natp}A_n}
+    \]
+
+    (2): By (1), 
+    \begin{align*}
+        \limv{n}\mu(A_n) &= \mu(A_1) - \limv{n}\mu(A_1 \setminus A_n) \\
+        &= \mu(A_1) - \mu\paren{A_1 \setminus \bigcup_{n \in \natp}A_n} = \mu\paren{\bigcap_{n \in \natp}A_n}
+    \end{align*}
+     
+\end{proof}