Added facts about vector measures.

src/fa/lc/convex.tex

@@ -99,10 +99,16 @@
         \item $[\cdot]$ is uniformly continuous.
         \item $[\cdot]$ is continuous.
         \item $[\cdot]$ is continuous at $0$.
+        \item $\bracs{x \in E| [x] < 1} \in \cn_E(0)$.
     \end{enumerate}
 \end{lemma}
 \begin{proof}
-    $(3) \Rightarrow (1)$: Let $\eps > 0$, then there exists $V \in \cn(0)$ such that $[x] < \eps$ for all $x \in V$. In which case, for any $x, y \in E$ with $x - y \in V$, $\abs{[x] - [y]} \le [x - y] < \eps$. By \autoref{proposition:tvs-uniform}, $[\cdot]$ is uniformly continuous.
+    $(4) \Rightarrow (1)$: Let $x, y \in E$ and $r > 0$. If
+    \[
+    x - y \in \bracs{x \in E|[x] < r} = r\bracs{x \in E|[x] < 1} \in \cn_E(0)
+    \]
+    
+    then $[x - y] < r$.
 \end{proof}