Adjusted the in measure version of DCT.

src/fa/lp/ui.tex

@@ -144,8 +144,8 @@
 \label{corollary:dct-filter}
     Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g, h \in L^p(X; \real)$ such that:
     \begin{enumerate}[label=(\alph*)]
-        \item $\fF \to g$ locally in measure.
-        \item There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$. 
+        \item[(M)] $\fF \to g$ locally in measure.
+        \item[(D)] There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$. 
     \end{enumerate}
     
     then $\fF \to f$ in $L^p(X; E)$. In particular, if $p = 1$, then
@@ -154,7 +154,7 @@
     \]
 \end{corollary}
 \begin{proof}
-    By \autoref{theorem:vitali-convergence}. 
+    Since (D) implies (UI) and (T) of the \hyperref[Vitali Convergence Theorem]{theorem:vitali-convergence}, the result follows from the Vitali Convergence Theorem. 
 \end{proof}
 
 \begin{lemma}[Scheffé]