Added MCT for convergence in measure.

src/fa/lp/ui.tex

@@ -50,7 +50,7 @@
 \label{theorem:vitali-convergence}
     Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, and $\fF \subset 2^{L^p(X; E)}$ be a filter, then $\fF$ is Cauchy in $L^p(X; E)$ if and only if:
     \begin{enumerate}
-        \item[(M)] $\fF$ is Cauchy in measure. 
+        \item[(M)] $\fF$ is locally Cauchy in measure. 
         \item[(UI)] For each $\eps > 0$, there exists $M \ge 0$ and $F \in \fF$ such that
         \[
         \sup_{f \in F}\int_{\bracs{\norm{f}_E \ge M}}\norm{f}_E^p d\mu < \eps
@@ -106,7 +106,7 @@
     
     Assume without loss of generality that $\mu(A) > 0$ and let $\delta = \eps\mu(A)^{-1/p}$. By (M), there exists $F_3 \in \fF$ with $F_3 \subset F_2$, such that for any $f, g \in F_3$, 
     \[
-    \mu\bracsn{\norm{f - g}_E \ge \delta} \le \paren{\frac{\eps}{2M}}^p
+    \mu(A \cap \bracsn{\norm{f - g}_E \ge \delta}) \le \paren{\frac{\eps}{2M}}^p
     \]
     
     In which case,
@@ -126,7 +126,7 @@
     Now, 
     \begin{align*}
         \normn{\one_{A \cap \bracsn{\norm{f - g}_E \ge \delta}}f}_{L^p(X; E)} &\le \normn{\one_{\bracsn{\norm{f}_E \ge M}}f}_{L^p(X; E)} \\
-        &+ \normn{\one_{\bracsn{\norm{f - g}_E \ge \delta, \norm{f}_E \le M}}f}_{L^p(X; E)} \\
+        &+ \normn{\one_{A \cap \bracsn{\norm{f - g}_E \ge \delta, \norm{f}_E \le M}}f}_{L^p(X; E)} \\
         &\le \eps + \eps/2 = 3\eps/2
     \end{align*}
 
@@ -140,3 +140,20 @@
     
 \end{proof}
 
+\begin{corollary}[Dominated Convergence Theorem (In Measure)]
+\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 $\fF \to g$ pointwise and locally in measure.
+        \item 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
+    \[
+    \lim_{f, \fF}\int f d\mu = \int g d\mu
+    \]
+\end{corollary}
+\begin{proof}
+    By \autoref{theorem:vitali-convergence}. 
+\end{proof}
+