Un-retracted some things.

src/fa/lp/ui.tex

@@ -48,7 +48,7 @@
 
 \begin{theorem}[Vitali Convergence Theorem]
 \label{theorem:vitali-convergence}
-    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a separable 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:
+    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 locally Cauchy in measure. 
         \item[(UI)] For each $\eps > 0$, there exists $M \ge 0$ and $F \in \fF$ such that
@@ -142,7 +142,7 @@
 
 \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 separable 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:
+    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[(M)] $\fF \to g$ locally in measure.
         \item[(D)] There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$.