From 4acc8fdf312dc173e6a6dc6713afe90e738eec03 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 28 Jun 2026 12:05:22 -0400 Subject: [PATCH] Un-retracted some things. --- src/fa/lp/ui.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/fa/lp/ui.tex b/src/fa/lp/ui.tex index ebad430..d2daa92 100644 --- a/src/fa/lp/ui.tex +++ b/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 $f \in F$.