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Adjusted the in measure version of DCT.

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Bokuan Li
2026-06-21 00:22:48 -04:00
parent 5ce01bd9f3
commit 8c5400bf88
1 changed files with 3 additions and 3 deletions
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src/fa/lp/ui.tex
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@@ -144,8 +144,8 @@
\label{corollary:dct-filter} \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: 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*)] \begin{enumerate}[label=(\alph*)]
\item $\fF \to g$ locally in measure. \item[(M)] $\fF \to g$ locally in measure.
\item There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$. \item[(D)] There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$.
\end{enumerate} \end{enumerate}
then $\fF \to f$ in $L^p(X; E)$. In particular, if $p = 1$, then then $\fF \to f$ in $L^p(X; E)$. In particular, if $p = 1$, then
@@ -154,7 +154,7 @@
\] \]
\end{corollary} \end{corollary}
\begin{proof} \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} \end{proof}
\begin{lemma}[Scheffé] \begin{lemma}[Scheffé]