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RETRACTION: SEPARABILITY REQUIRED TO DEFINE CONVERGENCE IN MEASURE

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Bokuan Li
2026-06-27 22:38:39 -04:00
parent ce52ac4b63
commit d8f56ef537
2 changed files with 9 additions and 9 deletions
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src/fa/lp/ui.tex
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@@ -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 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 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:
\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 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 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:
\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 $f \in F$.
@@ -159,7 +159,7 @@
\begin{lemma}[Scheffé]
\label{lemma:scheffe}
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 $g \in L^p(X; E)$, then $\fF \to g$ in $L^p(X; E)$ if and only if:
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 $g \in L^p(X; E)$, then $\fF \to g$ in $L^p(X; E)$ if and only if:
\begin{enumerate}
\item[(M)] $\fF \to g$ locally in measure.
\item[(N)] $\lim_{f, \fF} \norm{f}_{L^p(X; E)} = \norm{g}_{L^p(X; E)}$.