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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/measure/measurable-maps/in-measure.tex
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@@ -3,7 +3,7 @@
\begin{definition}[In Measure]
\label{definition:in-measure}
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$, let
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a separable metric space. For each $\eps, \delta > 0$, let
\[
U(\delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}
\]
@@ -33,7 +33,7 @@
\begin{definition}[Ky Fan Metric]
\label{definition:ky-fan}
Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and
Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a separable metric space, and
\[
\alpha: L^0(X; Y)^2 \to [0, \infty) \quad (f, g) \mapsto \inf\bracs{\eps > 0| \mu\bracs{d(f, g) > \eps} \le \eps} \wedge 1
\]
@@ -76,7 +76,7 @@
\begin{definition}[Locally In Measure]
\label{definition:locally-in-measure}
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$ and $A \in \cm$ with $\mu(A) < \infty$, let
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a separable metric space. For each $\eps, \delta > 0$ and $A \in \cm$ with $\mu(A) < \infty$, let
\[
U(A, \delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu(A \cap \bracs{d(f, g) > \delta}) < \eps}
\]
@@ -106,7 +106,7 @@
\begin{proposition}
\label{proposition:convergence-in-measure}
Let $(X, \cm, \mu)$ be a semifinite measure space, $(Y, d)$ be a metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, then $\fF$ is Cauchy in measure if and only if:
Let $(X, \cm, \mu)$ be a semifinite measure space, $(Y, d)$ be a separable metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, then $\fF$ is Cauchy in measure if and only if:
\begin{enumerate}
\item[(L)] $\fF$ is locally Cauchy in measure.
\item[(T)] For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that
@@ -135,7 +135,7 @@
\begin{lemma}
\label{lemma:ae-in-measure}
Let $(X, \cm, \mu)$ be a finite measure space, $(Y, d)$ be a metric space, and $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$ such that $f_n \to f$ almost everywhere, then $f_n \to f$ in measure.
Let $(X, \cm, \mu)$ be a finite measure space, $(Y, d)$ be a separable metric space, and $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$ such that $f_n \to f$ almost everywhere, then $f_n \to f$ in measure.
\end{lemma}
\begin{proof}
Let $\eps > 0$, then for almost every $x \in X$, there exists $N \in \natp$ such that $d(f_n(x), f(x)) < \eps$ for all $n \ge N$, so
@@ -151,7 +151,7 @@
\begin{theorem}
\label{theorem:cauchy-in-measure-limit}
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a complete metric space, then:
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a Polish space, then:
\begin{enumerate}
\item For any $\seq{f_n} \subset L^0(X; Y)$ that is Cauchy in measure, there exists $f \in L^0(X; Y)$ and a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ almost everywhere and in measure.
\item $L^0(X; Y)$ equipped with the uniform structure of convergence in measure is complete.