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Fixed typo in Cauchy in measure.

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Bokuan Li
2026-05-01 13:29:36 -04:00
parent 95829261c7
commit caf7790b15
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src/measure/measurable-maps/in-measure.tex
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@@ -36,7 +36,7 @@
\begin{proposition}[{{\cite[Theorem 2.30]{Folland}}}] \begin{proposition}[{{\cite[Theorem 2.30]{Folland}}}]
\label{proposition:cauchy-in-measure-limit} \label{proposition:cauchy-in-measure-limit}
Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a complete metric space, and $\seq{f_n}$ be Borel measurable functions from $X \to Y$, then: Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a complete metric space, and $\seq{f_n} \subset Y^X$ be a sequence Borel measurable functions from $X \to Y$ that is Cauchy in measure, then:
\begin{enumerate} \begin{enumerate}
\item There exists a Borel measurable function $f: X \to Y$ such that $f_n \to f$ in measure. \item There exists a Borel measurable function $f: X \to Y$ such that $f_n \to f$ in measure.
\item There exists a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ almost everywhere. \item There exists a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ almost everywhere.