Fixed typo.
All checks were successful
Compile Project / Compile (push) Successful in 43s

This commit is contained in:
Bokuan Li
2026-08-29 15:43:23 -04:00
parent fab7bb7af4
commit 926305e65c
2 changed files with 2 additions and 1 deletions

View File

@@ -1,6 +1,7 @@
\section{Strongly Measurable Functions} \section{Strongly Measurable Functions}
\label{section:strongly-measurable} \label{section:strongly-measurable}
\begin{definition}[Strongly Measurable Function] \begin{definition}[Strongly Measurable Function]
\label{definition:strongly-measurable} \label{definition:strongly-measurable}
Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $f: X \to E$, then the following are equivalent: Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $f: X \to E$, then the following are equivalent:

View File

@@ -37,7 +37,7 @@
\label{proposition:convergence-in-measure} \label{proposition:convergence-in-measure}
Let $(X, \cm, \cf, \mu)$ be a \hyperref[scaffolded]{definition:measure-scaffold} 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: Let $(X, \cm, \cf, \mu)$ be a \hyperref[scaffolded]{definition:measure-scaffold} 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} \begin{enumerate}
\item[(L)] $\fF$ is \hyperref[definition:locally-in-measure]{definition:locally-in-measure}. \item[(L)] $\fF$ is Cauchy \hyperref[locally in measure]{definition:locally-in-measure}.
\item[(T)] For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cf$ such that \item[(T)] For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cf$ such that
\[ \[
\sup_{f, g \in F}\mu(A^c \cap \bracs{d(f, g) > \delta}) < \eps \sup_{f, g \in F}\mu(A^c \cap \bracs{d(f, g) > \delta}) < \eps