diff --git a/src/measure/bochner-integral/strongly.tex b/src/measure/bochner-integral/strongly.tex index c491617..179ce67 100644 --- a/src/measure/bochner-integral/strongly.tex +++ b/src/measure/bochner-integral/strongly.tex @@ -1,6 +1,7 @@ \section{Strongly Measurable Functions} \label{section:strongly-measurable} + \begin{definition}[Strongly Measurable Function] \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: diff --git a/src/measure/measurable-maps/local-in-measure.tex b/src/measure/measurable-maps/local-in-measure.tex index c48ae6d..c589652 100644 --- a/src/measure/measurable-maps/local-in-measure.tex +++ b/src/measure/measurable-maps/local-in-measure.tex @@ -37,7 +37,7 @@ \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: \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 \[ \sup_{f, g \in F}\mu(A^c \cap \bracs{d(f, g) > \delta}) < \eps