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\section{Strongly Measurable Functions}
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\label{section:strongly-measurable}
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\begin{definition}[Strongly Measurable Function]
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\label{definition:strongly-measurable}
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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:
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\label{proposition:convergence-in-measure}
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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:
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\begin{enumerate}
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\item[(L)] $\fF$ is \hyperref[definition:locally-in-measure]{definition:locally-in-measure}.
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\item[(L)] $\fF$ is Cauchy \hyperref[locally in measure]{definition:locally-in-measure}.
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\item[(T)] For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cf$ such that
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\[
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\sup_{f, g \in F}\mu(A^c \cap \bracs{d(f, g) > \delta}) < \eps
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