Proposition 24.6.3.label 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:
- (L)
$\fF$ is locally Cauchy in measure.
- (T)
For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that
\[\sup_{f, g \in F}\mu(A^{c} \cap \bracs{d(f, g) > \delta}) < \eps\]
Proof. (L) + (T) $\Rightarrow$ (In Measure): Let $\eps, \delta > 0$. By (T) then there exists $F_{1} \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that
\[\sup_{f, g \in F_1}\mu(A^{c} \cap \bracs{d(f, g) > \delta}) < \eps\]
By (L), there exists $F_{2} \in \fF$ with $F_{2} \subset F_{1}$ such that
\[\sup_{f, g \in F_2}\mu(A \cap \bracs{d(f, g) > \delta}) < \eps\]
Therefore
\[\sup_{f, g \in F_2}\mu\bracs{d(f, g) > \delta}< 2\eps\]
$\square$
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