Definition 24.6.1 (In Measure).label Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$, let
\[U(\delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}\]
then
\[\fB = \bracs{U(\delta, \eps)|\eps, \delta > 0}\]
forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the uniform structure of convergence in measure on $\mathcal{L}^{0}(X; Y)$.
Proof. It is sufficient to check the conditions of Proposition 6.1.8:
- (FB1)
For each $\eps, \eps', \delta, \delta' > 0$,
\[U(\delta \wedge \delta', \eps \wedge \eps') \subset U(\delta, \eps) \cap U(\delta', \eps')\] - (UB3)
For each $\eps, \delta > 0$ and $f, g, h \in \mathcal{L}^{0}(X; Y)$,
\[\bracs{d(f, h) > \delta}\subset \bracs{d(f, g) > \delta}\cup \bracs{d(g, h) > \delta}\]so $U(\delta/2, \eps/2) \circ U(\delta/2, \eps/2) \subset U(\delta, \eps)$.
$\square$
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