Definition 24.6.3 (Locally In Measure).label Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$ and $A \in \cm$ with $\mu(A) < \infty$, let
\[U(A, \delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu(A \cap \bracs{d(f, g) > \delta}) < \eps}\]
then
\[\fB = \bracs{U(A, \delta, \eps)|\eps, \delta > 0, A \in \cm, \mu(A) < \infty}\]
forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the uniform structure of local 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 $\eps, \eps', \delta, \delta' > 0$ and $A, A' \in \cm$ with $\mu(A), \mu(A') < \infty$,
\[U(A \cup A', \delta \wedge \delta', \eps \wedge \eps') \subset U(A, \delta, \eps) \cap U(A', \delta', \eps')\] - (UB3)
For each $\eps, \delta > 0$, $A \in \cm$ with $\mu(A) < \infty$, 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 $U(A, \delta/2, \eps/2) \circ U(A, \delta/2, \eps/2) \subset U(A, \delta, \eps)$.
$\square$
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