Definition 26.8.1 (Locally In Measure*).label Let $(X, \cm, \cf, \mu)$ be a scaffolded measure space and $(Y, d)$ be a separable metric space. For each $\eps, \delta > 0$ and $A \in \cf$, let
then
forms a fundamental system of entourages for a uniformity.
The uniformity defined by $\fB$ is the uniform structure of local convergence in measure, and $\mathcal{L}_{\cf}^{0}(X; Y)$ denotes $\mathcal{L}^{0}(X; Y)$ equipped with this uniformity.
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 $\eps, \delta > 0$, $A \in \cf$, 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 $U(A, \delta/2, \eps/2) \circ U(A, \delta/2, \eps/2) \subset U(A, \delta, \eps)$.
$\square$
Post a Comment