Added uniform description for convergence in measure.

src/measure/measurable-maps/in-measure.tex

@@ -88,35 +88,21 @@
 
     forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the \textbf{uniform structure of local convergence in measure} on $\mathscr{M}(X; Y)$.
 \end{definition}
+\begin{proof}
+    It is sufficient to check the conditions of \autoref{proposition:fundamental-entourage-criterion}:
+    \begin{enumerate}
+        \item[(FB1)] For each $\eps, \eps', \delta, \delta' > 0$ and $A, A' \in \cm$ with $\mu(A), \mu(A') < \infty$, 
+        \[
+        U(A \cup A', \delta \wedge \delta', \eps \wedge \eps') \subset U(A, \delta, \eps) \cap U(A', \delta', \eps')
+        \]
+        \item[(UB3)] For each $\eps, \delta > 0$, $A \in \cm$ with $\mu(A) < \infty$, and $f, g, h \in \mathscr{M}(X; Y)$, 
+        \[
+        \bracs{d(f, h) > \delta} \subset \bracs{d(f, g) > \delta} \cup \bracs{d(g, h) > \delta}
+        \]
 
-
-
-
-
-\begin{definition}[Convergence in Measure]
-\label{definition:convergence-in-measure}
-    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, and $f$ be a $(\cm, \cb_Y)$-measurable function, then $\fF \to f$ \textbf{in measure} if for each $\eps > 0$, 
-    \[
-    \lim_{g, \fF}\mu(\bracs{d(f, g) > \eps}) = 0
-    \]
-
-    and $\fF \to f$ \textbf{locally in measure} if $\fF \to f$ in measure on every set of finite measure. 
-
-    Alternatively, if $\net{f}$ is a net of $(\cm, \cb_Y)$-measurable functions, then $f_\alpha \to f$ \textbf{in measure} if for each $\eps > 0$, 
-    \[
-    \lim_{\alpha \in A}\mu(\bracs{d(f, f_\alpha) > \eps}) = 0
-    \]
-
-    and $f_\alpha \to f$ \textbf{locally in measure} if $f_\alpha \to f$ in measure on every set of finite measure. 
-\end{definition}
-
-
-\begin{definition}[Cauchy in Measure]
-\label{definition:cauchy-in-measure}
-    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, then $\fF$ is \textbf{Cauchy in measure} if for every $\eps, \delta > 0$, there exists $A \in \fF$ such that $\mu(\bracs{d(f, g) > \delta}) < \eps$ for all $f, g \in A$. 
-
-    Alternatively, if $\net{f}$ is a net of $(\cm, \cb_Y)$-measurable functions, then $\net{f}$ is \textbf{Cauchy in measure} if for every $\eps, \delta > 0$, there exists $\alpha_0 \in A$ such that for each $\alpha, \beta \in A$ with $\alpha, \beta \ge \alpha_0$, $\mu(\bracs{d(f_\alpha, f_\beta) > \delta}) < \eps$. 
-\end{definition}
+        so $U(A, \delta/2, \eps/2) \circ U(A, \delta/2, \eps/2) \subset U(A, \delta, \eps)$. 
+    \end{enumerate}
+\end{proof}
 
 \begin{proposition}
 \label{proposition:convergence-in-measure}