Added a local to global characterisation of convergence in measure.

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

@@ -18,6 +18,7 @@
     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 $f, g \in A$. 
@@ -25,6 +26,35 @@
     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}
 
+\begin{proposition}
+\label{proposition:convergence-in-measure}
+    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:
+    \begin{enumerate}
+        \item[(L)] $\fF$ is locally Cauchy in measure. 
+        \item[(T)] For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that
+        \[
+        \sup_{f, g \in F}\mu(A^c \cap \bracs{d(f, g) > \delta}) < \eps
+        \]
+    \end{enumerate}
+\end{proposition}
+\begin{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
+    \[
+    \sup_{f, g \in F_2}\mu\bracs{d(f, g) > \delta} < 2\eps
+    \]
+\end{proof}
+
+
 \begin{lemma}
 \label{lemma:ae-in-measure}
     Let $(X, \cm, \mu)$ be a finite measure space, $(Y, d)$ be a metric space, and $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$ such that $f_n \to f$ almost everywhere, then $f_n \to f$ in measure.