Updated formulation of convergence in measure.

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

@@ -5,7 +5,7 @@
 \label{definition:in-measure}
     Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$, let
     \[
-    U(\delta, \eps) = \bracs{(f, g) \in \mathscr{M}(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}
+    U(\delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}
     \]
 
     then
@@ -13,7 +13,7 @@
     \fB = \bracs{U(\delta, \eps)|\eps, \delta > 0}
     \]
 
-    forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the \textbf{uniform structure of convergence in measure} on $\mathscr{M}(X; Y)$.
+    forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the \textbf{uniform structure of convergence in measure} on $\mathcal{L}^0(X; Y)$.
 \end{definition}
 \begin{proof}
     It is sufficient to check the conditions of \autoref{proposition:fundamental-entourage-criterion}:
@@ -22,7 +22,7 @@
         \[
         U(\delta \wedge \delta', \eps \wedge \eps') \subset U(\delta, \eps) \cap U(\delta', \eps')
         \]
-        \item[(UB3)] For each $\eps, \delta > 0$ and $f, g, h \in \mathscr{M}(X; Y)$, 
+        \item[(UB3)] For each $\eps, \delta > 0$ and $f, g, h \in \mathcal{L}^0(X; Y)$, 
         \[
         \bracs{d(f, h) > \delta} \subset \bracs{d(f, g) > \delta} \cup \bracs{d(g, h) > \delta}
         \]
@@ -35,19 +35,19 @@
 \label{definition:ky-fan}
     Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and
     \[
-    \alpha: \mathscr{M}(X; Y)^2 \to [0, \infty) \quad (f, g) \mapsto \inf\bracs{\eps > 0| \mu\bracs{d(f, g) > \eps} \le \eps} \wedge 1
+    \alpha: L^0(X; Y)^2 \to [0, \infty) \quad (f, g) \mapsto \inf\bracs{\eps > 0| \mu\bracs{d(f, g) > \eps} \le \eps} \wedge 1
     \]
 
     then:
     \begin{enumerate}
-        \item $\alpha$ is a metric on $\mathscr{M}(X; Y)$ modulo almost everywhere equality. 
-        \item $\alpha$ induces the uniform structure of convergence in measure on $\mathscr{M}(X; Y)$. 
+        \item $\alpha$ is a metric on $L^0(X; Y)$.
+        \item $\alpha$ induces the uniform structure of convergence in measure on $L^0(X; Y)$. 
     \end{enumerate}
 
-    The mapping $\alpha$ is the \textbf{Ky Fan metric} on $\mathscr{M}(X; Y)$. 
+    The mapping $\alpha$ is the \textbf{Ky Fan metric} on $L^0(X; Y)$. 
 \end{definition}
 \begin{proof}
-    (1): Let $f, g, h \in \mathscr{M}(X; Y)$, then
+    (1): Let $f, g, h \in \mathcal{L}^0(X; Y)$, then
     \begin{enumerate}
         \item[(M)] If $\alpha(f, g) = 0$, then by \hyperref[continuity from above]{proposition:measure-properties}, 
         \[
@@ -63,12 +63,12 @@
         so $\alpha(f, h) \le \alpha(f, g) + \alpha(g, h)$. 
     \end{enumerate}
 
-    so $\alpha$ is a metric on $\mathscr{M}(X; Y)$, modulo almost everywhere equality. 
+    so $\alpha$ is a metric on $\mathcal{L}^0(X; Y)$, modulo almost everywhere equality. 
 
-    (2): Let $f, g \in \mathscr{M}(X; Y)$. For any $\eps, \delta > 0$, if $\alpha(f, g) < \eps \wedge \delta$, then there exists $r \in (0, \eps \wedge \delta]$ such that $\mu\bracs{d(f, g) > r} \le r$. Thus
+    (2): Let $f, g \in \mathcal{L}^0(X; Y)$. For any $\eps, \delta > 0$, if $\alpha(f, g) < \eps \wedge \delta$, then there exists $r \in (0, \eps \wedge \delta]$ such that $\mu\bracs{d(f, g) > r} \le r$. Thus
     \[
-    \bracs{(f, g) \in \mathscr{M}(X; Y)|\alpha(f, g) < \eps \wedge \delta} \subset 
-    \bracs{(f, g) \in \mathscr{M}(X; Y)|\mu\bracs{d(f, g) >\delta} < \eps}
+    \bracs{(f, g) \in \mathcal{L}^0(X; Y)|\alpha(f, g) < \eps \wedge \delta} \subset 
+    \bracs{(f, g) \in \mathcal{L}^0(X; Y)|\mu\bracs{d(f, g) >\delta} < \eps}
     \]
 
     On the other hand, if $\mu\bracs{d(f, g) > \eps} \le \eps$, then $d(f, g) \le \eps$. Therefore $\alpha$ induces the uniform structure of convergence in measure. 
@@ -78,7 +78,7 @@
 \label{definition:locally-in-measure}
     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
     \[
-    U(A, \delta, \eps) = \bracs{(f, g) \in \mathscr{M}(X; Y)| \mu(A \cap \bracs{d(f, g) > \delta}) < \eps}
+    U(A, \delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu(A \cap \bracs{d(f, g) > \delta}) < \eps}
     \]
 
     then
@@ -86,7 +86,7 @@
     \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 \textbf{uniform structure of local convergence in measure} on $\mathscr{M}(X; Y)$.
+    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 $\mathcal{L}^0(X; Y)$.
 \end{definition}
 \begin{proof}
     It is sufficient to check the conditions of \autoref{proposition:fundamental-entourage-criterion}:
@@ -95,7 +95,7 @@
         \[
         U(A \cup A', \delta \wedge \delta', \eps \wedge \eps') \subset U(A, \delta, \eps) \cap U(A', \delta', \eps')
         \]
-        \item[(UB3)] For each $\eps, \delta > 0$, $A \in \cm$ with $\mu(A) < \infty$, and $f, g, h \in \mathscr{M}(X; Y)$, 
+        \item[(UB3)] For each $\eps, \delta > 0$, $A \in \cm$ with $\mu(A) < \infty$, and $f, g, h \in \mathcal{L}^0(X; Y)$, 
         \[
         \bracs{d(f, h) > \delta} \subset \bracs{d(f, g) > \delta} \cup \bracs{d(g, h) > \delta}
         \]
@@ -149,18 +149,16 @@
     \]
 \end{proof}
 
-
-\begin{proposition}[{{\cite[Theorem 2.30]{Folland}}}]
-\label{proposition:cauchy-in-measure-limit}
-    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a complete metric space, and $\seq{f_n} \subset Y^X$ be a sequence Borel measurable functions from $X \to Y$ that is Cauchy in measure, then:
+\begin{theorem}
+\label{theorem:cauchy-in-measure-limit}
+    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a complete metric space, then:
     \begin{enumerate}
-        \item There exists a Borel measurable function $f: X \to Y$ such that $f_n \to f$ in measure.
-        \item There exists a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ almost everywhere.
-        \item For any Borel measurable function $g: X \to Y$ such that $f_n \to g$ in measure, $f = g$ almost everywhere.
+        \item For any $\seq{f_n} \subset L^0(X; Y)$ that is Cauchy in measure, there exists $f \in L^0(X; Y)$ and a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ almost everywhere.
+        \item $L^0(X; Y)$ equipped with the uniform structure of convergence in measure is complete. 
     \end{enumerate}
-\end{proposition}
-\begin{proof}
-    (2): Since $\seq{f_n}$ is Cauchy in measure, there exists a subsequence $\seq{n_k}$ such that for each $k \in \natp$, $\mu(\bracsn{d(f_{n_k}, f_{n_{k+1}}) > 2^{-k}}) \le 2^{-k}$.
+\end{theorem}
+\begin{proof}[Proof, [{{\cite[Theorem 2.30]{Folland}}}]. ]
+    (1): Since $\seq{f_n}$ is Cauchy in measure, there exists a subsequence $\seq{n_k}$ such that for each $k \in \natp$, $\mu(\bracsn{d(f_{n_k}, f_{n_{k+1}}) > 2^{-k}}) \le 2^{-k}$.
     
     In this case, for any $K \in \natp$ and $j \ge k \ge K$, 
     \[
@@ -168,7 +166,7 @@
     \subset \bigcup_{\ell = K}^{\infty}\bracsn{d(f_{n_\ell}, f_{n_{\ell + 1}}) > 2^{-\ell}}
     \]
 
-    By monotonicity and subadditivity (\autoref{proposition:measure-properties}), 
+    By \hyperref[monotonicity and subadditivity]{proposition:measure-properties}, 
     \[
     \mu\paren{\bigcup_{j, k \ge K}\bracsn{d(f_{n_j}, f_{n_k}) > 2^{-K+1}}} 
     \le \sum_{\ell \ge K}\mu\bracsn{d(f_{n_\ell}, f_{n_{\ell + 1}}) > 2^{-\ell}}
@@ -187,12 +185,7 @@
 
     Thus, for almost every $x \in X$, there exists $K \in \natp$ such that $d(f_{n_j}(x), f_{n_k}(x)) < 2^{-K+1}$ for all $j, k \ge K$. Therefore $\seq{f_n(x)}$ is Cauchy for almost every $x$, and converges to a Borel measurable function $f: X \to Y$.
 
-    (1): Let $\eps, \delta > 0$, then there exists $N \in \natp$ such that $\mu(\bracs{d(f_m, f_n) > \delta/2}) < \eps/2$ for all $m, n \ge N$. Let $k \in \natp$ such that $n_k \ge N$ and $\mu(\bracs{d(f, f_{n_k}) > \delta/2}) < \eps/2$, then for any $m \ge N$,
-    \[
-    \mu\bracs{d(f_m, f) > \delta} \le \mu\bracs{d(f_m, f_{n_k}) > \delta/2} + \mu\bracs{d(f, f_{n_k}) > \delta/2} < \eps
-    \]
-
-    (3): By (2), there exists a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ and $f_{n_k} \to g$ almost everywhere, so $f = g$ almost everywhere. 
+    (2): Since the uniform structure of convergence in measure on $L^0(X; Y)$ is defined by the \hyperref[Ky Fan metric]{definition:ky-fan}, completeness follows from (1) and \autoref{proposition:complete-metric-space}. 
 \end{proof}
 
 \begin{theorem}[Monotone Convergence Theorem (in Measure)]

src/measure/radon/radon.tex

@@ -197,7 +197,7 @@
 \begin{proof}[Proof, {{\cite[Theorem 7.10]{Folland}}}. ]
     First assume that $f$ is bounded.
 
-    (1, bounded): If $f$ is bounded, then $f \in L^1(X; E)$. By \autoref{proposition:radon-cc-dense}, there exists $\seq{\phi_n} \subset C_c(X)$ such that $\phi_n \to f$ in $L^1(\mu)$. Since $\phi_n \to f$ in $L^1(\mu)$, $\phi_n \to f$ in measure by \autoref{proposition:lp-in-measure}. By taking a subsequence using \ref{proposition:cauchy-in-measure-limit}, assume without loss of generality that $\phi_n \to f$ almost everywhere. 
+    (1, bounded): If $f$ is bounded, then $f \in L^1(X; E)$. By \autoref{proposition:radon-cc-dense}, there exists $\seq{\phi_n} \subset C_c(X)$ such that $\phi_n \to f$ in $L^1(\mu)$. Since $\phi_n \to f$ in $L^1(\mu)$, $\phi_n \to f$ in measure by \autoref{proposition:lp-in-measure}. By taking a subsequence using \autoref{theorem:cauchy-in-measure-limit}, assume without loss of generality that $\phi_n \to f$ almost everywhere. 
 
     By \hyperref[Egoroff's Theorem]{theorem:egoroff}, there exists $A \subset \bracs{f \ne 0}$ such that $\phi_n \to f$ uniformly and $\mu(\bracs{f \ne 0} \setminus A) < \eps/3$. By \autoref{proposition:radon-regular-sigma-finite}, there exists $K \subset A$ compact such that $\mu(A \setminus K) < \eps/3$.