Added uniform description for convergence in measure.

src/fa/tvs/definition.tex

@@ -100,7 +100,7 @@
     \begin{enumerate}
         \item[(FB1)] For any $V, V' \in \fB$, there exists $W \in \fB$ with $W \subset V \cap V'$. In which case, $U_{V} \cap U_{V'} \supset U_W \in \mathfrak{V}$.
         \item[(UB1)] For any $x \in E$ and $V \in \fB$, $x - x = 0 \in V$, so $\Delta \subset U_V$.
-        \item[(UB2)] For any $V \in \fB$, by (TVB1), there exists $W \in \fB$ such that $W + W \subset V$. In which case, for any $x, y, z \in E$ with $x - y, y - z \in W$, $x - z \in V$. Therefore $U_W \circ U_W \subset U_V$.
+        \item[(UB3)] For any $V \in \fB$, by (TVB1), there exists $W \in \fB$ such that $W + W \subset V$. In which case, for any $x, y, z \in E$ with $x - y, y - z \in W$, $x - z \in V$. Therefore $U_W \circ U_W \subset U_V$.
     \end{enumerate}
 
     By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\fU$ on $E$ for which $\mathfrak{V}$ is a fundamental system of entourages.

src/measure/lebesgue-integral/non-negative.tex

@@ -25,7 +25,7 @@
 \label{lemma:lebesgue-non-negative-strict}
     Let $(X, \cm, \mu)$ be a measure space and $f \in \mathcal{L}^+(X, \cm)$. 
     \begin{enumerate}
-        \item For each $\phi \in \Sigma^+(X, \cm)$, denote $\phi \le_u f$ if there exists $\delta > 0$ such that $\phi + \delta \ge f$ on $\bracs{\phi > 0}$, then
+        \item For each $\phi \in \Sigma^+(X, \cm)$, denote $\phi \le_u f$ if there exists $\delta > 0$ such that $\phi + \delta \le f$ on $\bracs{\phi > 0}$, then
         \[
         \int f d\mu = \sup\bracs{\int \phi d\mu \bigg | \phi \in \Sigma^+(X, \cm), \phi \le_u f}
         \]

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

@@ -1,30 +1,108 @@
 \section{Convergence in Measure}
 \label{section:convergence-in-measure}
 
-\begin{definition}[Convergence in Measure]
+\begin{definition}[In Measure]
-\label{definition:convergence-in-measure}
+\label{definition: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 $\eps > 0$, 
+    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$, let
     \[
-    \lim_{g, \fF}\mu(\bracs{d(f, g) > \eps}) = 0
+    U(\delta, \eps) = \bracs{(f, g) \in \mathscr{M}(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}
     \]
 
-    and $\fF \to f$ \textbf{locally in measure} if $\fF \to f$ in measure on every set of finite measure. 
+    then
-
-    Alternatively, if $\net{f}$ is a net of $(\cm, \cb_Y)$-measurable functions, then $f_\alpha \to f$ \textbf{in measure} if for each $\eps > 0$, 
     \[
-    \lim_{\alpha \in A}\mu(\bracs{d(f, f_\alpha) > \eps}) = 0
+    \fB = \bracs{U(\delta, \eps)|\eps, \delta > 0}
     \]
 
-    and $f_\alpha \to f$ \textbf{locally in measure} if $f_\alpha \to f$ in measure on every set of finite measure. 
+    forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the \textbf{uniform structure of convergence in measure} on $\mathscr{M}(X; Y)$.
 \end{definition}
+\begin{proof}
+    It is sufficient to check the conditions of \autoref{proposition:fundamental-entourage-criterion}:
+    \begin{enumerate}
+        \item[(FB1)] For each $\eps, \eps', \delta, \delta' > 0$,  
+        \[
+        U(\delta \wedge \delta', \eps \wedge \eps') \subset U(\delta, \eps) \cap U(\delta', \eps')
+        \]
+        \item[(UB3)] For each $\eps, \delta > 0$ and $f, g, h \in \mathscr{M}(X; Y)$, 
+        \[
+        \bracs{d(f, h) > \delta} \subset \bracs{d(f, g) > \delta} \cup \bracs{d(g, h) > \delta}
+        \]
 
+        so $U(\delta/2, \eps/2) \circ U(\delta/2, \eps/2) \subset U(\delta, \eps)$. 
+    \end{enumerate}
+\end{proof}
 
-\begin{definition}[Cauchy in Measure]
+\begin{definition}[Ky Fan Metric]
-\label{definition:cauchy-in-measure}
+\label{definition:ky-fan}
-    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 $f, g \in A$. 
+    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and
+    \[
+    \alpha: \mathscr{M}(X; Y)^2 \to [0, \infty) \quad (f, g) \mapsto \inf\bracs{\eps > 0| \mu\bracs{d(f, g) > \eps} \le \eps}
+    \]
 
-    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 $\alpha, \beta \ge \alpha_0$, $\mu(\bracs{d(f_\alpha, f_\beta) > \delta}) < \eps$. 
+    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 $\mathscr{M}(X; Y)$. 
+    \end{enumerate}
+
+    The mapping $\alpha$ is the \textbf{Ky Fan metric} on $\mathscr{M}(X; Y)$. 
 \end{definition}
+\begin{proof}
+    (1): Let $f, g, h \in \mathscr{M}(X; Y)$, then
+    \begin{enumerate}
+        \item[(M)] If $\alpha(f, g) = 0$, then by \hyperref[continuity from above]{proposition:measure-properties}, 
+        \[
+        \mu\bracs{d(f, g) > 0} = \limv{n}\mu\bracs{d(f, g) > 1/n} \le \limv{n}\frac{1}{n} = 0
+        \]
+
+        so $f = g$ almost everywhere. 
+        \item[(PM3)] For each $\eps > 0$, 
+        \[
+        \bracs{d(f, h) > \eps} \subset \bracs{d(f, g) > \eps/2} \cup \bracs{d(g, h) > \eps/2}
+        \]
+
+        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. 
+
+    (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
+    \[
+    \bracs{(f, g) \in \mathscr{M}(X; Y)|\alpha(f, g) < \eps \wedge \delta} \subset 
+    \bracs{(f, g) \in \mathscr{M}(X; Y)|\mu\bracs{d(f, g) >\delta} < \eps}
+    \]
+
+    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. 
+\end{proof}
+
+\begin{definition}[Locally In Measure]
+\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
+    \[
+    U(A, \delta, \eps) = \bracs{(f, g) \in \mathscr{M}(X; Y)| \mu(A \cap \bracs{d(f, g) > \delta}) < \eps}
+    \]
+
+    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 \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 $f, g, h \in \mathscr{M}(X; Y)$, 
+        \[
+        \bracs{d(f, h) > \delta} \subset \bracs{d(f, g) > \delta} \cup \bracs{d(g, h) > \delta}
+        \]
+
+        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}

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

@@ -4,6 +4,8 @@
 \begin{definition}[Measurable Function]
 \label{definition:measurable-function}
     Let $(X, \cm)$ and $(Y, \cn)$ be measurable spaces and $f: X \to Y$ be a mapping, then $f$ is \textbf{$(\cm, \cn)$-measurable} if $f^{-1}(E) \in \cm$ for all $E \in \cn$.
+
+    The set $\mathscr{M}(X; Y)$ is the \textbf{space of measurable functions} from $X$ to $Y$. 
 \end{definition}
 
 

src/measure/notation.tex

@@ -12,6 +12,7 @@
     $\mathcal{B}_X$ & Borel $\sigma$-algebra on $X$. & \autoref{definition:borel-sigma-algebra} \\
     $\sigma(\{f_i \mid i \in I\})$ & $\sigma$-algebra generated by the maps $\{f_i\}$. & \autoref{definition:generated-sigma-algebra-function} \\
     $\bigotimes_{i \in I} \mathcal{M}_i$ & Product $\sigma$-algebra. & \autoref{definition:product-sigma-algebra} \\
+    $\mathscr{M}(X; Y)$ & Space of measurable functions from $X$ to $Y$. & \autoref{definition:measurable-function} \\
     $\chi_E = \mathbf{1}_E$ & Indicator function of $E$. & \autoref{definition:indicator-function} \\
     $\Sigma(X, \mathcal{M}; E)$ & Space of $E$-valued simple functions on $(X, \mathcal{M})$. & \autoref{definition:simple-function-standard-form} \\
     $\Sigma^+(X, \mathcal{M})$ & Space of non-negative simple functions. & \autoref{definition:simple-function-scalar} \\

src/topology/functions/set-systems.tex

@@ -53,7 +53,7 @@
         \]
 
         \item[(UB1)] For any $U \in \fU$, $\Delta \subset U$. Thus the diagonal in $X^T$ is in $E(S, U)$ for any $S \in \sigma$.
-        \item[(UB2)] For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \sigma$,
+        \item[(UB3)] For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \sigma$,
         \[
         E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
         \]

src/topology/groups/definition.tex

@@ -62,7 +62,7 @@
     \begin{enumerate}
         \item[(FB1)] For each $V, V' \in \cn_G(1)$, $V \cap V' \in \cn_G(0)$, so $U_{L, V \cap V'} = U_{L, V} \cap U_{L, V'}$.
         \item[(UB1)] For each $V \in \cn_G(1)$, $1 \in V$, so $\Delta \subset U_{L, V}$. 
-        \item[(UB2)] For each $V \in \cn_G(1)$, by (TG1), there exists $W \in \cn_G(1)$ such that $WW \subset V$. In which case, $U_{L, W} \circ U_{L, W} \subset U_{L, V}$. 
+        \item[(UB3)] For each $V \in \cn_G(1)$, by (TG1), there exists $W \in \cn_G(1)$ such that $WW \subset V$. In which case, $U_{L, W} \circ U_{L, W} \subset U_{L, V}$. 
     \end{enumerate}
 
     By \autoref{proposition:fundamental-entourage-criterion}, $\fB_L$ forms a fundamental system of entourages for a left translation-invariant uniformity $\fU_L$ on $G$. 

src/topology/uniform/completion.tex

@@ -46,7 +46,7 @@
     \begin{enumerate}
         \item[(FB1)] Let $\wh U, \wh V \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \subset U \cap V$. In which case, for any $(\fF, \mathfrak{G}) \in \wh W$, there exists $E \in \fF \cap \mathfrak{G}$ with $E \times E \subset W \subset U \cap V$. Thus $\wh W \subset \wh U \cap \wh V$.
         \item[(UB1)] Let $\wh U \in \wh \fB$ and $\fF \in \wh X$, then since $\fF$ is Cauchy, there exists $E \in \fF$ such that $E \times E \subset U$, so $(\fF, \fF) \in \wh U$.
-        \item[(UB2)] Let $\wh U \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \circ W \subset U$. For any $\fF, \mathfrak{G}, \mathfrak{H} \in \wh X$ such that $(\fF, \mathfrak{G}), (\mathfrak{G}, \mathfrak{H}) \in \wh W$, there exists $W$-small sets $E \in \fF \cap \mathfrak{G}$ and $F \in \fF \cap \mathfrak{H}$. Since $\mathfrak{G}$ is a filter, $E \cap F \ne \emptyset$ by (F2) and (F3). By \autoref{lemma:small-intersect}, $E \cup H$ is $W \circ W$-small and thus $U$-small. Using (F1), $E \cup H \in \fF \cap \mathfrak{H}$, so $(\fF, \mathfrak{H}) \in \wh U$. Therefore $\wh W \circ \wh W \subset U$.
+        \item[(UB3)] Let $\wh U \in \wh \fB$. By \autoref{lemma:symmetricfundamentalentourage}, there exists a symmetric entourage $W \in \fU$ such that $W \circ W \subset U$. For any $\fF, \mathfrak{G}, \mathfrak{H} \in \wh X$ such that $(\fF, \mathfrak{G}), (\mathfrak{G}, \mathfrak{H}) \in \wh W$, there exists $W$-small sets $E \in \fF \cap \mathfrak{G}$ and $F \in \fF \cap \mathfrak{H}$. Since $\mathfrak{G}$ is a filter, $E \cap F \ne \emptyset$ by (F2) and (F3). By \autoref{lemma:small-intersect}, $E \cup H$ is $W \circ W$-small and thus $U$-small. Using (F1), $E \cup H \in \fF \cap \mathfrak{H}$, so $(\fF, \mathfrak{H}) \in \wh U$. Therefore $\wh W \circ \wh W \subset U$.
     \end{enumerate}
 
     By \autoref{proposition:fundamental-entourage-criterion}, there exists a unique uniformity $\wh \fU \supset \wh \fB$. Moreover, $\wh \fB$ consists of symmetric entourages by construction.

src/topology/uniform/definition.tex

@@ -75,25 +75,27 @@
 
 \begin{proposition}
 \label{proposition:fundamental-entourage-criterion}
-    Let $X$ be a set and $\fB \subset 2^{X \times X}$ be a non-empty family of sets. If
+    Let $X$ be a set and $\fB \subset 2^{X \times X}$ be a non-empty family of sets such that
     \begin{enumerate}
         \item[(FB1)] For each $U, V \in \fB$, there exists $W \in \fB$ such that $W \subset U \cap V$.
         \item[(UB1)] For each $V \in \fB$, $\Delta \subset V$.
-        \item[(UB2)] For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
+        \item[(UB2)] For each $V \in \fB$, there exists $W \in \fB$ with $W \subset V^{-1}$. 
+        \item[(UB3)] For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
     \end{enumerate}
 
-    then there exists a unique uniformity $\fU \subset 2^{X \times X}$, which is given by
+    and let
     \[
     \fU = \bracs{U \subset X \times X| \exists V \in \fB: V \subset U}
     \]
 
+    then $\fU$ is the unique uniformity on $X$ such that $\fB$ is a fundamental system of entourages for $\fU$. 
 \end{proposition}
 \begin{proof}
     (F1): By definition of $\fU$.
 
     (F2): For any $U, V \in \fU$, there exists $U_0, V_0 \in \fB$ such that $U_0 \subset U$ and $V_0 \subset V$. By (FB1), there exists $W \in \fB$ with $W \subset U_0 \cap V_0 \subset U \cap V$. Thus $U \cap V \in \fU$.
 
-    (U1) and (U2): For any $U \in \fU$, there exists $U_0 \in \fB$ with $U_0 \subset U$. By (UB1), $\Delta \subset U_0 \subset U$. By (UB2), there exists $V_0 \in \fB \subset \fU$ with $V_0 \circ V_0 \subset U_0 \subset U$.
+    (U1), (U2), (U3): For any $U \in \fU$, there exists $U_0 \in \fB$ with $U_0 \subset U$. By (UB1), $\Delta \subset U_0 \subset U$. By (UB2) and (FB1), there exists $V_0 \in \fB$ with $V_0 \subset U_0 \cap U_0^{-1} \subset U$. By (UB3), there exists $W_0 \in \fB \subset \fU$ with $W_0 \circ W_0 \subset U_0 \subset U$.
 \end{proof}
 
 

src/topology/uniform/metric.tex

@@ -77,7 +77,7 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
         \]
 
         \item[(UB1)] For each $i \in I$, $d(x, x) = 0$ for all $x \in X$. Thus for any $i \in I$ and $r > 0$, $E(d_i, r)$ contains the diagonal.
-        \item[(UB2)] For each $J \subset I$ finite and $r > 0$,
+        \item[(UB3)] For each $J \subset I$ finite and $r > 0$,
         \begin{align*}
         \paren{\bigcap_{j \in J}E(d_j, r/2)} \circ \paren{\bigcap_{j \in J}E(d_j, r)} &\subset \bigcap_{j \in J}E(d_j, r/2) \circ E(d_j, r/2) \\
         &\subset \bigcap_{j \in J}E(d_j, r)

src/topology/uniform/uc.tex

@@ -54,7 +54,7 @@
     The collection $\fU$ is the \textbf{initial uniformity} on $X$ generated by $\seqi{f}$.
 \end{definition}
 \begin{proof}
-    (3): Since the diagonal is mapped to the diagonal and $\fB$ is closed under intersections, it is sufficient to verify (UB3) for $\fB$. Let $J \subset I$ be finite and $\bigcap_{j \in J}(f_j \times f_j)^{-1}(U_j) \in \fB$, then there exists $\bracs{V_j}_{j \in J}$ such that $V_j \circ V_j \subset U_j$ for each $j \in J$. In which case, for any $(x, y), (y, z) \in f_j^{-1}(V_j)$, $(f(x), f(y)), (f(y), f(z)) \in V_j$ and $(f(x), f(z)) \in U_j$. Thus $(f_j \times f_j)^{-1}(V_j) \circ (f_j \times f_j)^{-1}(V_j) \subset (f_j \times f_j)^{-1}(U_j)$, and
+    (3): Since the diagonal is mapped to the diagonal and $\fB$ is closed under intersections and inversions, it is sufficient to verify (UB3) for $\fB$. Let $J \subset I$ be finite and $\bigcap_{j \in J}(f_j \times f_j)^{-1}(U_j) \in \fB$, then there exists $\bracs{V_j}_{j \in J}$ such that $V_j \circ V_j \subset U_j$ for each $j \in J$. In which case, for any $(x, y), (y, z) \in f_j^{-1}(V_j)$, $(f(x), f(y)), (f(y), f(z)) \in V_j$ and $(f(x), f(z)) \in U_j$. Thus $(f_j \times f_j)^{-1}(V_j) \circ (f_j \times f_j)^{-1}(V_j) \subset (f_j \times f_j)^{-1}(U_j)$, and
     \[
     \paren{\bigcap_{j \in J}(f_j \times f_j)^{-1}(V_j)} \circ \paren{\bigcap_{j \in J}(f_j \times f_j)^{-1}(V_j)} \subset \bigcap_{j \in J}(f_j \times f_j)^{-1}(U_j)
     \]