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MAJOR RETRACTION IN UNIFORMITY DEFINING PROPOSITION.

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Bokuan Li
2026-06-21 21:56:28 -04:00
parent a57686be8f
commit f107df48bf
8 changed files with 97 additions and 11 deletions
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2
src/fa/tvs/definition.tex
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@@ -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.

86
src/measure/measurable-maps/in-measure.tex
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@@ -3,8 +3,92 @@
\begin{definition}[In Measure]
\label{definition:in-measure}
Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, then the uniform
Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$, let
\[
U(\delta, \eps) = \bracs{(f, g) \in \mathscr{M}(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}
\]
then
\[
\fB = \bracs{U(\delta, \eps)|\eps, \delta > 0}
\]
forms a fundamental system of entourages for a uniformity. The uniformity induced by $\fB$ is the \textbf{uniform structure of convergence in measure} on $\mathscr{M}(X; Y)$.
\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}[Ky Fan Metric]
\label{definition:ky-fan}
Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and
\[
\alpha: \mathscr{M}(X; Y)^2 \to [0, \infty) \quad (f, g) \mapsto \inf\bracs{\eps > 0| \mu\bracs{d(f, g) > \eps} \le \eps}
\]
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}

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src/topology/functions/set-systems.tex
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@@ -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 $S \in \sigma$,
\[
E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
\]

2
src/topology/groups/definition.tex
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@@ -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 $G$.

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src/topology/uniform/completion.tex
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@@ -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.

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src/topology/uniform/definition.tex
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@@ -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}

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src/topology/uniform/metric.tex
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@@ -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)

2
src/topology/uniform/uc.tex
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@@ -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
\[
\paren{\bigcap_{j \in J}(f_j \times f_j)^{-1}(V_j)} \circ \paren{\bigcap_{j \in J}(f_j \times f_j)^{-1}(V_j)} \subset \bigcap_{j \in J}(f_j \times f_j)^{-1}(U_j)
\]