From 0b24ab616ff94d96ea54efab3e829ced009ab4bb Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 21 Jan 2026 16:03:50 -0500 Subject: [PATCH] Added measurability in separable metric spaces. --- src/measure/index.tex | 1 + src/measure/lebesgue-integral/index.tex | 4 + src/measure/measurable-maps/index.tex | 3 + .../measurable-maps/measurable-maps.tex | 9 ++ src/measure/measurable-maps/metric.tex | 115 ++++++++++++++++++ src/measure/measurable-maps/real-valued.tex | 50 ++++++++ src/measure/measurable-maps/simple.tex | 34 ++++++ src/measure/measure/radon.tex | 6 +- src/measure/sets/algebra.tex | 80 ++++++++++++ src/topology/index.tex | 1 + src/topology/main/interiorclosureboundary.tex | 2 +- src/topology/metric/index.tex | 5 + src/topology/metric/metric.tex | 14 +++ src/topology/metric/set-distance.tex | 55 +++++++++ src/topology/uniform/metric.tex | 2 +- 15 files changed, 376 insertions(+), 5 deletions(-) create mode 100644 src/measure/lebesgue-integral/index.tex create mode 100644 src/measure/measurable-maps/metric.tex create mode 100644 src/measure/measurable-maps/real-valued.tex create mode 100644 src/measure/measurable-maps/simple.tex create mode 100644 src/topology/metric/index.tex create mode 100644 src/topology/metric/metric.tex create mode 100644 src/topology/metric/set-distance.tex diff --git a/src/measure/index.tex b/src/measure/index.tex index 3816288..8161296 100644 --- a/src/measure/index.tex +++ b/src/measure/index.tex @@ -4,3 +4,4 @@ \input{./src/measure/sets/index.tex} \input{./src/measure/measure/index.tex} \input{./src/measure/measurable-maps/index.tex} +\input{./src/measure/lebesgue-integral/index.tex} diff --git a/src/measure/lebesgue-integral/index.tex b/src/measure/lebesgue-integral/index.tex new file mode 100644 index 0000000..abacd27 --- /dev/null +++ b/src/measure/lebesgue-integral/index.tex @@ -0,0 +1,4 @@ +\chapter{The Lebesgue Integral} +\label{chap:lebesgue-integral} + +%\input{./src/measure/lebesgue-integral/simple.tex} diff --git a/src/measure/measurable-maps/index.tex b/src/measure/measurable-maps/index.tex index 6a6448f..ecc61dd 100644 --- a/src/measure/measurable-maps/index.tex +++ b/src/measure/measurable-maps/index.tex @@ -3,3 +3,6 @@ \input{./src/measure/measurable-maps/measurable-maps.tex} \input{./src/measure/measurable-maps/product.tex} +\input{./src/measure/measurable-maps/real-valued.tex} +\input{./src/measure/measurable-maps/simple.tex} +\input{./src/measure/measurable-maps/metric.tex} diff --git a/src/measure/measurable-maps/measurable-maps.tex b/src/measure/measurable-maps/measurable-maps.tex index 3ab680f..d62affe 100644 --- a/src/measure/measurable-maps/measurable-maps.tex +++ b/src/measure/measurable-maps/measurable-maps.tex @@ -16,6 +16,15 @@ Let $X, Y$ be topological spaces and $f: X \to Y$ be continuous, then $f$ is Borel measurable. \end{lemma} +\begin{lemma} +\label{lemma:measurable-function-generating-set} + Let $(X, \cm)$ and $(Y, \cn = \sigma(\ce))$ be measurable spaces, and $f: X \to Y$ such that $f^{-1}(\ce) \subset \cm$, then $f$ is $(\cm, \cn)$-measurable. +\end{lemma} +\begin{proof} + Let $\mathcal{F} = \bracs{E \in \ce| f^{-1}(E) \in \cm}$. +\end{proof} + + \begin{definition}[Generated $\sigma$-Algebra] \label{definition:generated-sigma-algebra-function} Let $X$ be a set, $\bracs{(Y_i, \cn_i)}_{i \in I}$ be measurable spaces, and $\seqi{f}$ with $f_i: X \to Y_i$ for each $i \in I$. The \textbf{$\sigma$-algebra generated by} $\seqi{f}$, diff --git a/src/measure/measurable-maps/metric.tex b/src/measure/measurable-maps/metric.tex new file mode 100644 index 0000000..03c145f --- /dev/null +++ b/src/measure/measurable-maps/metric.tex @@ -0,0 +1,115 @@ +\section{Measurable Functions into Metric Spaces} +\label{section:measurable-metric} + +\begin{proposition} +\label{proposition:metric-measurable-fibre-product} + Let $(X, \cm)$, $(Z, \cn)$ be measurable spaces, $\seqf{Y_j}$ separable metrisable topological spaces, $F: \prod_{j = 1}^n Y_j \to Z$ be a $(\cb_{\prod_{j = 1}^n Y_j}, \cn)$-measurable function. For any $\seqf{f_j}$ where for each $1 \le j \le n$, $f_j: X \to Y_j$ is $(\cm, \cb_{Y_j})$-measurable, the composition + \[ + X \to Z \quad x \mapsto F(f_1(x), \cdots, f_n(x)) + \] + is $(\cm, \cn)$-measurable. +\end{proposition} +\begin{proof} + By \ref{proposition:product-sigma-algebra-metric}, $\cb_{\prod_{j = 1}^n Y_j} = \bigotimes_{j = 1}^n \cb_{Y_j}$, so + \[ + X \to \prod_{j = 1}^n Y_j \quad x \mapsto (f_1(x), \cdots, f_n(x)) + \] + is $(\cm, \cb_{\prod_{j = 1}^n Y_j})$-measurable. Therefore the composition is $(\cm, \cn)$-measurable. +\end{proof} + +\begin{proposition} +\label{proposition:metric-measurables} + Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, $f, g: X \to Y$ be $(\cm, \cb_Y)$-measurable functions, then the following functions are measurable: + \begin{enumerate} + \item For any metric $d$ on $Y$, $x \mapsto d(f(x), f(y))$. + \item If $Y$ is a TVS over $K \in \RC$ and $\lambda \in K$, $\lambda f + g$. In particular, $\bracs{f = g} \in \cm$. + \item If $Y \in \RC$, $fg$. + \end{enumerate} +\end{proposition} +\begin{proof} + By \ref{proposition:metric-measurable-fibre-product}. +\end{proof} + +\begin{proposition} +\label{proposition:metric-measurable-compose} + Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, and $f: X \to Y$ be a function, then the following are equivalent: + \begin{enumerate} + \item $f$ is $(\cm, \cb_Y)$-measurable. + \item For each $\phi \in C(X; [0, 1])$, $\phi \circ f$ is $(\cm, \cb_\real)$-measurable. + \end{enumerate} +\end{proposition} +\begin{proof} + (2) $\Rightarrow$ (1): For each $U \subset X$ open, the function + \[ + d_{U^c}: X \to [0, 1] \quad x \mapsto d(x, U^c) \wedge 1 + \] + is continuous by \ref{proposition:set-distance-continuous}. By \ref{proposition:fattening-closure}, $\bracsn{d_{U^c} > 0} = U$. Thus $\bracs{f \in U} = \bracsn{d_{U^c} \circ f > 0}$. +\end{proof} + +\begin{proposition} +\label{proposition:metric-measurable-limit} + Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, and $\seq{f_n}$ be $(\cm, \cb_Y)$-measurable functions, then: + \begin{enumerate} + \item If $Y$ is completely metrisable, then $\bracsn{\limv{n}f_n \text{ exists}} \in \cm$. + \item If $f = \limv{n}f_n$ exists, then it is $(\cm, \cb_Y)$-measurable. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): Let $d$ be a complete metric on $Y$, then for any $x \in X$, $\limv{n}f_n(x)$ exists if and only if $\seq{f_n(x)}$ is Cauchy. In which case, + \[ + \bracs{\limv{n}f_n \text{ exists}} = \bigcap_{k \in \natp}\bigcup_{N \in \natp}\bigcap_{m \in \natp}\bigcap_{n \in \natp}\bracsn{d(f_m, f_n) < 1/k} + \] + is measurable by \ref{proposition:metric-measurables}. + + (2): For each $\phi \in C(X; [0, 1])$, $\phi \circ f = \limv{n}\phi \circ f_n$ is $(\cm, \cb_\real)$-measurable by \ref{proposition:limit-measurable}. Thus $f$ is $(\cm, \cb_\real)$-measurable by \ref{proposition:metric-measurable-compose}. +\end{proof} + + +\begin{proposition} +\label{proposition:measurable-simple-separable} + Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, $N: Y \to 2^Y$\footnote{This mapping is typically obtained as slices of the level sets of a continuous function $Y \times Y \to \real$.}, and $f: X \to Y$ such that + \begin{enumerate} + \item[(a)] For each $y \in Y$, $y \in \ol{N(y)^o}$. + \item[(b)] $\bigcap_{y \in Y}N(y) \ne \emptyset$. + \item[(c)] For any $y \in Y$, $\bracs{x \in X|y \in N(f(x))} \in \cm$. + \end{enumerate} + then the following are equivalent: + \begin{enumerate} + \item $f$ is $(\cm, \cb_Y)$-measurable. + \item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that + \begin{enumerate} + \item[(i)] For each $x \in X$ and $n \in \natp$, $f_n(x) \in N(f(x))$. + \item[(ii)] $f_n \to f$ pointwise. + \end{enumerate} + \item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that $f_n \to f$ pointwise. + \end{enumerate} +\end{proposition} +\begin{proof} + (1) $\Rightarrow$ (2): Let $\seq{y_n} \subset Y$ be a dense subset of $Y$. Assume without loss of generality that $y_1 \in \bigcap_{y \in Y}N(y)$. For each $N \in \nat$ and $x \in X$, let + \[ + C(N, x) = \bracs{1 \le n \le N|y_n \in N(f(x))} + \] + By assumption (b), $1 \in C(N, x) \ne \emptyset$. Let + \[ + k(N, x) = \min\bracs{n \in C(N, x) \bigg | d(f(x), y_n) = \min_{m \in C(N, x)}d(f(x), y_m)} + \] + then for any $k \in \natp$, + \[ + \bracs{x \in X|k(n, x) \le k} = \bigcup_{j = 1}^k\bracs{x \in X \bigg |y_j \in C(n, x), d(f(x), y_j) = \min_{m \in C(N, x)}d(f(x), y_m)} + \] + For each $1 \le m \le N$, $y \mapsto d(y, y_m)$ is continuous. Thus $x \mapsto d(f(x), y_m)$ and $x \mapsto \min_{m \in C(N, x)}d(f(x), y_m)$ are $(\cm, \cb_\real)$-measurable by \ref{proposition:limit-measurable} and assumption (c). By \ref{proposition:metric-measurables}, + \[ + \bracs{x \in X \bigg | d(f(x), y_j) = \min_{m \in C(N, x)}d(f(x), y_m)} \in \cm + \] + for each $1 \le j \le k$. Combining this with assumption (c) shows that $\bracs{x \in X|k(n, x) \le k} \in \cm$. + + Let $f_N(x) = y_{k(N, x)}$, then for each $N \in \natp$, $f_N(x) \subset \bracs{y_n|1 \le n \le N}$, so $f_N(x)$ is finitely valued. Since $x \mapsto k{(N, x)}$ is $(\cm, 2^\nat)$-measurable, $f_N$ is $(\cm, \cb_Y)$-measurable. + + Fix $x \in X$, then + \[ + f(x) \in \ol{N^o(f(x))} = \ol{\bracs{y_n| n \in \natp, y_n \in N^o(f(x))}} + \] + by assumption (a) and \ref{definition:dense}. Thus for any $\eps > 0$, there exists $N \in \nat$ such that $y_N \in N^o(f(x))$ and $d(f(x), y_N) < \eps$. In which case, $d(f(x), f_n(x)) < \eps$ for all $n \ge N$. Therefore $f_n \to f$ pointwise. + + (3) $\Rightarrow$ (1): By \ref{proposition:metric-measurable-limit}. +\end{proof} diff --git a/src/measure/measurable-maps/real-valued.tex b/src/measure/measurable-maps/real-valued.tex new file mode 100644 index 0000000..49afdb7 --- /dev/null +++ b/src/measure/measurable-maps/real-valued.tex @@ -0,0 +1,50 @@ +\section{Real-Valued Measurable Functions} +\label{section:real-complex-measurable} + +\begin{lemma} +\label{lemma:extended-real-measurable} + Let $(X, \cm)$ be a measurable space and $f: X \to \real$, then the following are equivalent: + \begin{enumerate} + \item $f$ is $(\cm, \cb_{\ol \real})$-measurable. + \item $f$ is $(\cm, \cb_{\real})$-measurable. + \end{enumerate} +\end{lemma} +\begin{proof} + (1) $\Rightarrow$ (2): $\cb_{\real} \subset \cb_{\ol \real}$. + + (2) $\Rightarrow$ (1): For any $E \subset \ol{\real}$, $f^{-1}(E) = f^{-1}(E \cap \real) \in \cm$. +\end{proof} + +\begin{proposition} +\label{proposition:limit-measurable} + Let $(X, \cm)$ be a measurable space, $\seq{f_n}$ be $(\cm, \cb_{\ol \real})$-measurable functions, then the following functions are $(\cm, \cb_{\ol \real})$-measurable: + \begin{enumerate} + \item $F = \sup_{n \in \natp}f_n$. + \item $f = \inf_{n \in \natp}f_n$. + \item $G = \limsup_{n \to \infty}f_n$. + \item $g = \limsup_{n \to \infty}f_n$. + \item $\limv{n}f_n$ (if it exists). + \end{enumerate} + In addition, if the above functions are $\real$-valued, then they are $(\cm, \cb_{\real})$-measurable. +\end{proposition} +\begin{proof} + (1): Let $\alpha \in \real$, then for any $x \in X$, $F(x) > \alpha$ if and only if there exists $n \in \natp$ such that $f_n(x) > \alpha$. Thus + \[ + \bracs{F > \alpha} = \bigcup_{n \in \natp}\bracs{f_n > \alpha} + \] + + (2): Let $\alpha \in \real$, then + \[ + \bracs{f < \alpha} = \bigcup_{n \in \natp}\bracs{f_n < \alpha} + \] + + By \ref{proposition:borel-sigma-extended-generators} and \ref{lemma:measurable-function-generating-set}, $F$ and $f$ are both $(\cm, \cb_{\ol \real})$-measurable. + + (3): $\limsup_{n \to \infty}f_n = \inf_{n \in \natp}\sup_{k \ge n}f_k$. + + (4): $\liminf_{n \to \infty}f_n = \sup_{n \in \natp}\inf_{k \ge n}f_k$. + + (5): If $\limv{n}f_n$ exists, then it is equal to (3) and (4). In which case, it is measurable. + + Finally, if the above functions are $\real$-valued, then they are $(\cm, \cb_\real)$-measurable by \ref{lemma:extended-real-measurable}. +\end{proof} diff --git a/src/measure/measurable-maps/simple.tex b/src/measure/measurable-maps/simple.tex new file mode 100644 index 0000000..211f7ff --- /dev/null +++ b/src/measure/measurable-maps/simple.tex @@ -0,0 +1,34 @@ +\section{Simple Functions} +\label{section:simple-function} + +\begin{definition}[Indicator Function] +\label{definition:indicator-function} + Let $(X, \cm)$ be a measurable space and $E \in \cm$, then the function + \[ + \chi_E = \one_E: X \to \bracs{0, 1} \quad x \mapsto \begin{cases} + 1 &x \in E \\ + 0 &x \not\in E + \end{cases} + \] + is the \textbf{characteristic/indicator function} of $E$. +\end{definition} + +\begin{definition}[Simple Function] +\label{definition:simple-function} + Let $(X, \cm)$ be a measurable space and $Y$ be a set, then a function $\phi: X \to Y$ is \textbf{simple/finitely valued} if: + \begin{enumerate} + \item $\phi(X)$ is finite. + \item For each $y \in \phi(X)$, $\phi^{-1}(y) \in \cm$. + \end{enumerate} +\end{definition} + +\begin{definition}[Standard Form] +\label{definition:simple-function-standard-form} + Let $(X, \cm)$ be a measurable space, $V$ be a vector space over $K \in \RC$, and $f: X \to Y$ be a simple function, then + \[ + f = \sum_{y \in f(X)}y\one_Y + \] + is the \textbf{standard form} of $f$. + + The set $\Sigma(X, \cm, V)$ is the vector space of all simple functions from $X$ to $Y$. +\end{definition} diff --git a/src/measure/measure/radon.tex b/src/measure/measure/radon.tex index 8a9b76c..3f721f3 100644 --- a/src/measure/measure/radon.tex +++ b/src/measure/measure/radon.tex @@ -5,8 +5,8 @@ \label{definition:radon-measure} Let $X$ be a LCH space and $\mu: \cb_X \to [0, \infty]$ be a Borel measure, then $\mu$ is a \textbf{Radon measure} if: \begin{enumerate} - \item For any $K \subset X$ compact, $\mu(K) < \infty$. - \item $\mu$ is outer regular on all Borel sets. - \item $\mu$ is inner regular on all open sets. + \item[(R1)] For any $K \subset X$ compact, $\mu(K) < \infty$. + \item[(R2)] $\mu$ is outer regular on all Borel sets. + \item[(R3')] $\mu$ is inner regular on all open sets. \end{enumerate} \end{definition} diff --git a/src/measure/sets/algebra.tex b/src/measure/sets/algebra.tex index 676ccae..885af49 100644 --- a/src/measure/sets/algebra.tex +++ b/src/measure/sets/algebra.tex @@ -72,7 +72,87 @@ Let $X$ be a set and $\ce \subset 2^X$, then the $\sigma$-algebra $\sigma(\ce)$ \textbf{generated by} $\ce$ is the smallest $\sigma$-algebra on $X$ containing $\ce$. \end{definition} + \begin{definition}[Borel $\sigma$-Algebra] \label{definition:borel-sigma-algebra} Let $(X, \topo)$ be a topological space, then the \textbf{Borel $\sigma$-algebra} $\cb_X$ on $X$ is the $\sigma$-algebra generated by $\topo$. \end{definition} + + +\begin{definition}[Borel $\sigma$-Algebra on $\ol{\real}$] +\label{definition:borel-sigma-algebra-extended} + The family + \[ + \cb_{\ol{\real}} = \bracsn{E \subset \ol \real| E \cap \real \in \cb_\real} + \] + is the \textbf{Borel $\sigma$-algebra} on $\ol{\real}$. +\end{definition} + + +\begin{proposition} +\label{proposition:borel-sigma-real-generators} + The following families of sets generate the Borel $\sigma$-algebra on $\real$: + \begin{enumerate} + \item $\bracs{(-\infty, a]| a \in \real}$. + \item $\bracs{(a, \infty)|a \in \real}$. + \item $\bracs{[a, \infty)| a \in \real}$. + \item $\bracs{(-\infty, a)| a \in \real}$. + \item $\bracs{[a, b)| -\infty < a < b < \infty}$. + \item $\bracs{[a, b]| -\infty < a < b < \infty}$. + \item $\bracs{(a, b]| -\infty < a < b < \infty}$. + \item $\bracs{(a, b)| -\infty < a < b < \infty}$. + \end{enumerate} +\end{proposition} +\begin{proof} + It is sufficient to show that the $\sigma$-algebra generated by any of the above two families coincide, and that the resulting $\sigma$-algebra is the Borel $\sigma$-algebra on $\real$. + + (1) $\to$ (2): For any $a \in \real$, $(a, \infty) = (-\infty, a)^c$. + + (2) $\to$ (3): For any $a \in \real$, $[a, \infty) = \bigcap_{n \in \natp}(a - 1/n, \infty)$. + + (3) $\to$ (4): For any $a \in \real$, $(-\infty, a) = [a, \infty)^c$. + + (4) $\to$ (5): For any $a, b \in \real$, $[a, b) = (-\infty, b) \cap (-\infty, a)^c$. + + (5) $\to$ (6): For any $a, b \in \real$, $[a, b]= \bigcap_{n \in \natp}(a - 1/n, b]$. + + (6) $\to$ (7): For any $a, b \in \real$, $(a, b] = \bigcup_{n \in \natp}[a + 1/n, b]$. + + (7) $\to$ (8): For any $a, b \in \real$, $(a, b) = \bigcup_{n \in \natp}(a, b - 1/n]$. + + (8) $\to$ (1): For any $a \in \real$, $(-\infty, a] = \bigcup_{n \in \natp}\bigcap_{k \in \natp}(-n, a + 1/k]$. + + For any $U \subset X$ open and $q \in U \cap \rational$, there exists $r_q > 0$ such that $(q - r_q, q + r_q) \subset U$. In which case, + \[ + U = \bigcup_{q \in U \cap \rational}(q - r_q, q + r_q) + \] + so (8) generates all open sets in $\real$. Conversely, every element of (8) is open, so the $\sigma$-algebra generated by (8) is the Borel $\sigma$-algebra on $\real$. +\end{proof} + +\begin{proposition} +\label{proposition:borel-sigma-extended-generators} + The following families of sets generate the Borel $\sigma$-algebra on $\ol \real$: + \begin{enumerate} + \item $\bracs{[-\infty, a]| a \in \real}$. + \item $\bracs{(a, \infty]|a \in \real}$. + \item $\bracs{[a, \infty]| a \in \real}$. + \item $\bracs{[-\infty, a)| a \in \real}$. + \end{enumerate} +\end{proposition} +\begin{proof} + (1) $\to$ (2): For any $a \in \real$, $(a, \infty] = [-\infty, a]^c$. + + (2) $\to$ (3): For any $a \in \real$, $[a, \infty] = \bigcap_{n \in \natp}(a - 1/n, \infty]$. + + (3) $\to$ (4): For any $a \in \real$, $[-\infty, a) = [a, \infty]^c$. + + (4) $\to$ (1): For any $a \in \real$, $[-\infty, a] = \bigcap_{n \in \natp}[-\infty, a + 1/n)$. + + By definition, all elements of (1), (2), (3), and (4) belong to $\cb_{\ol{\real}}$. Let $\cm$ be the $\sigma$-algebra generated by (1), (2), (3), and (4), then + \[ + \bracs{\infty} = \bigcap_{n \in \nat}[n, \infty] \quad \bracs{-\infty} = \bigcap_{n \in \nat}[-\infty, n] + \] + are elements of $\cm$. For any $a \in \real$, $(a, \infty) = (a, \infty] \setminus \bracs{\infty}$, so $\cm \supset \cb_\real$ by \ref{proposition:borel-sigma-real-generators}. + + In addition, for any $E \in \cb_{\ol{\real}}$, $E = (E \cap \real) \cup (E \setminus \real)$, where $E \cap \real \in \cb_\real$. Since $\bracs{\infty}, \bracs{-\infty} \in \cm$ and $\cb_\real \subset \cm$, $E \in \cm$ and $\cm = \cb_{\ol \real}$. +\end{proof} diff --git a/src/topology/index.tex b/src/topology/index.tex index 9065c28..d47ef26 100644 --- a/src/topology/index.tex +++ b/src/topology/index.tex @@ -4,3 +4,4 @@ \input{./src/topology/main/index.tex} \input{./src/topology/uniform/index.tex} \input{./src/topology/functions/index.tex} +\input{./src/topology/metric/index.tex} diff --git a/src/topology/main/interiorclosureboundary.tex b/src/topology/main/interiorclosureboundary.tex index 2ab16f1..4a881d9 100644 --- a/src/topology/main/interiorclosureboundary.tex +++ b/src/topology/main/interiorclosureboundary.tex @@ -63,7 +63,7 @@ \begin{definition}[Dense] \label{definition:dense} - Let $X$ be a topologicial space and $A \subset X$, then the following are equivalent: + Let $X$ be a topological space and $A \subset X$, then the following are equivalent: \begin{enumerate} \item $\ol{A} = X$. \item For every $\emptyset \ne U \subset X$ open, $A \cap U \ne \emptyset$. diff --git a/src/topology/metric/index.tex b/src/topology/metric/index.tex new file mode 100644 index 0000000..97a91cd --- /dev/null +++ b/src/topology/metric/index.tex @@ -0,0 +1,5 @@ +\chapter{Metric Spaces} +\label{chap:metric-space} + +\input{./src/topology/metric/metric.tex} +\input{./src/topology/metric/set-distance.tex} diff --git a/src/topology/metric/metric.tex b/src/topology/metric/metric.tex new file mode 100644 index 0000000..7fbd73e --- /dev/null +++ b/src/topology/metric/metric.tex @@ -0,0 +1,14 @@ +\section{Metrics} +\label{section:metric} + +\begin{definition}[Metric Space] +\label{definition:metric} + Let $X$ be a set and $d: X \times X$, then $d$ is a \textbf{metric} if: + \begin{enumerate} + \item[(PM1)] For any $x \in X$, $d(x, x) = 0$. + \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$. + \item[(PM2)] For any $x, y \in X$, $d(x, y) = d(y, x)$. + \item[(PM3)] For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$. + \end{enumerate} + The pair $(X, d)$ is a \textbf{metric space}, which comes with the metric uniformity induced by $d$, and the corresponding topology. +\end{definition} diff --git a/src/topology/metric/set-distance.tex b/src/topology/metric/set-distance.tex new file mode 100644 index 0000000..ac03be8 --- /dev/null +++ b/src/topology/metric/set-distance.tex @@ -0,0 +1,55 @@ +\section{Distance Between Sets} +\label{section:distance-between-sets} + +\begin{definition}[Distance Between Sets] +\label{definition:distance-betwene-sets} + Let $X$ be a set, $d: X \times X \to [0, \infty)$ be a pseudometric, and $A, B \subset X$, then + \[ + d(A, B) = \inf_{\substack{a \in A \\ b \in B}}d(a, b) + \] + is the \textbf{distance} between $A$ and $B$. +\end{definition} + +\begin{proposition} +\label{proposition:set-distance-continuous} + Let $(X, d)$ be a pseudometric space, $A \subset X$, and + \[ + d_A: X \to [0, \infty) \quad x \mapsto d(x, A) + \] + then $d_A \in UC(X; [0, \infty))$, where for any $x, y \in X$, $|d_A(x) - d_A(y)| \le d(x, y)$. +\end{proposition} +\begin{proof} + Let $x, y \in X$, then + \[ + d(y, A) = \inf_{a \in A}d(y, a) \le d(x, y) + \inf_{a \in A}d(x, a) = d(x, y) + d(x, A) + \] + As the argument is symmetric, $|d_A(x) - d_A(y)| \le d(x, y)$. +\end{proof} + +\begin{definition}[Fattening] +\label{definition:fattening} + Let $X$ be a set, $d: X \times X \to [0, \infty)$ be a pseudometric, $A \subset X$, and $\eps > 0$, then + \[ + B(A, \eps) = \bracs{x \in X|d(x, A) < \eps} = \bigcup_{a \in A}B(a, \eps) + \] + is the \textbf{$\eps$-fattening} of $A$ with respect to $d$. +\end{definition} + +\begin{proposition} +\label{proposition:fattening-closure} + Let $(X, d)$ be a metric space and $A \subset X$, then + \[ + \ol{A} = \bigcap_{\eps > 0}B(A, \eps) + \] +\end{proposition} +\begin{proof} + By \ref{proposition:uniformclosure}. +\end{proof} + +\begin{proposition} +\label{proposition:distance-compact} + Let $(X, d)$ be a metric space, $C \subset X$ be closed, and $K \subset X$ be compact. If $C \cap K = \emptyset$, then $d(K, C) > 0$. +\end{proposition} +\begin{proof} + Suppose that $d(K, C) = 0$, then there exists $\seq{(x_n, y_n)} \subset K \times C$ such that $d(x_n, y_n) \to 0$ as $n \to \infty$. By \ref{definition:compact}, there exists a subsequence $\seq{n_k}$ and $x \in K$ such that $x_{n_k} \to x$ as $k \to \infty$. In which case, $d(x, C) = 0$, and $x \in \ol{K}$ by \ref{proposition:fattening-closure}, so $K \cap C \ne \emptyset$. +\end{proof} diff --git a/src/topology/uniform/metric.tex b/src/topology/uniform/metric.tex index eb1f22c..0f403e7 100644 --- a/src/topology/uniform/metric.tex +++ b/src/topology/uniform/metric.tex @@ -13,7 +13,7 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \end{enumerate} If $d$ satisfies the above and \begin{enumerate} - \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y)$ + \item[(M)] For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$. \end{enumerate} then $d$ is a \textbf{metric}. \end{definition}