Added the Bochner integral.
This commit is contained in:
Bokuan Li
60
src/measure/bochner-integral/bochner.tex
Normal file
60
src/measure/bochner-integral/bochner.tex
Normal file
@@ -0,0 +1,60 @@
|
||||
\section{The Bochner Integral}
|
||||
\label{section:bochner-integral}
|
||||
|
||||
\begin{definition}[Bochner Integral]
|
||||
\label{definition:bochner-integral}
|
||||
Let $(X, \cm, \mu)$ be a measure space and $E$ be a Banach space over $K \in \RC$, then there exists a unique $I \in L(L^1(X; E); E)$ such that:
|
||||
\begin{enumerate}
|
||||
\item For any $x \in E$ and $A \in \cm$ with $\mu(A) < \infty$, $I(x \cdot \one_A) = x \cdot \mu(A)$.
|
||||
\item For all $f \in L^1(X; E)$, $\norm{If}_E \le \int \norm{f}_E d\mu$.
|
||||
\end{enumerate}
|
||||
|
||||
For any $f \in L^1(X; E)$, $If = \int f d\mu$ is the \textbf{Bochner integral} of $f$.
|
||||
\end{definition}
|
||||
\begin{proof}
|
||||
(1): For any $\phi \in \Sigma(X; E) \cap L^1(X; E)$, let
|
||||
\[
|
||||
I\phi = \sum_{y \in \phi(X)}^n y \cdot \mu\bracs{\phi = y}
|
||||
\]
|
||||
|
||||
For any $\lambda \in K$, if $\lambda \ne 0$, then the mapping $x \mapsto \lambda x$ is a bijection, so
|
||||
\[
|
||||
I(\lambda \phi) = \sum_{\lambda y \in \lambda \phi(X)}^n (\lambda y) \cdot \mu\bracs{\lambda \phi = \lambda y}
|
||||
= \lambda \sum_{y \in \phi(X)}y \cdot \mu\bracs{\phi = y} = \lambda I\phi
|
||||
\]
|
||||
|
||||
If $\lambda = 0$, then $I(\lambda \phi) = I(0) = 0$.
|
||||
|
||||
Let $\phi, \psi \in \Sigma(X; E) \cap L^1(X; E)$, then
|
||||
\begin{align*}
|
||||
I\phi + I\psi &= \sum_{y \in \phi(X)}y \cdot \mu\bracs{\phi = y} + \sum_{z \in \psi(X)}z \cdot \mu\bracs{\psi = z} \\
|
||||
&= \sum_{y \in \phi(X)} \sum_{z \in \psi(X)} (y + z) \cdot \mu\bracs{\phi = y, \psi = z} \\
|
||||
&= \sum_{y \in (\phi + \psi)(X)}\sum_{{z \in \phi(X) \atop {z' \in \psi(X) \atop z + z' = y}}}(z + z') \cdot \mu\bracsn{\phi = g, \psi = z'} \\
|
||||
&= \sum_{y \in (\phi + \psi)(X)}y \cdot \mu(\bracs{\phi + \psi = y}) = I\phi + I\psi
|
||||
\end{align*}
|
||||
|
||||
so $I$ is a linear operator on $\Sigma(X; E) \cap L^1(X; E)$ that satisfies (1).
|
||||
|
||||
(2): For any $\phi \in \Sigma(X; E) \cap L^1(X; E)$,
|
||||
\[
|
||||
\norm{I\phi}_E \le \sum_{y \in \phi(X)}\norm{y}_E \cdot \mu\bracs{\phi = y} = \int \norm{\phi}_E d\mu = \norm{\phi}_{L^1(X; E)}
|
||||
\]
|
||||
|
||||
By \autoref{proposition:lp-simple-dense}, $\Sigma(X; E) \cap L^1(X; E)$ is dense in $L^1(X; E)$. Therefore by the \hyperref[Linear Extension Theorem]{theorem:linear-extension-theorem-normed}, $I$ admits a unique norm-preserving extension to $L^1(X; E)$.
|
||||
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}[Dominated Convergence Theorem]
|
||||
\label{theorem:dct-bochner}
|
||||
Let $(X, \cm, \mu)$ be a measure spacs, $E$ be a Banach space over $K \in \RC$, $\seq{f_n} \subset L^1(X; E)$, and $f \in L^1(X; E)$. If
|
||||
\begin{enumerate}
|
||||
\item[(a)] $f_n \to f$ strongly pointwise.
|
||||
\item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$.
|
||||
\end{enumerate}
|
||||
|
||||
then $\int f d\mu = \limv{n}\int f_n d\mu$.
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
By the classical \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^1(X; E)$. Since $h \mapsto \int h d\mu$ is a bounded linear operator, $\int f d\mu = \limv{n}\int f_n d\mu$.
|
||||
\end{proof}
|
||||
|
||||
@@ -2,3 +2,4 @@
|
||||
\label{chap:bochner-integral}
|
||||
|
||||
\input{./strongly.tex}
|
||||
\input{./bochner.tex}
|
||||
|
||||
@@ -6,16 +6,22 @@
|
||||
Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $f: X \to E$, then the following are equivalent:
|
||||
\begin{enumerate}
|
||||
\item For each $\phi \in E^*$, $\phi \circ f$ is $(\cm, \cb_K)$-measurable and $f(X) \subset E$ is separable.
|
||||
\item $f$ is $(\cm, \cb_E)$ measurable and $f(X) \subset E$ is separable.
|
||||
\item $f$ is $(\cm, \cb_E)$-measurable and $f(X) \subset E$ is separable.
|
||||
\item There exists a sequence $\seq{f_n} \subset \Sigma(X, \cm; E)$ such that
|
||||
\begin{enumerate}
|
||||
\item[(a)] For each $n \in \natp$, $\norm{f_n}_E \le \norm{f}_E$.
|
||||
\item[(b)] $\norm{f_n(x) - f(x)}_E \to 0$ pointwise as $n \to \infty$.
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
|
||||
If the above holds, then $f$ is a \textbf{strongly measurable} function.
|
||||
\end{definition}
|
||||
\begin{proof}
|
||||
(1) $\Rightarrow$ (2): TODO
|
||||
(1) $\Rightarrow$ (2): First suppose that $E$ is separable. By \autoref{proposition:separable-banach-borel-sigma-algebra}, the Borel $\sigma$-algebra on $E$ coincides with the $\sigma$-algebra on $E$ generated by the weak topology. Thus if $\phi \circ f$ is $(\cm, \cb_K)$-measurable for all $\phi \in E^*$, then $f$ is $(\cm, \cb_E)$-measurable.
|
||||
|
||||
Now suppose that $E$ is arbitrary. Let $F \subset E$ be the closure of the linear span of $f(X)$, then $F$ is a separable closed subspace of $E$. For any $\phi \in F^*$, by the \hyperref[Hahn-Banch Theorem]{theorem:hahn-banach}, there exists an extension $\Phi \in E^*$ of $\phi$. In which case, since $f(X) \subset F$, for any Borel set $B \in \cb_{K}$, $\bracs{\phi \circ f \in B} = \bracs{\Phi \circ f \in B} \in \cm$. Thus $\phi \circ f$ is $(\cm, \cb_K)$-measurable for all $\phi \in E^*$.
|
||||
|
||||
By the separable case, $f$ is $(\cm, \cb_{F})$-measurable. Let $B \in \cb_E$, then $B \cap F \in \cb_F$ by \autoref{lemma:borel-induced}. Therefore $\bracs{f \in B} = \bracs{f \in B \cap F} \in \cm$, and $f$ is $(\cm, \cb_E)$-measurable.
|
||||
|
||||
(2) $\Rightarrow$ (3): By \autoref{proposition:measurable-simple-separable-norm}.
|
||||
|
||||
@@ -26,3 +32,21 @@
|
||||
|
||||
and each $f_n$ is finitely-valued, $f(X)$ is separable.
|
||||
\end{proof}
|
||||
|
||||
\begin{proposition}
|
||||
\label{proposition:strongly-measurable-properties}
|
||||
Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, then:
|
||||
\begin{enumerate}
|
||||
\item For any strongly measurable functions $f, g: X \to E$ and $\lambda \in K$, $\lambda f + g$ is strongly measurable.
|
||||
\item For any strongly measurable functions $\bracs{f_n: X \to E|n \in \natp}$ and $f: X \to E$, if $f_n \to f$ strongly pointwise, then $f$ is strongly measurable.
|
||||
\end{enumerate}
|
||||
|
||||
\end{proposition}
|
||||
\begin{proof}
|
||||
(1): Since $x \mapsto \lambda$ is continuous, $\lambda f$ is strongly measurable by (2) of \autoref{definition:strongly-measurable}.
|
||||
|
||||
By (3) of \autoref{definition:strongly-measurable}, there exists $\seq{f_n}, \seq{g_n} \subset \Sigma(X, \cm; E)$ such that $f_n \to f$ and $g_n \to g$ strongly pointwise. In which case, $\seq{f_n + g_n} \subset \Sigma(X, \cm; E)$ and $f_n + g_n \to f + g$ strongly pointwise. Therefore $f + g$ is also strongly measurable.
|
||||
|
||||
(2): By \autoref{proposition:metric-measurable-limit}, $f$ is $(\cm, \cb_E)$-measurable. Since $f(X) \subset \overline{\bigcup_{n \in \natp}}f_n(X)$, $f(X)$ is also separable, so $f$ is strongly measurable by (1) of \autoref{definition:strongly-measurable}.
|
||||
\end{proof}
|
||||
|
||||
|
||||
@@ -8,7 +8,7 @@
|
||||
|
||||
\begin{definition}[Integral of Non-Negative Simple Functions]
|
||||
\label{definition:lebesgue-simple}
|
||||
Let $(X, \cm, \mu)$ be a measure space and $f = \sum_{y \in f(X)}y \cdot \one_{\bracs{f = y}} \in \Sigma^+(X, \cm)$ be a non-negative simple function in standard form, then\footnote{With the convention that $0 \cdot \infty = 0$.}
|
||||
Let $(X, \cm, \mu)$ be a measure space and $f = \sum_{y \in f(X)}y \cdot \one_{\bracs{f = y}} \in \Sigma^+(X, \cm)$ be a non-negative simple function in standard form, then (with the convention that $0 \cdot \infty = 0$)
|
||||
\[
|
||||
\int f d\mu = \int f(x) \mu(dx) = \sum_{y \in f(X)}y \cdot \mu(\bracs{f = y})
|
||||
\]
|
||||
|
||||
@@ -71,7 +71,7 @@
|
||||
|
||||
\begin{proposition}
|
||||
\label{proposition:measurable-simple-separable}
|
||||
Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, and $N: Y \to 2^Y$\footnote{This mapping is typically obtained as slices of the level sets of a continuous function $Y \times Y \to \real$.} such that
|
||||
Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, and $N: Y \to 2^Y$ such that
|
||||
\begin{enumerate}
|
||||
\item[(a)] For each $y \in Y$, $y \in \ol{N(y)^o}$.
|
||||
\item[(b)] $\bigcap_{y \in Y}N(y) \ne \emptyset$.
|
||||
|
||||
@@ -13,7 +13,7 @@
|
||||
\label{lemma:kolmogorov-compact-sequence}
|
||||
Let $\seq{X_n}$ be topological spaces where for each $n \in \natp$,
|
||||
\begin{enumerate}
|
||||
\item[(a)] Every finite measure on $\prod_{j = 1}^n X_j$ is regular\footnote{A potential sufficient condition for this is that each $X_n$ is LCH where every open set is $\sigma$-compact. However, I have yet to verify if this condition persists over products.}.
|
||||
\item[(a)] Every finite measure on $\prod_{j = 1}^n X_j$ is regular.
|
||||
\item[(b)] $X_n$ is Hausdorff.
|
||||
\item[(c)] $X_n$ is separable.
|
||||
\end{enumerate}
|
||||
|
||||
@@ -72,90 +72,14 @@
|
||||
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}[Induced $\sigma$-Algebra]
|
||||
\label{definition:}
|
||||
Let $X$ be a set, $\cm \subset 2^X$ be a $\sigma$-algebra over $X$, and $E \subset X$, then the collection
|
||||
\[
|
||||
\cm_E = \bracs{A \cap E|A \in \cm}
|
||||
\]
|
||||
|
||||
\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 $\topo$.
|
||||
is the \textbf{$\sigma$-algebra on $E$ induced by $\cm$}.
|
||||
\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 \autoref{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}
|
||||
|
||||
125
src/measure/sets/borel.tex
Normal file
125
src/measure/sets/borel.tex
Normal file
@@ -0,0 +1,125 @@
|
||||
\section{The Borel $\sigma$-Algebra}
|
||||
\label{section:borel-sigma-algebra}
|
||||
|
||||
|
||||
\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 \autoref{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}
|
||||
|
||||
|
||||
|
||||
\begin{proposition}
|
||||
\label{proposition:separable-metric-borel-sigma-algebra}
|
||||
Let $X$ be a separable metric space, then the Borel $\sigma$-algebra on $X$ is generated by the following families of sets:
|
||||
\begin{enumerate}
|
||||
\item Open sets of $X$.
|
||||
\item $\bracs{B(x, r)|x \in X, r > 0}$.
|
||||
\item $\bracsn{\ol{B(x, r)}|x \in X, r > 0}$.
|
||||
\end{enumerate}
|
||||
|
||||
\end{proposition}
|
||||
\begin{proof}
|
||||
(1) $\subset$ (2): Let $U \subset X$ be open. By \autoref{definition:dense}, there exists a countable dense subset $S \subset U$. For each $x \in S$, let $r_x > 0$ such that $B(x, r) \subset U$, then $U = \bigcup_{x \in S}B(x, r_x)$ is a countable union of open balls.
|
||||
|
||||
(2) $\subset$ (3): For any $x \in X$ and $r > 0$, $B(x, r) = \bigcup_{n \in \natp}\overline{B(x, r - 1/n)}$ is a countable union of closed balls.
|
||||
|
||||
(3) $\subset$ (1): For each $x \in X$ and $r > 0$, $\overline{B(x, r)}$ is closed.
|
||||
\end{proof}
|
||||
|
||||
|
||||
\begin{lemma}
|
||||
\label{lemma:borel-induced}
|
||||
Let $X$ be a topological space and $Y \subset X$ be a subspace, then the Borel $\sigma$-algebra on $Y$ coincides with the $\sigma$-algebra on $Y$ induced by $\cb_X$.
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
Since $\bracsn{A \in \cb_X|A \cap Y \in \cb_Y}$ is a $\sigma$-algebra that contains all open sets in $X$, $\cb_Y$ contains the induced $\sigma$-algebra.
|
||||
|
||||
On the other hand, the induced $\sigma$-algebra contains all open sets in $Y$, so it contains $\cb_Y$.
|
||||
|
||||
Therefore the two $\sigma$-algebras coincide.
|
||||
\end{proof}
|
||||
|
||||
|
||||
@@ -2,6 +2,7 @@
|
||||
\label{chap:set-system}
|
||||
|
||||
\input{./algebra.tex}
|
||||
\input{./borel.tex}
|
||||
\input{./lambda.tex}
|
||||
\input{./elementary.tex}
|
||||
\input{./limits.tex}
|
||||
|
||||
Reference in New Issue
Block a user