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Enforced Borel sigma algebra convention.

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This commit is contained in:
Bokuan Li
2026-01-21 10:30:32 -05:00
parent d4187ecde4
commit 755c2d8ae7
3 changed files with 17 additions and 4 deletions
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3
src/measure/measure/index.tex
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@@ -5,7 +5,8 @@
\input{./src/measure/measure/complete.tex} \input{./src/measure/measure/complete.tex}
\input{./src/measure/measure/semifinite.tex} \input{./src/measure/measure/semifinite.tex}
\input{./src/measure/measure/sigma-finite.tex} \input{./src/measure/measure/sigma-finite.tex}
\input{./src/measure/measure/outer.tex}
\input{./src/measure/measure/regular.tex} \input{./src/measure/measure/regular.tex}
\input{./src/measure/measure/radon.tex}
\input{./src/measure/measure/outer.tex}
\input{./src/measure/measure/lebesgue-stieltjes.tex} \input{./src/measure/measure/lebesgue-stieltjes.tex}
\input{./src/measure/measure/kolmogorov.tex} \input{./src/measure/measure/kolmogorov.tex}

12
src/measure/measure/radon.tex Normal file
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@@ -0,0 +1,12 @@
\section{Radon Measures}
\label{section:radon-measure}
\begin{definition}[Radon Measure]
\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.
\end{enumerate}
\end{definition}

6
src/measure/measure/regular.tex
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@@ -3,7 +3,7 @@
\begin{definition}[Inner Regular] \begin{definition}[Inner Regular]
\label{definition:inner-regular} \label{definition:inner-regular}
Let $X$ be a topological space and $\mu: \cb_X \to [0, \infty]$ be a measure, then $\mu$ is \textbf{inner regular} if for any $E \in \cb_X$, Let $X$ be a topological space, $\mu: \cb_X \to [0, \infty]$ be a Borel measure, and $E \in \cb_X$, then $\mu$ is \textbf{inner regular} on $E$ if
\[ \[
\mu(E) = \sup\bracs{\mu(K)| K \subset E, K \text{ compact}} \mu(E) = \sup\bracs{\mu(K)| K \subset E, K \text{ compact}}
\] \]
@@ -11,7 +11,7 @@
\begin{definition}[Outer Regular] \begin{definition}[Outer Regular]
\label{definition:outer-regular} \label{definition:outer-regular}
Let $X$ be a topological space and $\mu: \cb_X \to [0, \infty]$ be a measure, then $\mu$ is \textbf{outer regular} if for any $E \in \cb_X$, Let $X$ be a topological space, $\mu: \cb_X \to [0, \infty]$ be a Borel measure, and $E \in \cb_X$, then $\mu$ is \textbf{outer regular} on $E$ if
\[ \[
\mu(E) = \sup\bracs{\mu(U)| U \supset A, U \text{ open}} \mu(E) = \sup\bracs{\mu(U)| U \supset A, U \text{ open}}
\] \]
@@ -19,7 +19,7 @@
\begin{definition}[Regular] \begin{definition}[Regular]
\label{definition:regular-measure} \label{definition:regular-measure}
Let $X$ be a topological space and $\mu: \cb_X \to [0, \infty]$ be a measure, then $\mu$ is \textbf{regular} if it is inner regular and outer regular. Let $X$ be a topological space and $\mu: \cb_X \to [0, \infty]$ be a measure, then $\mu$ is \textbf{regular} if it is inner regular and outer regular on all Borel sets.
\end{definition} \end{definition}
\begin{theorem}[{{\cite[Theorem 7.8]{Folland}}}] \begin{theorem}[{{\cite[Theorem 7.8]{Folland}}}]