Enforced Borel sigma algebra convention.

This commit is contained in:
Bokuan Li
2026-01-21 10:31:25 -05:00
parent 755c2d8ae7
commit bc9927a326
4 changed files with 5 additions and 2 deletions

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@@ -9,4 +9,5 @@
\input{./src/measure/measure/radon.tex}
\input{./src/measure/measure/outer.tex}
\input{./src/measure/measure/lebesgue-stieltjes.tex}
\input{./src/measure/measure/radon.tex}
\input{./src/measure/measure/kolmogorov.tex}

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@@ -19,7 +19,7 @@
\end{enumerate}
Let $\bracs{\mu_{I}| I \subset \natp \text{ finite}}$ be consistent Borel probability measures, then for any $\seq{B_n}$ where:
\begin{enumerate}
\item[(d)] For each $n \in \nat$, $B_n \in \cb(\prod_{j = 1}^n X_j)$.
\item[(d)] For each $n \in \nat$, $B_n \in \cb_{\prod_{j = 1}^n X_j}$.
\item[(e)] For each $n \in \nat$, $B_{n+1} \subset B_n \times X_{n+1}$.
\item[(f)] There exists $\eps > 0$ such that $\mu_{[n]}(B_n) > \eps$ for all $n \in \natp$.
\end{enumerate}

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@@ -3,7 +3,7 @@
\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:
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.

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@@ -0,0 +1,2 @@
\section{Riesz Representation Theorem}
\label{section:riesz-radon}