From 755c2d8ae7adc5e9819e2e30c0e9199d234f6b37 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Wed, 21 Jan 2026 10:30:32 -0500
Subject: [PATCH] Enforced Borel sigma algebra convention.

---
 src/measure/measure/index.tex   |  3 ++-
 src/measure/measure/radon.tex   | 12 ++++++++++++
 src/measure/measure/regular.tex |  6 +++---
 3 files changed, 17 insertions(+), 4 deletions(-)
 create mode 100644 src/measure/measure/radon.tex

diff --git a/src/measure/measure/index.tex b/src/measure/measure/index.tex
index f1a8f19..b91ccf3 100644
--- a/src/measure/measure/index.tex
+++ b/src/measure/measure/index.tex
@@ -5,7 +5,8 @@
 \input{./src/measure/measure/complete.tex}
 \input{./src/measure/measure/semifinite.tex}
 \input{./src/measure/measure/sigma-finite.tex}
-\input{./src/measure/measure/outer.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/kolmogorov.tex}
diff --git a/src/measure/measure/radon.tex b/src/measure/measure/radon.tex
new file mode 100644
index 0000000..d467060
--- /dev/null
+++ b/src/measure/measure/radon.tex
@@ -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}
diff --git a/src/measure/measure/regular.tex b/src/measure/measure/regular.tex
index bca7c5d..c46a888 100644
--- a/src/measure/measure/regular.tex
+++ b/src/measure/measure/regular.tex
@@ -3,7 +3,7 @@
 
 \begin{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
     \[
     \mu(E) = \sup\bracs{\mu(K)| K \subset E, K \text{ compact}}
     \]
@@ -11,7 +11,7 @@
 
 \begin{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
     \[
     \mu(E) = \sup\bracs{\mu(U)| U \supset A, U \text{ open}}
     \]
@@ -19,7 +19,7 @@
 
 \begin{definition}[Regular]
 \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}
 
 \begin{theorem}[{{\cite[Theorem 7.8]{Folland}}}]