From 76107709a8604077e3b1a91f26b5b979872d1437 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Tue, 16 Jun 2026 13:42:54 -0400
Subject: [PATCH] Added a description of weak-* convergence in the case of
 bounded Borel functions.

---
 refs.bib                   | 14 +++++++-------
 src/measure/vector/fin.tex | 21 ++++++++++++++++++++-
 2 files changed, 27 insertions(+), 8 deletions(-)

diff --git a/refs.bib b/refs.bib
index 16abb18..05b13af 100644
--- a/refs.bib
+++ b/refs.bib
@@ -216,11 +216,11 @@
   doi     = {10.1007/BF02771592}
 }
 
-@MISC {StackRadonDual,
-    TITLE = {How to understand C(X)'' = bounded Borel measurable functions?},
-    AUTHOR = {GEdgar (https://math.stackexchange.com/users/442/gedgar)},
-    HOWPUBLISHED = {Mathematics Stack Exchange},
-    NOTE = {URL:https://math.stackexchange.com/q/392719 (version: 2013-05-15)},
-    EPRINT = {https://math.stackexchange.com/q/392719},
-    URL = {https://math.stackexchange.com/q/392719}
+@misc {StackRadonDual,
+    title = {How to understand C(X)'' = bounded Borel measurable functions?},
+    author = {GEdgar (https://math.stackexchange.com/users/442/gedgar)},
+    howpublished = {Mathematics Stack Exchange},
+    note = {URL:https://math.stackexchange.com/q/392719 (version: 2013-05-15)},
+    eprint = {https://math.stackexchange.com/q/392719},
+    url = {https://math.stackexchange.com/q/392719}
 }
\ No newline at end of file
diff --git a/src/measure/vector/fin.tex b/src/measure/vector/fin.tex
index b229793..5c8bbfb 100644
--- a/src/measure/vector/fin.tex
+++ b/src/measure/vector/fin.tex
@@ -93,9 +93,28 @@ In any case, the above example shows that a linear functional on $M(X, \cm; \com
     \]
 \end{proof}
 
+
+Despite the fact that it does not cover the full dual space, the bounded Borel functions still forms a subspace where weak-* convergence has a convenient description. 
+
 \begin{proposition}
 \label{proposition:measure-l-infinity-dominated-convergence}
-    Let $
+    Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mathscr{M} \subset M(X, \cm; E)$ be a closed subspace such that:
+    \begin{enumerate}
+        \item[(P)] For each $x \in X$, $\bracs{x} \in \cm$, and the delta mass $\delta_x$ is in $\mathscr{M}$.
+    \end{enumerate}
+
+    then for any sequence $f_n: X \to E^*$ of bounded measurable functions and $f: X \to E^*$ be a bounded measurable function, the following are equivalent:
+    \begin{enumerate}
+        \item For each $\mu \in \mathscr{M}$, $\limv{n}\int f_n d\mu = \int f d\mu$. 
+        \item For each $x \in X$, $\limv{n}f_n(x) = f(x)$, and $\sup_{n \in \natp}\norm{f_n}_u < \infty$. 
+    \end{enumerate}
 \end{proposition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): By (P), for each $x \in X$, $\limv{n}f_n(x) = f(x)$. By the \hyperref[Uniform Boundedness Principle]{theorem:uniform-boundedness},
+    \[
+    \sup_{n \in \natp}\norm{f_n}_u \le \sup_{n \in \natp}\norm{f_n}_{\mathscr{M}^*} < \infty
+    \]
 
+    (2) $\Rightarrow$ (1): By the \hyperref[Dominated Convergence Theorem]{theorem:dct-bochner-vector}. 
+\end{proof}