From a57686be8fb4514943ca017e70a92d08a9293c17 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Sun, 21 Jun 2026 21:08:13 -0400
Subject: [PATCH] Added notation for space of measurable functions.

---
 src/measure/measurable-maps/in-measure.tex      | 8 ++++++++
 src/measure/measurable-maps/measurable-maps.tex | 2 ++
 src/measure/notation.tex                        | 1 +
 3 files changed, 11 insertions(+)

diff --git a/src/measure/measurable-maps/in-measure.tex b/src/measure/measurable-maps/in-measure.tex
index b275f65..a8e2108 100644
--- a/src/measure/measurable-maps/in-measure.tex
+++ b/src/measure/measurable-maps/in-measure.tex
@@ -1,6 +1,14 @@
 \section{Convergence in Measure}
 \label{section:convergence-in-measure}
 
+\begin{definition}[In Measure]
+\label{definition:in-measure}
+    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, then the uniform
+\end{definition}
+
+
+
+
 \begin{definition}[Convergence in Measure]
 \label{definition:convergence-in-measure}
     Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, and $f$ be a $(\cm, \cb_Y)$-measurable function, then $\fF \to f$ \textbf{in measure} if for each $\eps > 0$, 
diff --git a/src/measure/measurable-maps/measurable-maps.tex b/src/measure/measurable-maps/measurable-maps.tex
index 1723fde..714b5ef 100644
--- a/src/measure/measurable-maps/measurable-maps.tex
+++ b/src/measure/measurable-maps/measurable-maps.tex
@@ -4,6 +4,8 @@
 \begin{definition}[Measurable Function]
 \label{definition:measurable-function}
     Let $(X, \cm)$ and $(Y, \cn)$ be measurable spaces and $f: X \to Y$ be a mapping, then $f$ is \textbf{$(\cm, \cn)$-measurable} if $f^{-1}(E) \in \cm$ for all $E \in \cn$.
+
+    The set $\mathscr{M}(X; Y)$ is the \textbf{space of measurable functions} from $X$ to $Y$. 
 \end{definition}
 
 
diff --git a/src/measure/notation.tex b/src/measure/notation.tex
index 36badc1..2168869 100644
--- a/src/measure/notation.tex
+++ b/src/measure/notation.tex
@@ -12,6 +12,7 @@
     $\mathcal{B}_X$ & Borel $\sigma$-algebra on $X$. & \autoref{definition:borel-sigma-algebra} \\
     $\sigma(\{f_i \mid i \in I\})$ & $\sigma$-algebra generated by the maps $\{f_i\}$. & \autoref{definition:generated-sigma-algebra-function} \\
     $\bigotimes_{i \in I} \mathcal{M}_i$ & Product $\sigma$-algebra. & \autoref{definition:product-sigma-algebra} \\
+    $\mathscr{M}(X; Y)$ & Space of measurable functions from $X$ to $Y$. & \autoref{definition:measurable-function} \\
     $\chi_E = \mathbf{1}_E$ & Indicator function of $E$. & \autoref{definition:indicator-function} \\
     $\Sigma(X, \mathcal{M}; E)$ & Space of $E$-valued simple functions on $(X, \mathcal{M})$. & \autoref{definition:simple-function-standard-form} \\
     $\Sigma^+(X, \mathcal{M})$ & Space of non-negative simple functions. & \autoref{definition:simple-function-scalar} \\