Adjusted notation for spaces of measurable functions.

src/measure/measurable-maps/measurable-maps.tex

@@ -4,10 +4,16 @@
 \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}
 
+\begin{definition}[Space of Measurable Functions]
+\label{definition:measurable-function-space}
+    Let $(X, \cm)$ and $(Y, \cn)$ be measurable spaces, then the set $\mathscr{L}^0(X, \cm; Y) = \mathcal{L}^0(X; Y)$ is the \textbf{space of measurable functions} from $X$ to $Y$. 
+
+    For any measure $\mu$ on $(X, \cm)$, the space $L^0(X, \cm, \mu; Y) = L^0(X; Y)$ is the space of measurable functions from $X$ to $Y$, modulo almost everywhere equality. 
+\end{definition}
+
+
 
 \begin{definition}[Borel Measurable]
 \label{definition:borel-measurable-function}

src/measure/notation.tex

@@ -12,7 +12,8 @@
     $\mathcal{B}_X$ & 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} \\
+    $\mathcal{L}^0(X; Y)$ & Space of measurable functions from $X$ to $Y$. & \autoref{definition:measurable-function-space} \\
+    $L^0(X; Y)$ & Space of measurable functions from $X$ to $Y$, modulo almost everywhere equality. & \autoref{definition:measurable-function-space} \\
     $\chi_E = \mathbf{1}_E$ & 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} \\