Added simple function approximation.

src/measure/measurable-maps/metric.tex

@@ -67,13 +67,13 @@
 
 \begin{proposition}
 \label{proposition:measurable-simple-separable}
-    Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, $N: Y \to 2^Y$\footnote{This mapping is typically obtained as slices of the level sets of a continuous function $Y \times Y \to \real$.}, and $f: X \to Y$ such that
+    Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, and $N: Y \to 2^Y$\footnote{This mapping is typically obtained as slices of the level sets of a continuous function $Y \times Y \to \real$.} such that
     \begin{enumerate}
         \item[(a)] For each $y \in Y$, $y \in \ol{N(y)^o}$.
         \item[(b)] $\bigcap_{y \in Y}N(y) \ne \emptyset$.
-        \item[(c)] For any $y \in Y$, $\bracs{x \in X|y \in N(f(x))} \in \cm$.
+        \item[(c)] For any $y_0 \in Y$, $\bracs{y \in Y|y_0 \in N(y)} \in \cb_Y$.
     \end{enumerate}
-    then the following are equivalent:
+    Then, for any $f: X \to Y$, the following are equivalent:
     \begin{enumerate}
         \item $f$ is $(\cm, \cb_Y)$-measurable.
         \item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that
@@ -113,3 +113,28 @@
 
     (3) $\Rightarrow$ (1): By \ref{proposition:metric-measurable-limit}.
 \end{proof}
+
+\begin{proposition}
+\label{proposition:measurable-simple-separable-norm}
+    Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_E)$ be a separable normed space, and $f: X \to E$, then the following are equivalent:
+    \begin{enumerate}
+        \item $f$ is $(\cm, \cb_E)$-measurable.
+        \item There exists simple functions $\seq{f_n}$ such that $\abs{f_n} \le \abs{f}$ for all $n \in \natp$, and $f_n \to f$ pointwise.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    Let
+    \[
+    N: E \to 2^E \quad y \mapsto B_E(0, \norm{y}_E)
+    \]
+    then
+    \begin{enumerate}
+        \item[(a)] $y \in \ol{B_E(0, \norm{y}_E)}$.
+        \item[(b)] $0 \in \bigcap_{y \in E}N(y)$.
+        \item[(c)] For any fixed $y_0 \in E$,
+        \[
+        \bracs{y \in E|y_0 \in N(y)} = \bracs{y \in E|\norm{y_0}_E \le \norm{y}_E} \in \cb_E
+        \]
+    \end{enumerate}
+    By \ref{proposition:measurable-simple-separable}, (1) and (2) are equivalent.
+\end{proof}

src/measure/measurable-maps/simple.tex

@@ -15,7 +15,7 @@
 
 \begin{definition}[Simple Function]
 \label{definition:simple-function}
-    Let $(X, \cm)$ be a measurable space and $Y$ be a set, then a function $\phi: X \to Y$ is \textbf{simple/finitely valued} if:
+    Let $(X, \cm)$ be a measurable space and $Y$ be a set, then a function $\phi: X \to Y$ is \textbf{simple} if:
     \begin{enumerate}
         \item $\phi(X)$ is finite.
         \item For each $y \in \phi(X)$, $\phi^{-1}(y) \in \cm$.
@@ -26,7 +26,7 @@
 \label{definition:simple-function-standard-form}
     Let $(X, \cm)$ be a measurable space, $V$ be a vector space over $K \in \RC$, and $f: X \to Y$ be a simple function, then
     \[
-    f = \sum_{y \in f(X)}y\one_Y
+    f = \sum_{y \in f(X)}y \cdot \one_{\bracs{f \in Y}}
     \]
     is the \textbf{standard form} of $f$.