15.4 Simple Functions
Definition 15.4.1 (Indicator Function). Let $(X, \cm)$ be a measurable space and $E \in \cm$, then the function
\[\chi_{E} = \one_{E}: X \to \bracs{0, 1}\quad x \mapsto \begin{cases}1 &x \in E \\
0 &x \not\in E\end{cases}\]
is the characteristic/indicator function of $E$.
Definition 15.4.2 (Simple Function). Let $(X, \cm)$ be a measurable space and $Y$ be a set, then a function $\phi: X \to Y$ is simple if:
$\phi(X)$ is finite.
For each $y \in \phi(X)$, $\phi^{-1}(y) \in \cm$.
Definition 15.4.3 (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
\[f = \sum_{y \in f(X)}y \cdot \one_{\bracs{f = y}}\]
is the standard form of $f$.
The set $\Sigma(X, \cm; V)$ is the space of $V$-valued simple functions on $(X, \cm)$, which forms a vector space.