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.