\begin{definition}[Definition:Integral of Positive Simple Function]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $\EE^+$ denote the space of positive simple functions.
Let $f: X \to \R, f \in \EE^+$ be a positive simple function.
Suppose that $f$ admits the following standard representation:
:$\ds f = \sum_{i \mathop = 0}^n a_i \chi_{E_i}$
where $a_0 := 0$, and $\chi$ denotes characteristic function.
Then the '''$\mu$-integral of $f$''', denoted $\map {I_\mu} f$, is defined by:
:$\ds \map {I_\mu} f := \sum_{i \mathop = 0}^n a_i \map \mu {E_i}$
\end{definition}