\begin{definition}[Definition:Integral of Positive Measurable Function]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
We define the '''$\mu$-integral of positive measurable functions''', denoted $\ds \int \cdot \rd \mu: \MM_{\overline \R}^+ \to \overline \R_{\ge 0}$, as:
:$\forall f \in \MM_{\overline \R}^+: \ds \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$
where:
:$\MM_{\overline \R}^+$ denotes the space of positive $\Sigma$-measurable functions
:$\overline \R_{\ge 0}$ denotes the positive extended real numbers
:$\sup$ is a supremum in the extended real ordering
:$\map {I_\mu} g$ denotes the $\mu$-integral of the positive simple function $g$
:$g \le f$ denotes pointwise inequality
:$\EE^+$ denotes the space of positive simple functions
\end{definition}