\begin{definition}[Definition:Integral Sign]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let:
:$\ds \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$
denote the $\mu$-integral of the positive measurable function $f$.
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.
The symbol:
:$\ds \int \ldots \rd \mu$
is called the '''integral sign'''.
Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated.
\end{definition}