Definition:Integral Sign

Definition
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.