Definition:Integral Sign

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let:
 * $\displaystyle \int f \ \mathrm d \mu := \sup \, \left\{{I_\mu \left({g}\right): g \le f, g \in \mathcal E^+}\right\}$

denote the $\mu$-integral of the positive measurable function $f$. where:


 * $\mathcal M_{\overline \R}^+$ 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
 * $I_\mu \left({g}\right)$ denotes the $\mu$-integral of the positive simple function $g$
 * $g \le f$ denotes pointwise inequality
 * $\mathcal E^+$ denotes the space of positive simple functions

The symbol:
 * $\displaystyle \int \ldots \ \mathrm d \mu$

is called the integral sign.

Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated.