Definition:Integral of Positive Measurable Function

Definition

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

Integral Sign

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.

Also known as

Sometimes it is convenient to indicate the integration variable explicitly. In these cases, one may write one of:

$\ds \int \map f x \map \mu {\d x}$
$\ds \int \map f x \map {\d \mu} x$

in place of $\ds \int f \rd \mu$.

Sometimes it is also presentationally convenient to write $\map \mu f$.

It should be noted that this is abuse of notation, since a measure does not take functions as arguments.