Definition:Integral of Positive Measurable Function

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

Define the $\mu$-integral of positive measurable functions, denoted $\displaystyle \int \cdot \rd \mu: \mathcal M_{\overline \R}^+ \to \overline \R_{\ge 0}$, by:

$\forall f \in \mathcal M_{\overline \R}^+: \displaystyle \int f \rd \mu := \sup \, \left\{{I_\mu \left({g}\right): g \le f, g \in \mathcal E^+}\right\}$


$\mathcal M_{\overline \R}^+$ 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

Integral Sign

The symbol:

$\displaystyle \int \ldots \rd \mu$

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:

$\displaystyle \int f \left({x}\right) \, \mu \left({\rd x}\right)$
$\displaystyle \int f \left({x}\right) \rd \mu \left({x}\right)$

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

Sometimes it is also presentationally convenient to write $\mu \left({f}\right)$.

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

Also see