Definition:Integral of Positive Measurable Function

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

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


 * $\forall f \in \MM_{\overline \R}^+: \displaystyle \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$

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

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


 * $\displaystyle \int \map f x \map \mu {\d x}$
 * $\displaystyle \int \map f x \map {\d \mu} x$

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

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.

Also see

 * Definition:Integral of Positive Simple Function


 * Integral of Characteristic Function: Corollary
 * Integral of Positive Measurable Function is Positive Homogeneous
 * Integral of Positive Measurable Function is Additive
 * Integral of Positive Measurable Function is Monotone