Definition:Integral of Positive Measurable Function

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

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


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

where:


 * $\mathcal{M}_{\overline{\R}}^+$ denotes the space of positive $\Sigma$-measurable functions
 * $\overline{\R}_{\ge0}$ 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 positive simple functions
 * $g \le f$ denotes pointwise inequality
 * $\mathcal{E}^+$ 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:


 * $\displaystyle \int f \left({x}\right) \, \mu \left({\mathrm dx}\right)$
 * $\displaystyle \int f \left({x}\right) \, \mathrm d\mu \left({x}\right)$

in place of $\displaystyle \int f \, \mathrm d\mu$.

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

 * 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