# Definition:Integral of Positive Measurable Function

Jump to navigation
Jump to search

## Definition

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:

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

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

### 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

- 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

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next) $9.4$