# Definition:Integral of Positive Measurable Function

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## 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

### 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:

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

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

- 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) $9.4$