# Definition:Integrable Function/Measure Space

## Definition

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

Let $\MM_{\overline \R}$ denote the space of $\Sigma$-measurable, extended real-valued functions .

Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function.

Then $f$ is said to be **$\mu$-integrable** if and only if:

- $\ds \int f^+ \rd \mu < +\infty$

and

- $\ds \int f^- \rd \mu < +\infty$

where $f^+$, $f^-$ are the positive and negative parts of $f$, respectively.

The integral signs denote $\mu$-integration of positive measurable functions.

### Complex Function

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

Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.

Let $f : X \to \C$ be a $\Sigma/\map \BB C$-measurable function.

Let $\map \Re f : X \to \R$ and $\map \Im f : X \to \R$ be the real part and imaginary part of $f$ respectively.

We say that $f$ is **$\mu$-integrable** if and only if $\map \Re f$ and $\map \Im f$ are integrable.

## Also known as

When no ambiguity arises, one may also simply speak of **integrable functions**.

To emphasize $X$ or $\Sigma$, also **$X$-integrable function** and **$\Sigma$-integrable function** are encountered.

Any possible ambiguity may be suppressed by the phrasing **$\struct {X, \Sigma, \mu}$-integrable functions**, but this is usually too cumbersome.

## Also see

- Definition:Integral of Integrable Function, justifying the name
**integrable function** - Definition:Space of Integrable Functions

- Characterization of Integrable Functions, demonstrating other ways to verify
**$\mu$-integrability**.

- Results about
**integrable functions**can be found**here**.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $10.1$