# Definition:Integrable Function

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## Contents

## Definition

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

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

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

- $\displaystyle \int f^+ \, \mathrm d\mu < +\infty$

and

- $\displaystyle \int f^- \, \mathrm d\mu < +\infty$

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

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

## 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 **$\left({X, \Sigma, \mu}\right)$-integrable functions**, but this is usually too cumbersome.

## Also see

- Integral of Integrable Function, justifying the name
**integrable function** - Space of Integrable Functions
- Characterization of Integrable Functions, demonstrating other ways to verify
**$\mu$-integrability**.

## Sources

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