Definition:Integrable Function

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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:

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


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

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