Definition:Integrable Function

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 iff:


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

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

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

Because $f = f^+ - f^-$ and, more importantly, $\left|{f}\right| = f^+ + f^-$, then the above can be written concisely as
 * $\displaystyle \int \left|{f}\right| \, \mathrm d\mu < +\infty$

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