Definition:Integrable Function/Measure Space

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

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

Then $f$ is said to be $\mu$-integrable :


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

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.

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.