Definition:Integrable Function/Measure Space

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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 if and only if:

$\displaystyle \int f^+ \rd \mu < +\infty$

and

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

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


Sources