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summable function (Definition)

A measurable function $f : \Omega \to \reals$ where $(\Omega, \mathcal{A}, \mu)$ is a measure space is said to be summable if the Lebesgue integral of the absolute value of $f$ exists and is finite, \begin{equation*} \int_{\Omega} |f| d\mu < +\infty \end{equation*}An alternative way of expressing this condition is to assert that $f \in L^1(\Omega)$

Note that some authors distinguish between integrable and summable: an integrable function is one for which the above integral exists; a summable function is one for which the integral exists and is finite.




"summable function" is owned by ehremo. [ full author list (2) ]
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See Also: Lebesgue integrable

Keywords:  summable Lebesgue integrable

Attachments:
support of integrable function is $\sigma$-finite (Theorem) by asteroid
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Cross-references: integral, function, finite, absolute value, Lebesgue integral, measure space, measurable function
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This is version 5 of summable function, born on 2008-07-13, modified 2009-01-12.
Object id is 10783, canonical name is SummableFunction.
Accessed 1416 times total.

Classification:
AMS MSC28A25 (Measure and integration :: Classical measure theory :: Integration with respect to measures and other set functions)

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