PlanetMath: locally integrable function

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locally integrable function

(Definition)

Definition Suppose that is an open set in $\mathbb{R}^n$ , and $f\colon U\to\mathbb{C}$ is a Lebesgue measurable function. If the Lebesgue integral

$\displaystyle \int_K \vert f\vert dx$

is finite for all compact subsets

, then

is locally integrable. The set of all such functions is denoted by $L^1_{\scriptsize {\mbox{loc}}}(U)$ .

Example

$L^1(U)\subset L^1_{\scriptsize {\mbox{loc}}}(U)$ , where is the set of (globally) integrable functions.
Continuous functions are locally integrable.
The function for $x\neq 0$ and is not locally integrable.

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"locally integrable function" is owned by matte.

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Attachments:: fundamental lemma of calculus of variations (Theorem) by matte

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Cross-references: continuous functions, compact subsets, finite, Lebesgue integral, function, Lebesgue measurable, open set
There are 7 references to this entry.

This is version 8 of locally integrable function, born on 2003-07-08, modified 2006-01-16.
Object id is 4430, canonical name is LocallyIntegrableFunction.
Accessed 5216 times total.

Classification:

AMS MSC:

28B15 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Set functions, measures and integrals with values in ordered spaces)

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