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[parent] locally integrable function (Definition)

Definition Suppose that $ U$ 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 $ K$ in $ U$, then $ f$ is locally integrable. The set of all such functions is denoted by $ L^1_{\scriptsize {\mbox{loc}}}(U)$.

Example

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



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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
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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 MSC28B15 (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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