Locally integrable function

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In mathematics, a locally integrable function is a function which is integrable on any compact set of its domain of definition.

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[edit] Formal definition

Formally, let Ω be an open set in the Euclidean space \scriptstyle\mathbb{R}^n and \scriptstyle f:\Omega\to\mathbb{C} be a Lebesgue measurable function. If the Lebesgue integral

\int_K | f| dx \,

is finite for all compact subsets K in Ω, then f is called locally integrable. The set of all such functions is denoted by

L^1_{loc}(\Omega)

[edit] Properties

Theorem. Every function f belonging to Lp(Ω), \scriptstyle 1\leq p\leq+\infty, where Ω is an open subset of \scriptstyle\mathbb{R}^n is locally integrable. To see this, consider the characteristic function \scriptstyle\chi_K of a compact subset K of Ω: then, for \scriptstyle p\leq+\infty

\left|{\int_\Omega|\chi_K|^q dx}\right|^{1/q}=\left|{\int_K dx}\right|^{1/q}=|\mu(K)|^{1/q}<+\infty

where

Then by Hölder's inequality

{\int_K|f|dx}={\int_\Omega|f\chi_K|dx}\leq\left|{\int_\Omega|f|^p dx}\right|^{1/p}\left|{\int_K dx}\right|^{1/q}=\|f\|_p|\mu(K)|^{1/q}<+\infty

therefore

f\in L^1_{loc}(\Omega)

Note that since the following inequality is true

{\int_K|f|dx}={\int_\Omega|f\chi_K|dx}\leq\left|{\int_K|f|^p dx}\right|^{1/p}\left|{\int_K dx}\right|^{1/q}=\|f\|_p|\mu(K)|^{1/q}<+\infty

the thesis is true also for functions f belonging only to Lp(K) for each compact subset K of Ω.

[edit] Examples

  • The constant function 1 defined on the real line is locally integrable but not globally integrable. More generally, continuous functions and constants are locally integrable.
  • The function f(x) = 1 / x for \scriptstyle x\neq 0 and f(0) = 0 is not locally integrable.

[edit] Applications

Locally integrable functions play a prominent role in distribution theory. Also they occur in the definition of various classes of functions and function spaces, like functions of bounded variation.

[edit] See also

[edit] References

This article incorporates material from Locally integrable function on PlanetMath, which is licensed under the GFDL.

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