Locally integrable function: Definition from Answers.com

In mathematics, a locally integrable function is a function which is integrable on any compact set of its domain of definition. Their importance lies on the fact that we do not care about their behavior at infinity.

1 Formal definition
2 Properties
3 Examples
4 Applications
5 See also
6 References
7 External links

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 of $f$ is such that

$\int_K | f| \mathrm{d}x <+\infty\,$

i.e. it is finite for all compact subsets $K$ in $Ω$ , then $f$ is called locally integrable. The set of all such functions is denoted by $\scriptstyle L^1_{loc}(\Omega)$ :

$L^1_{loc}(\Omega)=\left\{f:\Omega\to\mathbb{C}\text{ measurable }\left|\int_K | f| \mathrm{d}x <+\infty\ \forall K\in {\mathcal{P}_0(\Omega)}\right.\right\}$

where $\scriptstyle{\mathcal{P}_0(\Omega)}$ is the set of all compact subsets of the set $Ω$ .

Properties

Theorem. Every function $f$ belonging to $L p (Ω)$ , $\scriptstyle 1\leq p\leq+\infty$ , 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

$q$ is the positive number such that $1 / p + 1 / q = 1$ for a given $\scriptstyle 1\leq p\leq+\infty$
$μ(K)$ is the Lebesgue measure of the compact set $K$

Then by Hölder's inequality, the product $\scriptstyle f\chi_K$ is integrable i.e. belongs to $L 1 (K)$ and

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

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 theorem is true also for functions $f$ belonging only to $L p (K)$ for each compact subset $K$ of $Ω$ .

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

$f(x)= \begin{cases} 1/x &x\neq 0\\ 0 & x=0 \end{cases}$

is not locally integrable near

x = 0

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.

References

Strichartz, Robert S. (2003), A Guide to Distribution Theory and Fourier Transforms (2^nd ed.), River Edge, NJ: World Scientific Publishers, ZBL 1029.46039, ISBN 981-238-430-8, http://books.google.it/books?id=T7vEOGGDCh4C&printsec=frontcover&dq=A+Guide+to+Distribution+Theory+and+Fourier+Transforms#v=onepage&q=&f=false

External links

Vinogradova, I.A. (2001), "Locally integrable function", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/L/l060460.htm
Todd Rowland, "Locally integrable" from MathWorld.

This article incorporates material from Locally integrable function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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Locally integrable function

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