Locally integrable function

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

Formal definition

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

:$int\_K\; |\; f|\; dx\; ,$

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

:$L^1\_\{loc\}(Omega)$

Properties

Theorem. Every function $f$ belonging to $L^p(Omega)$, $scriptstyle\; 1leq\; pleq+infty$, where $Omega$ is an open subset of $scriptstylemathbb\{R\}^n$ is locally integrable. To see this, consider the characteristic function $scriptstylechi\_K$ of a compact subset $K$ of $Omega$: then, for $scriptstyle\; pleq+infty$

:

where
*$q$ is the positive number such that $1/p+1/q=1$ for a given $scriptstyle\; 1leq\; pleq+infty$
*$mu(K)$ is the Lebesgue measure of the compact set $K$Then by Hölder's inequality

:

therefore

:$fin\; L^1\_\{loc\}(Omega)$

Note that since the following inequality is true

:

the thesis is true also for functions $f$ belonging only to $L^p(K)$ for each compact subset $K$ of $Omega$.

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 $scriptstyle\; x\; eq\; 0$ and $f(0)=0$ is not locally integrable.

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.

See also

*Compact set
*Distributions
*Lebesgue integral
*Lebesgue measure
*"L"^"p"(Ω) space

References

*Harvrefcol
Surname = Strichartz
Given = Robert S.
Title = A Guide to Distribution Theory and Fourier Transforms
Publisher = World Scientific Publishers
Year = 2003, ISBN 981-238-430-8.