Locally integrable function

In mathematics, a locally integrable function is a function which is integrable on any compact set.

Formally, let $U$ be an open set in the Euclidean space $\mathbb {R} ^{n}$ and

$f\colon U\to \mathbb {C}$

be a Lebesgue measurable function. If the Lebesgue integral

$\int _{K}|f|dx\,$

is finite for all compact subsets $K$ in $U$, then $f$ is called locally integrable. The set of all such functions is denoted by $L_{loc}^{1}(U).$

Examples

• Every (globally) integrable function on $U$ is locally integrable, that is,

$L^{1}(U)\subset L_{loc}^{1}(U)$ (see Lp space).

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

Uses

Locally integrable functions play a proeminent role in distribution theory.

References

• Robert S Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific, 2003. ISBN 9812384308.

Locally integrable function at PlanetMath.