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 Rⁿ and

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

$\int_K | f| dx \,$

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

$L^1_{loc}(U).$

[edit] Examples

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

More generally, every p-power integrable function (1 ≤ p ≤ ∞) on U is locally integrable:

$L^p(U)\subset L^1_{loc}(U)$ .

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.

Locally integrable functions play a prominent role in distribution theory.

Robert S Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific, 2003. ISBN 981-238-430-8.

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