Locally integrable function
From Wikipedia, the free encyclopedia
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
be a Lebesgue measurable function. If the Lebesgue integral
is finite for all compact subsets K in U, then f is called locally integrable. The set of all such functions is denoted by
[edit] Examples
- Every (globally) integrable function on U is locally integrable, that is,
-
(see Lp space).
- More generally, every p-power integrable function (1 ≤ p ≤ ∞) on U is locally integrable:
-
.
- 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
and f(0) = 0 is not locally integrable.
[edit] Uses
Locally integrable functions play a prominent role in distribution theory.
[edit] References
- 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.
