locally integrable function
Definition Suppose that $U$ is an open set in ${\mathbb{R}}^{n}$, and $f:U\to \u2102$ is a Lebesgue measurable function. If the Lebesgue integral^{}
$${\int}_{K}f\mathit{d}x$$ 
is finite for all compact subsets $K$ in $U$, then $f$ is locally integrable. The set of all such functions is denoted by ${L}_{\text{loc}}^{1}(U)$.
Example

1.
${L}^{1}(U)\subset {L}_{\text{loc}}^{1}(U)$, where ${L}^{1}(U)$ is the set of (globally) integrable functions.

2.
Continuous functions^{} are locally integrable.

3.
The function $f(x)=1/x$ for $x\ne 0$ and $f(0)=0$ is not locally integrable.
Title  locally integrable function 

Canonical name  LocallyIntegrableFunction 
Date of creation  20130322 13:44:19 
Last modified on  20130322 13:44:19 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  11 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 28B15 