locally integrable function

Definition Suppose that $U$ is an open set in $\mathbb{R}^{n}$ , and $f\colon U\to\mathbb{C}$ is a Lebesgue measurable function. If the Lebesgue integral

\int_{K}|f|dx

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

Example

1.

$L^{1}(U)\subset L^{1}_{\scriptsize{\mbox{loc}}}(U)$ , 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 $x\neq 0$ and $f(0)=0$ is not locally integrable.

Title	locally integrable function
Canonical name	LocallyIntegrableFunction
Date of creation	2013-03-22 13:44:19
Last modified on	2013-03-22 13:44:19
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	11
Author	matte (1858)
Entry type	Definition
Classification	msc 28B15