# 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. 1.

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

2. 2.

Continuous functions are locally integrable.

3. 3.

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

Title locally integrable function LocallyIntegrableFunction 2013-03-22 13:44:19 2013-03-22 13:44:19 matte (1858) matte (1858) 11 matte (1858) Definition msc 28B15