locally integrable function

Definition Suppose that $U$ is an open set in ${ℝ}^n$, and $f:U\to ℂ$ 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 $L_{\text{loc}}^1(U)$.

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	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