locally integrable function

DefinitionSuppose 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 $f$ islocally integrable. The set of allsuch 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.