Definition Suppose that $U$ is an open set in $\sR^n$ , and $f\colon U\to\sC$ 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.