Locally integrable function

A measurable scalar or finite-dimensional vector-valued function $f$ on an open set $U\subseteq\mathbb R^n$ is locally integrable, written $f\in L^1_{\mathrm{loc}}(U)$ , if

\int_K |f(x)|\,dx<\infty \quad\text{for every [[topology/compact-set|compact]] }K\subset U.

Here $|f|$ is absolute value or the Euclidean norm. As with Lebesgue spaces, functions are identified almost everywhere.

Local versus global

The constant function one lies in $L^1_{\mathrm{loc}}(\mathbb R^n)$ but not in $L^1(\mathbb R^n)$ . Every continuous function is locally integrable, and Hölder's inequality gives $L^p_{\mathrm{loc}}\subseteq L^1_{\mathrm{loc}}$ for $p\ge1$ . Local integrability is sufficient to integrate $f$ against any smooth compactly supported test function, giving a regular distribution.