What is $L_{loc}^1$? I know what $L^1$ is. I saw this on page 15 in the book Variational Analysis in Sobolev and BV Spaces.
Let us first consider a function $f\in L_{loc}^1(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^N$. One cannot, for an arbitrary $x\in \Omega$, give a meaning to $f(x)$.