What norms can we define on $L^p_{\mathrm{loc}}$ ?or What is the most commonly used norm on $L^p_{\mathrm{loc}}$.It is tempting to define $$\|f\|_{L^p_{\mathrm{loc}}}:=\sup_{K\;\text{is compact}}{\|f\|_{L^{p}(K)}}$$
But this can be infinite for $f\in L^p_{\mathrm{loc}}$.