Characterization of $L_{\mathrm{loc} }^p(\mathbb{R}^n)$ functions

Question

I would like to know if, when it comes to checking if $f\in L_{\mathrm{loc}}^p(\mathbb{R}^n)$, instead of checking if $f\in L^p(K)$ for every open $K\subset\subset \mathbb{R}^n$, if it suffices to check that $f\in L^p(K)$ for some class of $K\subset\subset \mathbb{R}^n$ that is nice (i.e., something like $\partial K$ of class $C^1$, etc.). That is, I want to know if there exist nice sets $K\subset\subset \mathbb{R}^n$ such that if $f\in L^p(K)$ for all such $K$, then $f\in L_{\mathrm{loc}}^p(\mathbb{R}^n)$.