Any ideas on how to get started with this?
Let $\rho \in L^1(\mathbb{R}^N)$ with $\int \rho=1$. Set $\rho_n(x)=n^N \rho(nx)$. Let $f \in L^p(\mathbb{R}^N)$. Show that $\rho_n \star f \to f$ in $L^p(\mathbb{R}^N)$.
The proof of Theorem 4.22 would almost go through for this case. The problem is that I don't know anything about the support of the $\rho_n$'s.This seems to be crucial for the proof of Proposition 4.21 which is used to prove the theorem.