Consider the following statement:
If $(\rho_{\epsilon})_{\epsilon > 0}$ is an approximation identity and $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ uniformly over $\Omega$. Where $\Omega$ is possibly any $\neq \varnothing$ subset of $\Bbb R^d$.
I know that if $f \in L^{\infty}(\Bbb R^d) \cap C(\Bbb R^d)$, then $\rho_{\epsilon} * f \to f$ over any compact in $\Bbb R^d$. The proof uses the fact that if $K$ is a compact, then $f$ is uniformly continuous on $K + \overline{B(0,1)}$, etc. Obviously, one can't proceed in the same way to prove the above statement.
Frankly, I am not sure whether or not the statement is true, but it is written in my notebook, so there is a good chance that the professor said it.
Is the statement true? If so, any clues on how to prove it?