Let $C_c^\infty$ denotes the set of real valued function with compact support.
Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$.
If $f$ were in $L^1$, the result would follow through Riesz theorem as $ \|(f \ast g_n)(x)-f(x) \|_1 \to 0 \tag{1} $ when $g_n=ng(nt)$ and for any $ g \in C_c^\infty$.
I don't think Riesz theorem applies in $L^1_\text{loc}$.
I tried to express $(f \ast g_n)(x)-f(x)$ in many ways, but none seems to lead me to the results.
Riesz Theorem: If $f_n$ goes to $f$ in $L^p(\mathbb R)$ then there exists a subsequence of $f_n$ converging pointwise a.e. to $f$.
This question is related.