General version of the fundamental lemma of calculus of variations

Let the measure $\mu$ on $\mathcal{B}(\mathbb{R}^n)$ be a general Borel measure.

Let $f:\mathbb{R^n} \to \mathbb{R}$ be a locally integrable function and $\int f\varphi d\mu =0$ for any smooth function $\varphi $ with compact support.

Then, I want to show that $f=0, \mu$-a.e. holds.

I think that using topological regularity for measure $A\mapsto \int _{A} |f|$ and smooth version of Urysohn’s lemma would work, but I am not sure.