If $f|A < 0$ the show that exists a open set $O$ such that $A \subset O$ and $f <0$ $\mu-$ a.e. in $O$

Question

Let $\mu$ be a Radon measure in $\mathbb R^n$ and $A \subset \mathbb R^n$ such that $\mu(A)>0$ suppose that $f: \mathbb{R}^n \to \mathbb R$ and $f|_A < 0$ then is possible prove that exists a open set $O$ such that $A \subset O$ and $f<0$ $\mu-$a.e in $O$? I try to constructed these set using the approximation of the measure of $A$ by open sets more especifically the regularity of $\mu$ but i cant conclude the existence of these $O$ any hint or help i will be very grateful