Does the sequence $\frac{1}{\mu([-1/n,1/n])} \chi_{[-1/n,1/n]}(x)$ converges to $\delta(0)$ for any "nice measure" $\mu$?

Question

Let $\mu$ be a sufficiently "nice" Borel probability measure on $\mathbb{R}$. Any Radon probability measure or centered Gaussian measure will do.

Now, consider the sequence of functions \begin{equation} \phi_n(x):=\frac{1}{\mu([-1/n,1/n])} \chi_{[-1/n,1/n]}(x) \end{equation}

Then, for any bounded smooth function $f$ on $\mathbb{R}$, I wonder if \begin{equation} \int_{\mathbb{R}} \phi_n(x) f(x)d\mu(x) \to f(0) \end{equation} as $n \to \infty$.

When $\mu$ is just a Lebesgue measure, I am aware that this is a well-known result. I am curious about extending to different meausures.