Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function such that $f(x)$ is positive in a small punctured neighborhood of $x=0$ but $f(0)=0$.

Now, define a collection of centered Gaussian measures on $\mathbb{R}$ as \begin{equation} d\mu_a(x):=\frac{1}{\sqrt{2\pi}f(a)} e^{-\frac{1}{2}\frac{x^2}{[f(a)]^2}} \end{equation} where $a$ is any sufficiently small nonzero real number.

Then, it is well-known that $d\mu_a \to \delta(0)$ in the sense of probability measures as $a \to 0$.

Theerfore, for any smooth function $F : \mathbb{R} \to \mathbb{R}$, we have \begin{equation} \int_{\mathbb{R}}F(x)d\mu_a(x) \to F(0) \end{equation} as $a \to 0$.

However, I wonder what will happen to the following limit: \begin{equation} \frac{1}{a^2}\int_{\mathbb{R}}[F(x)-F(0)]^2d\mu_a(x) \end{equation} as $a \to \infty$. Since $\frac{f(a)}{a} \to f'(0)$, it is clear that \begin{equation} \frac{1}{a^2}\int_{\mathbb{R}}x^2d\mu_a(x)= \Bigl( \frac{f(a)}{a} \Bigr)^2 \to [f'(0)]^2 \end{equation}

However, is it possible to generalize to any smooth $F$ as in the above? This kind of issue is completely new to me and I can't quite see how to handle..

Could anyone please provide some insights?