This question is an extension of the one I posted on ME: https://math.stackexchange.com/questions/4701500/if-alpha-nx-int-lvert-x-y-rvert-leq-1-n-lvert-x-y-rvert2-d-muy

It might be elementary for here, but I would deeply appreciate any help.

Let for each $n \in \mathbb{N}$, let $r_n : \mathbb{R}^m \to (0,\infty)$ be a sequence of smooth functions that converges to zero "pointwise" as $n \to \infty$.

Also let $\mu$ be a sufficiently regular Borel probability measure on $\mathbb{R}^m$. For concreteness, we can think of the stadard normal Gaussian measure.

Now, define \begin{equation} \alpha_n(x):=\int_{\lVert x-y \lVert \leq r_n(x)} \lVert x-y \rVert^2 d\mu(y) \end{equation} for each $n$ and let $A_n := \int_{\mathbb{R}^m} \alpha_n(x) d\mu(x)$.

Then, I suspect that for any bounded real-valued smooth function $F$ on $\mathbb{R}^m$, we have \begin{equation} \frac{1}{A_n}\int_{\mathbb{R}^m} F(x) \alpha_n(x) d\mu(x) \to \int_{\mathbb{R}^m} F(x) d\mu(x) \end{equation} as $n \to \infty$.

However, I cannot find a way to justify this guess rigorously. I tried to use convolution theorems but they do not fit in the above formula. Perhaps it is related to ergodicity?

Could anyone please help me?