Energy in doubling measure metric spaces

Question

Let $(X,\mu, d)$ be a metric measure space where $\mu$ is a doubling measure. For a relatively compact set $U\in X$ consider the following quantity $$I(U,\mu,d)=\int_U \int_U \log^2(d(x,y)) d\mu(x) d\mu(y). $$ Are there examples of metric measure spaces with relatively compact subsets for which $I$ isnt finite?

Answer 1

(What follows does not claim to be a complete answer, but rather a possible reasoning line for a proof that such integrals converge in the general case.)

In the Euclidean $\mathbb{R}^n$ case this should be true for any relatively compact subset $U$, because the following integrals converge for all $R \in \mathbb{R}^+$ and all $\alpha \geq 0$ (and in particular for the non-negative integers $\alpha := n-1$, with $n\geq 1$): \begin{equation} \int_0^R \log^2(r) r^{\alpha} dr = \frac{R^{α + 1} \big((α + 1)^2 \log^2(R) - 2 (α + 1) \log(R) + 2\big)}{(α + 1)^3} \end{equation} On a general doubling space $X$ we can invoke Assouad's Theorem to see that there exist constants $n$ and $\epsilon \in (0,1)$ such that the $\epsilon$-snowflaked version of $X$, with metric $(d_X)^\epsilon$, can be embedded into Euclidean $\mathbb{R}^n$ by a bi-Lipschitz function $f:X \rightarrow \mathbb{R}^n$. In particular, a relatively compact $U \subset X$ would be mapped homeomorphically onto a relatively compact $f(U) \subset \mathbb{R}^n$.

At this point we could perform the integration on $f(U)$ (going through the bi-Lipschitz homeomorphism $f^{-1}:f(U) \rightarrow U$), use bi-Lipschitzianity and observe that the following integrals also converge, for all $n\geq 1$ and all $\epsilon \in (0,1)$: \begin{equation} \int_0^R \log^2(r^\epsilon) r^{\epsilon(n-1)} dr = \epsilon^2\int_0^R \log^2(r) r^{\epsilon(n-1)} dr \end{equation}