this post is concerned with functionals defined in measures. Consider the following functional


$$\mathcal{W}[\mu]=-\int\_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$


were we define $-\log\vert0\vert=+\infty$ and $\mu$ is a finite, non negative Radon measure in $\mathbb{R}$. It is clear that if $\mu$ does given mass to points of $\mathbb{R}$, then the functional above is $\infty$, also is easy to build absolutely continuous measures with respect to the Lebesgue measure were this functional is finit. The question is: There are singular measures (obviously without atoms) such that the functional is finit? I have tried of build an example using the Cantor set and a suitable Hausdorff measure restricted to the Cantor set, but I can not limit such integral, therefore I think that such measure exist. Best regads.



On the domain of functionals in measure with singular kernels
It is well known (to those who know it well) that the Hausdorff dimension of a set is closely related to its capacities. More precisely, if we define the capacitary dimension of a set $A\subset\mathbb R^n$ as the infimum of those $s>0$ for which
$$
I\_s(\mu) = \int \!\!\!\int |x-y|^{-s}\, d\mu(x)d\mu(y)=\infty
$$
for all probability measures $\mu$ supported by $A$, then this dimension equals the Hausdorff dimension for a Borel set $A$. See *Mattila, Geometry of sets and measures in Euclidean space,* Theorem 8.9.


In particular, this means that any Borel set $A$ with $\dim A>0$ supports a measure of the kind you are looking for.


