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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.

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    $\begingroup$ I think this is a perfectly reasonable question. It may have a known answer, and that answer may be immediate to people who know certain things, but I hardly feel it is common knowledge for all analysts, or "immediate from the definition" $\endgroup$
    – Yemon Choi
    Commented Jun 4, 2015 at 1:09

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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.

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