In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well defined for all measures and may takes values in $[-\infty,+\infty]$. To avoid this annoying fact, one typically restrict to measures which integrate the logarithm around infinity, that is which satisfy the condition (C)
$$ \int \log(1+|x|)\mu(dx)<+\infty,$$
so that $I(\mu)>-\infty$ thanks to $|x-y|\leq (1+|x|)(1+|y|) $. A fondamental fact is that if $\mu,\nu$ both satisfy (C), have finite logarithmic energy and $\mu(\mathbb{C})=\nu(\mathbb{C})$, then $I(\mu-\nu)\geq0$ and equality holds iff $\mu=\nu$ (we extend naturally the definition of $I$ to signed measures). I understand the condition (C) to be convenient, but maybe not sharp (one can imagine $\mu$ which does not satisfies (C) but with finite logarithmic energy). This leads to my first question :


* What can we says about $I(\mu-\nu)$ when (at least one of) the measures do not satisfy (C) ?


This question is moreover motivated by the appearance of the logarithmic energies in random matrix theory (in large deviations rate functions) and in free probability (reinterpreted up to a sign as a non-commutative entropy by Voiculescu). In this setting, $I(\mu-\nu)$ is a natural candidate for a relative free entropy, and questions of geometric nature are bothering me : Let $A\_{c}$ be the set of signed measures $\mu$ with finite logarithmic energy acting on $\mathbb{C}$ with total mass $\mu(\mathbb{C})=c$ such that if $\mu$ has Jordan decomposition $\mu^+-\mu^-$, then both $\mu^+$ and $\mu^-$ satisfy (C). Then the previous fact yields that $A\_0$ is a pre-Hilbert space with scalar product 
$$ I(\mu,\nu)=\iint\log\frac{1}{|x-y|}\mu(dx)\nu(dy).$$ Note that $A\_0$ is not complete since it is clearly not closed. Moreover, it is not hard to check that $A\_0$ acts by translations on $A\_1$ which inherits of a structure of affine space, leading to a metric on probability measures with finite logarithmic energy satisfying (C). It is now time for my second question :


* What relations may exist between this metric and metrics compatible with the weak topology ? (e.g Prohorov's ? Levy / Bounded Lipshitz ? Wasserstein's ? ...) Or total variation norm ?



What do we actually know about logarithmic energy ?
Concerning the first question: We have $I(\mu-\nu)>0$ whenever $I(\mu-\nu)$ is defined, finite or not, and $\mu$,$\nu$ are different signed Radon measures with equal total masses (or rather charges). This, with any finite dimensional Hilbert space in place of the plane, is Example 3.3 in 
<http://www.ams.org/journals/tran/1997-349-08/S0002-9947-97-01966-1/home.html> ,
where the assumption $\sigma\neq0$ is missing.


