Equilibrium measure

Remarks

Such a measure exists, because the probability measures on $K$ form a weak-star compact set and $I$ is lower semicontinuous for this topology. It is unique, because the energy is strictly convex on probability measures of finite energy. So one speaks of the equilibrium measure of $K$.

Example: let $K$ be the unit circle and let $\nu$ be normalized arc length $d\theta / 2\pi$. For $\abs{z} \le 1$ the mean value $\frac{1}{2\pi} \int_0^{2\pi} \log \abs{z - e^{i\theta}} \, d\theta$ equals $\log \max(\abs{z}, 1) = 0$. So the potential of $\nu$ vanishes on $K$, and $I(\nu) = 0 = V(K)$, which agrees with capacity $1$.

Non-example: the unit point mass $\delta_1$ is a probability measure on the unit circle, but $I(\delta_1) = \log(1/0) = \infty \ne V(K)$.

In $\R^n$ with $n \ge 3$ the same definition is used with the Newtonian kernel $\abs{x - y}^{2-n}$ in place of $\log(1/\abs{z - w})$.