Energy of a measure

Summary

The energy of a positive measure is the double integral of a potential-theoretic kernel against the measure in both variables. Typical kernels are $\abs{x - y}^{2-n}$ in $\R^n$ for $n \ge 3$ and $\log(1/\abs{x - y})$ in the plane. Physically it is the electrostatic energy of the charge distribution described by the measure. Minimising the energy over probability measures on a compact set defines its capacity and its equilibrium measure. Sets that carry no nonzero measure of finite energy are the negligible sets of potential theory.