Potential of a measure

Summary

The potential of a measure $\mu$ with respect to a kernel $k$ is the function $x \mapsto \int k(x, y) \, d\mu(y)$. It describes the field of a distribution of mass or charge $\mu$. Here $k(x, y)$ is the effect at $x$ of a unit mass at $y$. With the Newtonian kernel one gets the Newtonian potential, with the logarithmic kernel the logarithmic potential, and with $\abs{x - y}^{\alpha - n}$ the Riesz potentials. Energy, capacity, equilibrium measures and balayage are all defined through potentials of measures.