Potential of a measure

Remarks

The integral is well defined: a lower semicontinuous function is bounded below on the compact set $\set{x} \times \operatorname{supp} \mu$, and $\mu$ is finite. Example: for the Dirac measure $\mu = \delta_a$ one gets $U_k^\mu(x) = k(x, a)$. Example: with the Newtonian kernel on $\R^3$ and $\mu$ the uniform probability measure on the unit sphere, $U_k^\mu(x) = 1$ for $\abs{x} \le 1$ and $1/\abs{x}$ for $\abs{x} \ge 1$. Non-example: $k(x, y) = -1/\abs{x - y}$ on $\R^3$ off the diagonal is not a kernel in this sense. It tends to $-\infty$ at the diagonal, so no value in $(-\infty, +\infty]$ on the diagonal makes it lower semicontinuous. The potential $U_k^\mu$ is itself lower semicontinuous, by Fatou's lemma. Standard kernels are the Newtonian kernel, the logarithmic kernel and the Riesz kernels $\abs{x - y}^{\alpha - n}$ with $0 < \alpha < n$. Signed measures are handled by writing $\mu = \mu^+ - \mu^-$ where both potentials are finite.