Mutual energy

Remarks

The mutual energy is symmetric, $I(\mu, \nu) = I(\nu, \mu)$, and $I(\mu, \mu)$ is the energy $I(\mu)$. It equals $\int U^\mu \, d\nu$, where $U^\mu$ is the potential of $\mu$. For a signed measure $\lambda = \mu - \nu$ with $I(\mu)$, $I(\nu)$ and $I(\mu, \nu)$ finite, one puts $I(\lambda) = I(\mu) - 2 I(\mu, \nu) + I(\nu)$.

Example: for Dirac measures $\delta_a$ and $\delta_b$ with $a \ne b$ in $\R^3$, $I(\delta_a, \delta_b) = 1/\abs{a - b}$. For the uniform probability measure $\sigma$ on the unit sphere in $\R^3$ and $\delta_0$, $I(\sigma, \delta_0) = U^\sigma(0) = 1$.

Non-example: for $\mu = \nu = \delta_0$, the integrand is $k(0) = +\infty$ on the only point of the support, so $I(\mu, \nu) = +\infty$ is not a finite mutual energy.