Mutual Energy

Let $\Omega$ be a Space with Measure $\mu\geq 0$ , and let $\Phi(P,Q)$ be a real function on the Product Space $\Omega\times\Omega$ . When

$\displaystyle (\mu,nu)$	$\textstyle =$	$\displaystyle \int\!\!\!\int \Phi(P,Q)\,d\mu(Q)\,d\nu(P)$
	$\textstyle =$	$\displaystyle \int \Phi(P,\mu)\,d\nu(P)$

exists for measures $\mu,\nu\geq 0$ , $(\mu,\nu)$ is called the mutual energy. $(\mu,\mu)$ is then called the Energy.

See also Energy

References

Iyanaga, S. and Kawada, Y. (Eds.). ``General Potential.'' §335.B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1038, 1980.