This is from the page https://mathworld.wolfram.com/MutualEnergy.html
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
$$(\mu,\nu)=\iint\Phi(P,Q)\,\mathrm{d}\mu(Q)\mathrm{d}\nu(P)=\int\Phi(P,\mu)\,\mathrm{d}\nu(P)$$
exists for measures $\mu,\nu\geq 0$, $(\mu,\nu)$ is called the mutual energy. $(\mu,\mu)$ is then called the energy.
I'm just trying to wrap my head around how a definition of energy works with linear product spaces and measures.