The Tensor-Averaged Running Cost of a Configuration-Space Cost

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Tensor powers of measures in $\mathcal{P}_{2}(\mathbb{R}^{d})$ lie in $\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments, and $\tfrac{1}{N}$ is the multiplicative inverse of $N$, read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background, which exists by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field.

(Tensor-averaged cost)¶ Let $c:\mathbb{R}^{dN}\to\mathbb{R}$ be bounded and Borel, hence integrable with respect to every probability measure on $\mathbb{R}^{dN}$. The tensor-averaged cost of $c$ is the function