The Tensor Power of a Probability Measure on Euclidean Space

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $q,N\in\mathbb{N}$, with the block maps $\mathfrak{p}_{k}$ of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks and finite products $\prod_{k=1}^{N}$.

(Tensor power)¶ For $\rho\in\mathcal{P}(\mathbb{R}^{q})$, the $N$-th tensor power $\rho^{\otimes N}\in\mathcal{P}(\mathbb{R}^{qN})$ is the unique probability measure on $\mathbb{R}^{qN}$ with

which exists and is unique by Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §tensor.