Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used (the letter $P$ below denotes a probability measure on a configuration space), let $q,N\in\mathbb{N}$. The block maps $\mathfrak{p}_{k}:\mathbb{R}^{qN}\to\mathbb{R}^{q}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks, Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, so that $\mathfrak{p}_{k}^{-1}(B)\in\mathcal{B}(\mathbb{R}^{qN})$ for $B\in\mathcal{B}(\mathbb{R}^{q})$; $\prod_{k=1}^{N}$ is the finite product.

1. (Tensor powers)¶ For every $\rho\in\mathcal{P}(\mathbb{R}^{q})$ there is exactly one $P\in\mathcal{P}(\mathbb{R}^{qN})$ such that

2. (Average of the block marginals)¶ For every $P\in\mathcal{P}(\mathbb{R}^{qN})$ the function

is a probability measure on $\mathbb{R}^{q}$. For every Borel $f:\mathbb{R}^{q}\to[0,\infty]$,

and a Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrable with respect to $A_{P}$ exactly when each $f\circ\mathfrak{p}_{k}$ is integrable with respect to $P$, the same identity then holding.