The One-Particle Marginal of a Probability Measure on the Configuration Space

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 $q,N\in\mathbb{N}$, with the block maps $\mathfrak{p}_{k}$ of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks.

(One-particle marginal)¶ For $P\in\mathcal{P}(\mathbb{R}^{qN})$, the one-particle marginal $P^{[1]}\in\mathcal{P}(\mathbb{R}^{q})$ is the probability measure

the average of the laws $(\mathfrak{p}_{k})_{\#}P$ of the $N$ particles; it is a probability measure by Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average, whose integration identity reads $\int f\,dP^{[1]}=\frac{1}{N}\sum_{k=1}^{N}\int f\circ\mathfrak{p}_{k}\,dP$.