Mean-Square Deviation of the Empirical Average of a Bounded Borel Function under a Tensor Power

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, let $q,N\in\mathbb{N}$, let $\rho\in\mathcal{P}(\mathbb{R}^{q})$ with tensor power $\rho^{\otimes N}\in\mathcal{P}(\mathbb{R}^{qN})$, let $\mathfrak{p}_{k}:\mathbb{R}^{qN}\to\mathbb{R}^{q}$, $k\in[N]$, be the block maps of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks, and let $\mu^{N}_{x}$ be the empirical measure of $x\in\mathbb{R}^{qN}$. In real arithmetic $N$ stands for its image under the canonical map, which is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $\frac{1}{N}$ means $N^{-1}$. Let $\varphi:\mathbb{R}^{q}\to\mathbb{R}$ be bounded and Borel, let $\varphi^{2}$ denote its pointwise square, which is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and bounded by the square of a bound for $\varphi$, and put

integrals of bounded Borel functions as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Then the following hold.

1. (The empirical average)¶ For every $x\in\mathbb{R}^{qN}$ the function $\varphi$ is integrable with respect to $\mu^{N}_{x}$, and

The function $g:\mathbb{R}^{qN}\to\mathbb{R}$, $g(x)=\int_{\mathbb{R}^{q}}\varphi\,d\mu^{N}_{x}$, is Borel and bounded.

2. (Mean-square deviation)¶