Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $x_{k}$, the maps $p_{n},p_{n}^{*},P_{n},Q_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and the pairs of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs, the following hold.

1. (Coordinate maps)¶ For all $n\in\mathbb{N}$, $x,x'\in X$ and $y\in\mathbb{R}^{n}$: $p_{n}(p_{n}^{*}(y))=y$, $|p_{n}^{*}(y)|=\lVert y\rVert$, $|P_{n}x|=\lVert p_{n}(x)\rVert$, $|x|^{2}=|P_{n}x|^{2}+|Q_{n}x|^{2}$, and $\lVert p_{n}(x)-p_{n}(x')\rVert\le|x-x'|$. Consequently each coordinate function $x\mapsto x_{k}$, as a map into $(\mathbb{R},d_{\mathbb{R}})$, the maps $p_{n}:(X,d)\to(\mathbb{R}^{n},d_{E})$ and $p_{n}^{*}:(\mathbb{R}^{n},d_{E})\to(X,d)$, with $d_{E}$ as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, and the maps $P_{n},Q_{n}:(X,d)\to(X,d)$, are Lipschitz with constant $1$, hence Borel; in particular $|Q_{n}x|\le|x|$.

2. (Generation)¶ $\mathcal{B}(X)$ is the $\sigma$-algebra generated by the sets $\{x\in X:x_{k}<s\}$ with $k\in\mathbb{N}$ and $s\in\mathbb{R}$.

3. (Measurability by coordinates)¶ Let $(\Omega,\mathcal{F})$ be a measurable space and $Y:\Omega\to X$. Then $Y$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$ if and only if, for every $k\in\mathbb{N}$, the function $\omega\mapsto\langle Y(\omega),e_{k}\rangle$ is measurable with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ of the real line, which is that of $(\mathbb{R},d_{\mathbb{R}})$ by claim 2 of Borel Measurability and Bounded Integration on a Metric Space.

4. (Determination by finite-dimensional projections)¶ Let $\mu,\nu$ be Borel measures on $(X,d)$ with $\mu(X)=\nu(X)<\infty$ and $(p_{n})_{\#}\mu=(p_{n})_{\#}\nu$ for every $n\in\mathbb{N}$. Then $\mu=\nu$.

5. (Product $\sigma$-algebra)¶ $\mathcal{B}(X\times X)$ is the product $\sigma$-algebra $\mathcal{B}(X)\otimes\mathcal{B}(X)$, and $\pi_{1},\pi_{2}$ are Borel.

6. (Pairing)¶ For a measurable space $(\Omega,\mathcal{F})$ and maps $S,T:\Omega\to X$ measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$, the map $(S,T)$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X\times X)$.

7. (Product measure)¶ For Borel measures $\mu,\nu$ on $(X,d)$ of finite total mass, which are $\sigma$-finite, the product measure $\mu\otimes\nu$ is a Borel measure on $X\times X$ with $(\pi_{1})_{\#}(\mu\otimes\nu)=\nu(X)\,\mu$ and $(\pi_{2})_{\#}(\mu\otimes\nu)=\mu(X)\,\nu$.