Proof of Head and Tail of a Diagonal Gaussian Measure on a Hilbert Space are Independent

Each result cited is universally quantified over the data in its own statement.

Throughout, $\rho$ denotes the product metric on $\mathbb{R}^{n}\times X$, a metric by claim 1 of The Product Metric is a Metric; since $d_{E}(u,u')=\lVert u-u'\rVert$ by Euclidean Space and Lebesgue Measure: Standing Notation §space, it is given by

$$\rho\bigl((u,w),(u',w')\bigr)=\max\{\lVert u-u'\rVert,\,|w-w'|\}\qquad(u,u'\in\mathbb{R}^{n},\ w,w'\in X).$$

As recalled in the statement, its Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ by The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product. For maps $T:S_{1}\to S_{2}$ and $R:S_{2}\to S_{3}$ between measurable spaces, each measurable, and a measure $\lambda$ on $S_{1}$, one has $(R\circ T)^{-1}(C)=T^{-1}(R^{-1}(C))$ for every measurable $C\subseteq S_{3}$, so that the image measures of claim 1 of Image Measures, Measures with Densities, and Change of Variables satisfy

$$(R\circ T)_{\#}\lambda=R_{\#}(T_{\#}\lambda).\tag{$\dagger$}$$

Step 1 (continuity of $h_{n}$). Let $x,x'\in X$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, $\lVert p_{n}(x)-p_{n}(x')\rVert\le|x-x'|$, and $Q_{n}$ is Lipschitz with constant $1$, so $|Q_{n}x-Q_{n}x'|\le|x-x'|$. Hence $\rho(h_{n}(x),h_{n}(x'))\le|x-x'|$. Given $x\in X$ and a real $\varepsilon>0$, the choice $\delta=\varepsilon$ gives $\rho(h_{n}(x),h_{n}(x'))<\varepsilon$ whenever $|x-x'|<\delta$, so $h_{n}$ is continuous on $X$, and it is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. This proves the first assertion of claim 1.

Step 2 (continuity of $\Phi_{n}$ and $\Phi_{n}\circ h_{n}=\mathrm{id}_{X}$). For $u,u'\in\mathbb{R}^{n}$ the definition $p_{n}^{*}(y)=\sum_{k=1}^{n}y_{k}e_{k}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates gives $p_{n}^{*}(u)-p_{n}^{*}(u')=p_{n}^{*}(u-u')$, and $|p_{n}^{*}(u-u')|=\lVert u-u'\rVert$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. By the triangle inequality for the norm of $X$, for $(u,w),(u',w')\in\mathbb{R}^{n}\times X$,

$$|\Phi_{n}(u,w)-\Phi_{n}(u',w')|\le|p_{n}^{*}(u-u')|+|w-w'|=\lVert u-u'\rVert+|w-w'|\le2\rho\bigl((u,w),(u',w')\bigr).$$

Given a real $\varepsilon>0$, the choice $\delta=\varepsilon/2$ shows that $\Phi_{n}$ is continuous on $\mathbb{R}^{n}\times X$, and it is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. For $x\in X$, since $P_{n}=p_{n}^{*}\circ p_{n}$ and $Q_{n}x=x-P_{n}x$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates,

$$\Phi_{n}(h_{n}(x))=p_{n}^{*}(p_{n}(x))+Q_{n}x=P_{n}x+(x-P_{n}x)=x .$$

Step 3 (a random series with law $\gamma_{c}$). The standard normal distribution $N$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ by Standard Normal Distribution. By Existence of Independent and Identically Distributed Sequences applied to $N$ there are a probability space $(\Omega,\mathcal{F},P)$ and an independent and identically distributed sequence $(\xi_{k})_{k\in\mathbb{N}}$ of random variables on it with common distribution $N$, that is, an independent sequence of standard normal random variables. For this sequence let $Y_{m}$ ($m\in\mathbb{N}$), $G$ and $Y$ be as in The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections, so that $Y_{m}(\omega)=\sum_{k=1}^{m}\sqrt{c_{k}}\,\xi_{k}(\omega)\,e_{k}$. By The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections §convergence, $G\in\mathcal{F}$ and $P(G)=1$. By The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections §random-element, $Y$ is a random element of $(X,d)$, and $\langle Y(\omega),e_{k}\rangle=\sqrt{c_{k}}\,\xi_{k}(\omega)$ for every $\omega\in G$ and $k\in\mathbb{N}$; for $\omega\notin G$, $Y(\omega)=0_{X}$ by definition. By The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections §unique and Diagonal Gaussian Measures on a Hilbert Space §measure, the law $P_{Y}$ of Random Element of a Metric Space and Its Law, given by $P_{Y}(C)=P(Y^{-1}(C))$ for $C\in\mathcal{B}(X)$, equals $\gamma_{c}$, and by Diagonal Gaussian Measures on a Hilbert Space §measure $(p_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}}$.

Step 4 (two independent $\sigma$-algebras). Let $\mathcal{G}_{1}=\sigma(\xi_{k}:1\le k\le n)$ and $\mathcal{G}_{2}=\sigma(\xi_{k}:k\in\mathbb{N},\ k>n)$ be the generated $\sigma$-algebras, sub-$\sigma$-algebras of $\mathcal{F}$. The index sets $\{1,\dots,n\}$ and $\{k\in\mathbb{N}:k>n\}$ are nonempty and disjoint, so by Grouping Lemma for Independent Random Variables the family $(\mathcal{G}_{1},\mathcal{G}_{2})$ is independent in the sense of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras; in particular

$$P(E\cap E')=P(E)\,P(E')\qquad\text{for all }E\in\mathcal{G}_{1},\ E'\in\mathcal{G}_{2}.\tag{1}$$

For $k\le n$ and a Borel set $B'\subseteq\mathbb{R}$ the set $\xi_{k}^{-1}(B')$ belongs to the generating family of $\mathcal{G}_{1}$, hence to $\mathcal{G}_{1}$ by claim 2 of Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra; so $\xi_{k}$ is measurable with respect to $\mathcal{G}_{1}$. Likewise $\xi_{k}$ is measurable with respect to $\mathcal{G}_{2}$ for $k>n$.

Step 5 (the head is $\mathcal{G}_{1}$-measurable on $G$). For $\omega\in\Omega$ and $j\in\mathbb{N}$, orthonormality of $(e_{k})$ gives $\langle Y_{n}(\omega),e_{j}\rangle=\sum_{k=1}^{n}\sqrt{c_{k}}\,\xi_{k}(\omega)\langle e_{k},e_{j}\rangle$, which equals $\sqrt{c_{j}}\,\xi_{j}(\omega)$ if $j\le n$ and $0$ if $j>n$. By claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions each of these functions of $\omega$ is measurable with respect to $\mathcal{G}_{1}$, so by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §measurable, applied to the measurable space $(\Omega,\mathcal{G}_{1})$, the map $Y_{n}$ is measurable with respect to $\mathcal{G}_{1}$ and $\mathcal{B}(X)$. Since $p_{n}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, the map $U=p_{n}\circ Y_{n}:\Omega\to\mathbb{R}^{n}$ is measurable with respect to $\mathcal{G}_{1}$ and $\mathcal{B}(\mathbb{R}^{n})$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and $U(\omega)=(\sqrt{c_{1}}\,\xi_{1}(\omega),\dots,\sqrt{c_{n}}\,\xi_{n}(\omega))$. For $\omega\in G$, Step 3 gives $p_{n}(Y(\omega))=(\langle Y(\omega),e_{k}\rangle)_{k=1}^{n}=U(\omega)$.

Step 6 (the tail is $\mathcal{G}_{2}$-measurable). For $m>n$ let $S_{m}=\sum_{k=n+1}^{m}c_{k}\,\xi_{k}^{2}$, a function on $\Omega$ measurable with respect to $\mathcal{G}_{2}$ by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Fix $\omega\in\Omega$. The partial sums $\sum_{k=1}^{m}c_{k}\xi_{k}(\omega)^{2}$ form a nondecreasing sequence of real numbers, since $c_{k}>0$ by Variance Sequences and Their Truncations §variances, so they converge if and only if they are bounded above; for $m>n$ they equal $\sum_{k=1}^{n}c_{k}\xi_{k}(\omega)^{2}+S_{m}(\omega)$. Hence $\omega\in G$ if and only if there is $M\in\mathbb{N}$ with $S_{m}(\omega)\le M$ for every $m>n$, that is,

$$G=\bigcup_{M\in\mathbb{N}}\ \bigcap_{m>n}S_{m}^{-1}\bigl((-\infty,M]\bigr)\in\mathcal{G}_{2},$$

each $(-\infty,M]$ being a Borel set. Next, for $x\in X$ and $k\in\mathbb{N}$, orthonormality gives $\langle Q_{n}x,e_{k}\rangle=x_{k}-\sum_{j=1}^{n}x_{j}\langle e_{j},e_{k}\rangle$, which is $0$ for $k\le n$ and $x_{k}$ for $k>n$. Since $Y(\omega)=0_{X}$ for $\omega\notin G$ and $Q_{n}0_{X}=0_{X}-p_{n}^{*}(p_{n}(0_{X}))=0_{X}$, Step 3 yields, for every $\omega\in\Omega$,

$$\langle Q_{n}Y(\omega),e_{k}\rangle=0\quad(k\le n),\qquad\langle Q_{n}Y(\omega),e_{k}\rangle=\mathbf{1}_{G}(\omega)\sqrt{c_{k}}\,\xi_{k}(\omega)\quad(k>n).$$

By claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and $G\in\mathcal{G}_{2}$, all these functions are measurable with respect to $\mathcal{G}_{2}$, so by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §measurable, applied to $(\Omega,\mathcal{G}_{2})$, the map $Q_{n}\circ Y$ is measurable with respect to $\mathcal{G}_{2}$ and $\mathcal{B}(X)$.

Step 7 (rectangles). Let $A\in\mathcal{B}(\mathbb{R}^{n})$ and $B\in\mathcal{B}(X)$, and set $E_{1}=(p_{n}\circ Y)^{-1}(A)$, $E_{1}'=U^{-1}(A)$ and $E_{2}=(Q_{n}\circ Y)^{-1}(B)$. Then $E_{1}\in\mathcal{F}$, since $p_{n}\circ Y$ is measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space; $E_{1}'\in\mathcal{G}_{1}$ by Step 5; and $E_{2}\in\mathcal{G}_{2}$ by Step 6. By Step 5, $E_{1}\cap G=E_{1}'\cap G$. Since $P(\Omega\setminus G)=0$, every $E\in\mathcal{F}$ satisfies $P(E)=P(E\cap G)+P(E\setminus G)=P(E\cap G)$, by additivity and monotonicity of $P$. Therefore, using (1),

$$P(E_{1}\cap E_{2})=P(E_{1}\cap E_{2}\cap G)=P(E_{1}'\cap E_{2}\cap G)=P(E_{1}'\cap E_{2})=P(E_{1}')\,P(E_{2})=P(E_{1})\,P(E_{2}),$$

where the last equality uses $P(E_{1}')=P(E_{1}'\cap G)=P(E_{1}\cap G)=P(E_{1})$. Now $h_{n}^{-1}(A\times B)=p_{n}^{-1}(A)\cap Q_{n}^{-1}(B)$, so $Y^{-1}(h_{n}^{-1}(A\times B))=E_{1}\cap E_{2}$, $Y^{-1}(p_{n}^{-1}(A))=E_{1}$ and $Y^{-1}(Q_{n}^{-1}(B))=E_{2}$. Since $P_{Y}=\gamma_{c}$ by Step 3,

$$(h_{n})_{\#}\gamma_{c}(A\times B)=P_{Y}\bigl(h_{n}^{-1}(A\times B)\bigr)=P(E_{1}\cap E_{2})=P(E_{1})\,P(E_{2})=\gamma_{c}\bigl(p_{n}^{-1}(A)\bigr)\,\gamma_{c}\bigl(Q_{n}^{-1}(B)\bigr)=\gamma_{c^{(n)}}(A)\,\bigl((Q_{n})_{\#}\gamma_{c}\bigr)(B),$$

the last step by $(p_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}}$ (Step 3).

Step 8 (claim 1). By Step 1 and claim 1 of Image Measures, Measures with Densities, and Change of Variables, $(h_{n})_{\#}\gamma_{c}$ is a measure on $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$, and $(Q_{n})_{\#}\gamma_{c}$ is a probability measure on $\mathcal{B}(X)$, $Q_{n}$ being Borel. Moreover $\gamma_{c^{(n)}}$ is a probability measure on $\mathcal{B}(\mathbb{R}^{n})$ by Diagonal Gaussian Measures on Euclidean Space §measure, read with $n$ in place of $d$ as in Variance Sequences and Their Truncations §truncations. The probability measures $\gamma_{c^{(n)}}$ and $(Q_{n})_{\#}\gamma_{c}$ are finite, hence $\sigma$-finite, so by Existence and Uniqueness of the Product Measure there is exactly one measure on $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ whose value on every measurable rectangle $A\times B$ is $\gamma_{c^{(n)}}(A)\,((Q_{n})_{\#}\gamma_{c})(B)$, namely $\gamma_{c^{(n)}}\otimes(Q_{n})_{\#}\gamma_{c}$; by Step 7, $(h_{n})_{\#}\gamma_{c}$ is such a measure. Hence $(h_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}}\otimes(Q_{n})_{\#}\gamma_{c}$, which proves claim 1.

Step 9 (claim 2). By Step 2, $\Phi_{n}$ is continuous, hence Borel, and $\Phi_{n}\circ h_{n}=\mathrm{id}_{X}$. Since $(\mathrm{id}_{X})_{\#}\gamma_{c}=\gamma_{c}$, ($\dagger$) and claim 1 give

$$\gamma_{c}=(\Phi_{n}\circ h_{n})_{\#}\gamma_{c}=(\Phi_{n})_{\#}\bigl((h_{n})_{\#}\gamma_{c}\bigr)=(\Phi_{n})_{\#}\bigl(\gamma_{c^{(n)}}\otimes(Q_{n})_{\#}\gamma_{c}\bigr),$$

which proves claim 2.