Proof of The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections

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

Throughout, $X$, $\langle\cdot,\cdot\rangle$, $|\cdot|$, $d$, $0_{X}$ and $(e_{k})_{k\in\mathbb{N}}$ are as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space, and $\mathbf{1}_{A}$ denotes the indicator of a set $A$. For $k\in\mathbb{N}$ write $s_{k}=\sqrt{c_{k}}$. Since $c_{k}>0$ by Variance Sequences and Their Truncations §variances, we have $s_{k}\ge0$ and $s_{k}^{2}=c_{k}\neq0$, so $s_{k}>0$. Claims 1 to 3 are proved for an arbitrary probability space $(\Omega,\mathcal{F},P)$ and an arbitrary independent sequence $(\xi_{k})_{k\in\mathbb{N}}$ of standard normal random variables on it, fixed through Step 9.

We use one elementary fact about real sequences, recorded as $(\ast)$: if a sequence $(a_{j})_{j\in\mathbb{N}}$ of real numbers satisfies $a_{j}\le a_{j+1}$ for every $j$ and converges to a real number $L$, then $a_{N}\le L$ for every $N$. Indeed, if $a_{N}>L$ for some $N$, put $\varepsilon=a_{N}-L>0$; for every $j\ge N$ we have $a_{j}\ge a_{N}$, hence $|a_{j}-L|\ge a_{j}-L\ge\varepsilon$, which contradicts the convergence of $(a_{j})$ to $L$ in the sense of Limit of a Sequence of Real Numbers, since that provides some $j\ge N$ with $|a_{j}-L|<\varepsilon$.

Step 1. (The partial sums.) For $n\in\mathbb{N}$ and $\omega\in\Omega$ put

$$S_{n}(\omega)=\sum_{k=1}^{n}c_{k}\,\xi_{k}(\omega)^{2}.$$

Each $\xi_{k}^{2}$ is measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, so $S_{n}$ is a random variable by claim 2 there. Each $S_{n}$ is nonnegative, and $S_{n+1}=S_{n}+c_{n+1}\xi_{n+1}^{2}\ge S_{n}$ pointwise. By claim 1 of Moments and Stability of the Standard Normal Distribution, $\mathbb{E}[\xi_{k}^{2}]=1$; since $\xi_{k}^{2}$ is nonnegative, this expectation is the integral $\int_{\Omega}\xi_{k}^{2}\,dP$ of the nonnegative function $\xi_{k}^{2}$. Indeed, by Expectation, Variance, and Moments the expectation of the integrable variable $\xi_{k}^{2}$ is its Lebesgue integral, which by Integrable Function and the Lebesgue Integral is the integral of its positive part minus the integral of its negative part; since $\xi_{k}^{2}\ge0$, its positive part is $\xi_{k}^{2}$ itself and its negative part is the zero function, whose integral is $0$ by the null-integral clause (applied with the empty null set). Hence the two readings of $\mathbb{E}[\xi_{k}^{2}]$ coincide. Applying claim 1 of Linearity and Monotonicity of the Lebesgue Integral (additivity, and homogeneity with the constants $c_{k}\in[0,\infty)$) finitely many times gives

$$\int_{\Omega}S_{n}\,dP=\sum_{k=1}^{n}c_{k}\qquad(n\in\mathbb{N}).$$

Let $\tau=\sum_{k=1}^{\infty}c_{k}$, a real number because the series converges (Variance Sequences and Their Truncations §variances). Since every $c_{k}$ is positive, domination by the sum gives $\sum_{k=1}^{n}c_{k}\le\tau$ for every $n$. Hence $\int_{\Omega}S_{n}\,dP\le\tau$ for every $n$.

Step 2. ($G$ is an event of probability one.) The functions $S_{n}$, read as maps into $[0,\infty]$, are measurable in the sense of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable and nondecreasing in $n$. By Monotone Convergence Theorem, the function $S:\Omega\to[0,\infty]$, $S(\omega)=\sup_{n}S_{n}(\omega)$, is measurable and $\int_{\Omega}S\,dP=\sup_{n}\int_{\Omega}S_{n}\,dP\le\tau<\infty$ by Step 1. By finiteness almost everywhere, the set $D=\{\omega\in\Omega:S(\omega)=\infty\}$ belongs to $\mathcal{F}$ and $P(D)=0$.

We show $G=\Omega\setminus D$. For each $\omega$, the numbers $S_{n}(\omega)$ are the partial sums of the series $\sum_{k=1}^{\infty}c_{k}\xi_{k}(\omega)^{2}$ (Series of Real Numbers §partial-sums). If $\omega\notin D$, then $S(\omega)$ is a real number bounding the set $\{S_{n}(\omega):n\in\mathbb{N}\}$ from above, and this sequence is nondecreasing, so it converges by the monotone convergence of bounded nondecreasing real sequences; thus $\omega\in G$. Conversely, if $\omega\in G$, the partial sums converge to some real $L$, so $S_{n}(\omega)\le L$ for every $n$ by $(\ast)$, whence $S(\omega)\le L<\infty$ and $\omega\notin D$. Therefore $G=\Omega\setminus D$ belongs to $\mathcal{F}$, a $\sigma$-algebra being closed under complements (Sigma-Algebra and Measurable Space), and $P(G)=P(\Omega)-P(D)=1$ by the difference formula for measures.

Step 3. (Convergence of $Y_{n}(\omega)$ and the coordinates of the limit.) Fix $\omega\in G$ and put $a_{k}=s_{k}\xi_{k}(\omega)$ for $k\in\mathbb{N}$. Then $a_{k}^{2}=s_{k}^{2}\xi_{k}(\omega)^{2}=c_{k}\xi_{k}(\omega)^{2}$, so the series $\sum_{k=1}^{\infty}a_{k}^{2}$ converges, by the definition of $G$. The space $X$ is a real Hilbert space playing the role of $H$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space), and $(e_{k})_{k\in\mathbb{N}}$, being an orthonormal basis, is in particular an orthonormal sequence (Orthonormal Basis of a Real Hilbert Space §basis). By the Riesz-Fischer criterion, the series $\sum_{k=1}^{\infty}a_{k}e_{k}$ converges in $X$, and its sum $z$ satisfies $\langle z,e_{j}\rangle=a_{j}$ for every $j\in\mathbb{N}$. The $n$-th partial sum of this series is $\sum_{k=1}^{n}a_{k}e_{k}=Y_{n}(\omega)$ (Series in a Real Inner Product Space §partial-sums), so by Series in a Real Inner Product Space §convergent the sequence $(Y_{n}(\omega))_{n\in\mathbb{N}}$ converges in $(X,d)$, with limit $z$, the limit being unique as recorded there. Together with Step 2 this proves claim 1. It also shows that $Y$ is well defined, that $Y(\omega)=z$ for $\omega\in G$, and hence

$$\langle Y(\omega),e_{j}\rangle=s_{j}\,\xi_{j}(\omega)\qquad(\omega\in G,\ j\in\mathbb{N}),$$

which is the identity asserted in claim 2.

Step 4. ($Y$ is a random element.) For $\omega\notin G$ we have $Y(\omega)=0_{X}$, and $\langle0_{X},e_{k}\rangle=0$: since $0_{X}+0_{X}=0_{X}$, additivity in the first argument (Real Inner Product Space §inner-product) gives $\langle0_{X},e_{k}\rangle=\langle0_{X},e_{k}\rangle+\langle0_{X},e_{k}\rangle$. Combining with Step 3, for every $k\in\mathbb{N}$ and every $\omega\in\Omega$,

$$\langle Y(\omega),e_{k}\rangle=s_{k}\,\xi_{k}(\omega)\,\mathbf{1}_{G}(\omega).$$

Since $G\in\mathcal{F}$ by Step 2, the indicator $\mathbf{1}_{G}$ is measurable by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $s_{k}\xi_{k}$ is measurable by claim 2 there, and their product is measurable by claim 3 there. By measurability by coordinates, $Y$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$, the Borel $\sigma$-algebra of $(X,d)$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures). So $Y$ is a random element of $(X,d)$ in the sense of Random Element of a Metric Space and Its Law, and claim 2 is proved.

Step 5. (The scaled variables.) Fix $k\in\mathbb{N}$ and put $V_{k}=s_{k}\xi_{k}$, a random variable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $B\subseteq\mathbb{R}$ write $s_{k}^{-1}B=\{s_{k}^{-1}y:y\in B\}$. For every real $x$ we have $s_{k}x\in B$ if and only if $x\in s_{k}^{-1}B$: if $s_{k}x=y\in B$ then $x=s_{k}^{-1}y$, and if $x=s_{k}^{-1}y$ with $y\in B$ then $s_{k}x=y$. Hence $\{V_{k}\in B\}=\{\xi_{k}\in s_{k}^{-1}B\}$. If $B$ is a Borel set, then $s_{k}^{-1}B$ is a Borel subset of $\mathbb{R}^{1}$ by claim 1 of Scaling of Lebesgue Measure and the Lebesgue Integral on $\mathbb{R}^n$ (with dimension $1$ and the nonzero real $s_{k}^{-1}$), and $\mathcal{B}(\mathbb{R}^{1})=\mathcal{B}(\mathbb{R})$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets with $m=1$, its $\sigma$-algebra $\mathcal{B}_{1}$ being $\mathcal{B}(\mathbb{R})$ as recorded in that lemma. Two consequences follow.

First, for every $n\in\mathbb{N}$ the random variables $V_{1},\dots,V_{n}$ are independent: for Borel sets $B_{1},\dots,B_{n}$ the events $\{V_{k}\in B_{k}\}=\{\xi_{k}\in s_{k}^{-1}B_{k}\}$ ($k\le n$) are independent, because $\xi_{1},\dots,\xi_{n}$, a finite subfamily of an independent sequence, are independent and the sets $s_{k}^{-1}B_{k}$ are Borel (Independence of Events and of Random Variables).

Second, since $\xi_{k}$ has distribution $N$ (Standard Normal Distribution), the distribution $\nu_{k}$ of $V_{k}$ (Distribution and Cumulative Distribution Function of a Random Variable) is given by

$$\nu_{k}(B)=P(\xi_{k}\in s_{k}^{-1}B)=N(s_{k}^{-1}B)\qquad(B\in\mathcal{B}(\mathbb{R})).$$

Step 6. (Each $\nu_{k}$ is a one-dimensional diagonal Gaussian measure.) Let $\lambda$ be the Lebesgue measure on $\mathcal{B}(\mathbb{R})$ of Existence of Lebesgue Measure on the Real Line; Lebesgue measure $\lambda_{1}$ on $\mathbb{R}^{1}$ is $\lambda$ itself (Lebesgue Measure on $\mathbb{R}^n$), and we read $\mathbb{R}^{1}$ as $\mathbb{R}$. The one-tuple $(c_{k})$ is a variance vector in dimension $1$ (The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances); let $\gamma_{(c_{k})}$ be the corresponding diagonal Gaussian measure, so $\gamma_{(c_{k})}(B)=\int_{\mathbb{R}}\mathbf{1}_{B}\rho_{(c_{k})}\,d\lambda$. We show $\nu_{k}=\gamma_{(c_{k})}$.

By the definition of the weighted square and of the density in dimension $1$, the weighted square of $y\in\mathbb{R}$ is $y^{2}/c_{k}$ and $Z_{(c_{k})}=\log(\kappa s_{k})$, so $\rho_{(c_{k})}(y)=\exp\bigl(-y^{2}/(2c_{k})-\log(\kappa s_{k})\bigr)$. By claims 1 and 2 of Basic Properties of the Exponential Function and the identity $\exp(\log t)=t$ of The Natural Logarithm,

$$\rho_{(c_{k})}(y)=(\kappa s_{k})^{-1}h(y),\qquad h(y)=\exp\bigl(-y^{2}/(2c_{k})\bigr)\qquad(y\in\mathbb{R}).$$

Here $\kappa$ is, by The Diagonal Gaussian Density on Euclidean Space and Its Notation, the total mass of claim 2 of The Gaussian Weight Defines a Probability Distribution, namely the integral $\int_{\mathbb{R}}g\,d\lambda$ of $g(x)=\exp(-x^{2}/2)$ against $\lambda$, with the integral of Lebesgue Integral of a Nonnegative Measurable Function; this is the same integral of the same function against the same measure that Standard Normal Distribution uses as its normalising constant, so $N(A)=\kappa^{-1}\int_{\mathbb{R}}\mathbf{1}_{A}\,g\,d\lambda$ for $A\in\mathcal{B}(\mathbb{R})$.

Fix $B\in\mathcal{B}(\mathbb{R})$ and let $f=\mathbf{1}_{B}h=\kappa s_{k}\,\mathbf{1}_{B}\,\rho_{(c_{k})}$. It is nonnegative and Borel, because $\rho_{(c_{k})}$ is Borel by regularity of the density and by claims 1 to 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $x\in\mathbb{R}$, using $s_{k}^{2}=c_{k}$ and the equivalence of Step 5,

$$f(s_{k}x)=\mathbf{1}_{B}(s_{k}x)\exp\bigl(-s_{k}^{2}x^{2}/(2c_{k})\bigr)=\mathbf{1}_{s_{k}^{-1}B}(x)\,g(x).$$

Claim 2 of Scaling of Lebesgue Measure and the Lebesgue Integral on $\mathbb{R}^n$, with dimension $1$ and the nonzero real $s_{k}$, gives $\int_{\mathbb{R}}f(s_{k}x)\,d\lambda(x)=s_{k}^{-1}\int_{\mathbb{R}}f\,d\lambda$. Using Step 5 and claim 1 of Linearity and Monotonicity of the Lebesgue Integral for the constant factors,

$$\nu_{k}(B)=N(s_{k}^{-1}B)=\kappa^{-1}\int_{\mathbb{R}}\mathbf{1}_{s_{k}^{-1}B}\,g\,d\lambda=\kappa^{-1}s_{k}^{-1}\int_{\mathbb{R}}f\,d\lambda=\kappa^{-1}s_{k}^{-1}\,\kappa s_{k}\int_{\mathbb{R}}\mathbf{1}_{B}\,\rho_{(c_{k})}\,d\lambda=\gamma_{(c_{k})}(B).$$

Step 7. (Factorisation of the density.) Let $2\le m$. For $u\in\mathbb{R}^{m-1}$ and $v\in\mathbb{R}$ let $\iota(u,v)=\iota^{m-1,1}(u,v)=(u_{1},\dots,u_{m-1},v)$ be the concatenation, as described in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets. The truncations $c^{(m-1)}$ and $c^{(m)}$ are variance vectors (Variance Sequences and Their Truncations §truncations). By the definition of the weighted square and of $Z_{c}$, splitting off the last term of each finite sum,

$$|\iota(u,v)|_{c^{(m)}}^{2}=\sum_{i=1}^{m-1}\frac{u_{i}^{2}}{c_{i}}+\frac{v^{2}}{c_{m}}=|u|_{c^{(m-1)}}^{2}+\frac{v^{2}}{c_{m}},\qquad Z_{c^{(m)}}=Z_{c^{(m-1)}}+\log(\kappa s_{m}),$$

so claim 1 of Basic Properties of the Exponential Function, together with the formula for $\rho_{(c_{m})}$ in Step 6, gives

$$\rho_{c^{(m)}}(\iota(u,v))=\rho_{c^{(m-1)}}(u)\,\rho_{(c_{m})}(v).$$

Step 8. (The law of the vector $(V_{1},\dots,V_{n})$.) Let $\mathcal{B}_{r}$ ($r\ge1$) be the iterated product $\sigma$-algebras of Joint Distribution, Expectations, and Block Independence for Independent Random Variables. They are defined by the same recursion and the same identification of $\mathbb{R}^{r-1}\times\mathbb{R}$ with $\mathbb{R}^{r}$ as the $\sigma$-algebras $\mathcal{B}_{r}$ of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, so $\mathcal{B}_{r}=\mathcal{B}(\mathbb{R}^{r})$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; that identification is $(u,v)\mapsto\iota(u,v)$ of Step 7. Fix $n\in\mathbb{N}$ (so $1\le n$, as $\mathbb{R}^{n}$ requires). We show by induction on $m\in\{1,\dots,n\}$ that $\gamma_{c^{(m)}}=\nu_{1}\otimes\cdots\otimes\nu_{m}$ as measures on $\mathcal{B}_{m}=\mathcal{B}(\mathbb{R}^{m})$, the right-hand side being the iterated product of Joint Distribution, Expectations, and Block Independence for Independent Random Variables.

For $m=1$, $c^{(1)}=(c_{1})$ and this is Step 6 with $k=1$. Let $2\le m\le n$ and assume the claim for $m-1$. By definition, $\nu_{1}\otimes\cdots\otimes\nu_{m}$ is the product measure $(\nu_{1}\otimes\cdots\otimes\nu_{m-1})\otimes\nu_{m}$ of Existence and Uniqueness of the Product Measure on $\mathcal{B}_{m-1}\otimes\mathcal{B}(\mathbb{R})$, transported to $\mathbb{R}^{m}$ along $\iota$; both factors are probability measures, hence $\sigma$-finite. Transporting $\gamma_{c^{(m)}}$ to $\mathcal{B}_{m-1}\otimes\mathcal{B}(\mathbb{R})$ along the same identification, the uniqueness assertion of Existence and Uniqueness of the Product Measure shows that it suffices to prove

$$\gamma_{c^{(m)}}(\iota(A\times B))=\gamma_{c^{(m-1)}}(A)\,\nu_{m}(B)\qquad(A\in\mathcal{B}(\mathbb{R}^{m-1}),\ B\in\mathcal{B}(\mathbb{R})),$$

since the right-hand side equals $(\nu_{1}\otimes\cdots\otimes\nu_{m-1})(A)\,\nu_{m}(B)$ by the induction hypothesis. Fix such $A,B$. The set $\iota(A\times B)$ belongs to $\mathcal{B}(\mathbb{R}^{m})$ by the product clause of the Euclidean pairs toolkit, so $F=\mathbf{1}_{\iota(A\times B)}\rho_{c^{(m)}}$ is a nonnegative Borel function (regularity and Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions). Since $\iota$ is a bijection, $\mathbf{1}_{\iota(A\times B)}(\iota(u,v))=\mathbf{1}_{A}(u)\mathbf{1}_{B}(v)$, and by Step 7

$$F(\iota(u,v))=\bigl(\mathbf{1}_{A}(u)\rho_{c^{(m-1)}}(u)\bigr)\,\mathbf{1}_{B}(v)\rho_{(c_{m})}(v).$$

By iterated integration over the two factors with $q=m-1$ and $p=1$ (recall $\lambda_{1}=\lambda$), and claim 1 of Linearity and Monotonicity of the Lebesgue Integral applied first with the constant $\mathbf{1}_{A}(u)\rho_{c^{(m-1)}}(u)\in[0,\infty)$ and then with the constant $\nu_{m}(B)\in[0,1]$,

$$\gamma_{c^{(m)}}(\iota(A\times B))=\int_{\mathbb{R}^{m}}F\,d\lambda_{m}=\int_{\mathbb{R}^{m-1}}\mathbf{1}_{A}(u)\rho_{c^{(m-1)}}(u)\,\gamma_{(c_{m})}(B)\,\lambda_{m-1}(du)=\nu_{m}(B)\int_{\mathbb{R}^{m-1}}\mathbf{1}_{A}\rho_{c^{(m-1)}}\,d\lambda_{m-1}=\nu_{m}(B)\,\gamma_{c^{(m-1)}}(A),$$

where $\gamma_{(c_{m})}(B)=\nu_{m}(B)$ by Step 6 and the first and last equalities are the definition of the diagonal Gaussian measure. This completes the induction.

By Step 5, $V_{1},\dots,V_{n}$ are independent random variables with distributions $\nu_{1},\dots,\nu_{n}$. By claim 1 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables with $r=n$, the map $V:\Omega\to\mathbb{R}^{n}$, $V(\omega)=(V_{1}(\omega),\dots,V_{n}(\omega))$, is measurable with respect to $\mathcal{F}$ and $\mathcal{B}_{n}=\mathcal{B}(\mathbb{R}^{n})$, and its distribution satisfies, by the induction with $m=n$,

$$P(V\in C)=(\nu_{1}\otimes\cdots\otimes\nu_{n})(C)=\gamma_{c^{(n)}}(C)\qquad(C\in\mathcal{B}(\mathbb{R}^{n})).$$

Step 9. (Claim 3.) By claim 1 of Image Measures, Measures with Densities, and Change of Variables, applied to the measurable map $Y$ of Step 4, the law $P_{Y}$ of Random Element of a Metric Space and Its Law is a measure on $\mathcal{B}(X)$ with $P_{Y}(X)=P(\Omega)=1$; it is thus a Borel measure on $(X,d)$ of total mass $1$, that is, $P_{Y}\in\mathcal{P}(X)$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures).

Fix $n\in\mathbb{N}$. The map $p_{n}$ is Borel by the Lipschitz property of the coordinate maps, so by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward $(p_{n})_{\#}P_{Y}(C)=P_{Y}(p_{n}^{-1}(C))=P(E_{C})$ for $C\in\mathcal{B}(\mathbb{R}^{n})$, where $E_{C}=Y^{-1}(p_{n}^{-1}(C))\in\mathcal{F}$. For $\omega\in G$, by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and Step 3,

$$p_{n}(Y(\omega))=\bigl(\langle Y(\omega),e_{1}\rangle,\dots,\langle Y(\omega),e_{n}\rangle\bigr)=\bigl(s_{1}\xi_{1}(\omega),\dots,s_{n}\xi_{n}(\omega)\bigr)=V(\omega),$$

so $E_{C}\cap G=V^{-1}(C)\cap G$. Now $P(\Omega\setminus G)=0$ by Step 2 and the difference formula. For every event $E$, the events $E\cap G$ and $E\setminus G$ are disjoint with union $E$, and $P(E\setminus G)\le P(\Omega\setminus G)=0$ by monotonicity, so $P(E)=P(E\cap G)$ by finite additivity. Applying this to $E_{C}$ and to $V^{-1}(C)$, both events by Step 8, and using Step 8,

$$(p_{n})_{\#}P_{Y}(C)=P(E_{C}\cap G)=P(V^{-1}(C)\cap G)=P(V^{-1}(C))=\gamma_{c^{(n)}}(C)\qquad(C\in\mathcal{B}(\mathbb{R}^{n})).$$

Thus $(p_{n})_{\#}P_{Y}=\gamma_{c^{(n)}}$ for every $n$, proving claim 3.

Step 10. (Claim 4.) Existence: the standard normal distribution $N$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Standard Normal Distribution), so Existence of Independent and Identically Distributed Sequences provides a probability space and an independent sequence of random variables on it, each with distribution $N$, that is, standard normal. For this space and sequence, claim 3 (Steps 1 to 9) shows that $\mu=P_{Y}$ belongs to $\mathcal{P}(X)$ and satisfies $(p_{n})_{\#}\mu=\gamma_{c^{(n)}}$ for every $n\in\mathbb{N}$.

Uniqueness: if $\mu,\mu'\in\mathcal{P}(X)$ both satisfy $(p_{n})_{\#}\mu=\gamma_{c^{(n)}}=(p_{n})_{\#}\mu'$ for every $n$, then $\mu$ and $\mu'$ are Borel measures on $(X,d)$ with $\mu(X)=\mu'(X)=1<\infty$ and equal push-forwards under every $p_{n}$, so $\mu=\mu'$ by determination by finite-dimensional projections.

Finally, for every probability space and every independent sequence of standard normal random variables on it, the law $P_{Y}$ of the corresponding map $Y$ belongs to $\mathcal{P}(X)$ and has $(p_{n})_{\#}P_{Y}=\gamma_{c^{(n)}}$ for every $n$, by claim 3; by uniqueness it is the measure $\mu$. This proves claim 4.