The Gaussian-Tail Extension of a Probability Measure on the Rescaled Head

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $n\in\mathbb{N}$, with $\tau_{n}$ and $\Psi_{n}$ as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §tails, and let $d_{\times}$ be the product metric on $\mathbb{R}^{n}\times X$.

1. ($\Psi_{n}$ is Borel)¶ The map $\Psi_{n}$ is Lipschitz, hence continuous by A Lipschitz Map is Uniformly Continuous, from $(\mathbb{R}^{n}\times X,d_{\times})$ to $(X,d)$: for $(u,w),(u',w')\in\mathbb{R}^{n}\times X$, the triangle inequality, $|e_{k}|=1$ and $0<a_{k}^{1/2}$ give

$$|\Psi_{n}(u,w)-\Psi_{n}(u',w')|\le\sum_{k=1}^{n}a_{k}^{1/2}|u_{k}-u'_{k}|+|w-w'|\le\Bigl(1+\sum_{k=1}^{n}a_{k}^{1/2}\Bigr)\,d_{\times}\bigl((u,w),(u',w')\bigr),$$

since $|u_{k}-u'_{k}|\le\lVert u-u'\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate and each of $\lVert u-u'\rVert$ and $|w-w'|$ is at most $d_{\times}((u,w),(u',w'))$ by Product Metric on the Cartesian Product of Two Metric Spaces. Hence $\Psi_{n}$ is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, the Borel $\sigma$-algebra of $\mathbb{R}^{n}\times X$ being $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §tails.

2. (The product measure)¶ For $\lambda\in\mathcal{P}(\mathbb{R}^{n})$, the measures $\lambda$ and $\tau_{n}$ are finite, hence $\sigma$-finite, so the product measure $\lambda\otimes\tau_{n}$ on $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ exists and is unique by that theorem, and it is a probability measure, since $(\lambda\otimes\tau_{n})(\mathbb{R}^{n}\times X)=\lambda(\mathbb{R}^{n})\,\tau_{n}(X)=1$ by the defining identity on rectangles.

3. (Gaussian-tail extension)¶ For $\lambda\in\mathcal{P}(\mathbb{R}^{n})$, the Gaussian-tail extension of $\lambda$ at level $n$ is the probability measure

$$E_{n}(\lambda)=(\Psi_{n})_{\#}(\lambda\otimes\tau_{n})\in\mathcal{P}(X),$$

the image measure of claim 1 of that lemma.