Diagonal Gaussian Measures on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinate maps $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and push-forwards as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, let $c$ be a variance sequence, with truncations $c^{(n)}$ and diagonal Gaussian measures $\gamma_{c^{(n)}}$ on $\mathbb{R}^{n}$ ($n\in\mathbb{N}$).

(Diagonal Gaussian measure)¶ The diagonal Gaussian measure on $X$ with variances $c$, written $\gamma_{c}$, is the unique $\mu\in\mathcal{P}(X)$ such that $(p_{n})_{\#}\mu=\gamma_{c^{(n)}}$ for every $n\in\mathbb{N}$; it exists and is unique 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.