Structure theorem for Gaussian measures

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In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is the only way to obtain a Gaussian measure on a separable Banach space. It was proved in 1977 by Kallianpur-Sato-Stefan and Dudley-Feldman-le Cam.

[edit] Statement of the theorem

Let $γ$ be a strictly positive Gaussian measure on a separable Banach space $E$ . Then there exists a separable Hilbert space $H$ and a map $i : H \to E$ such that $i : H \to E$ is an abstract Wiener space with $\gamma = i_{*} \left( \gamma^{H} \right)$ , where $γ H$ is the canonical Gaussian cylinder set measure on $H$ .