Structure theorem for Gaussian measures

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.

Statement of the theorem

Let $\gamma$ 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 $\gamma ^{H}$ is the canonical Gaussian cylinder set measure on $H$.