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