Structure theorem for Gaussian measures
In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kallianpur-Sato-Stefan and Dudley-Feldman-le Cam.
There is the earlier result due to H. Satô which proves that "any Gaussian measure on a separable Banach space is an abstract Wiener measure in the sense of L. Gross". The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general topological vector space.Let γ be a strictly positive Gaussian measure on a separable Banach space. Then there exists a separable Hilbert space and a map i : H → E such that i : H → E is an abstract Wiener space with γ = i∗, where γH is the canonical Gaussian cylinder set measure on H.
