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 essentially the only way to obtain a strictly positive 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 → E such that i : H → E is an abstract Wiener space with γ = i_∗(γ^H), where γ^H is the canonical Gaussian cylinder set measure on H.