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.

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.

References

• Dudley, Richard M.; Feldman, Jacob and Le Cam, Lucien (1971). "On seminorms and probabilities, and abstract Wiener spaces". Annals of Mathematics. Second Series 93: 390–408. ISSN 0003-486X.  MR0279272