Structure theorem for Gaussian measures
{{Short description|Mathematical theorem}}
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 (1969) [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118797795 H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure], 1969. 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.
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∗(γH), where γH is the canonical Gaussian cylinder set measure on H.
References
{{reflist}}
- {{cite journal
| last = Dudley
| first = Richard M. |author2=Feldman, Jacob |author3=Le Cam, Lucien
| title = On seminorms and probabilities, and abstract Wiener spaces
| journal = Annals of Mathematics |series=Second Series
| volume = 93
| year = 1971
| issue = 2 | pages = 390–408
| issn = 0003-486X
| doi=10.2307/1970780
| jstor = 1970780 | mr=0279272}}
{{Analysis in topological vector spaces}}
{{Measure theory}}
{{Banach spaces}}