| 词条 | Structure theorem for Gaussian measures |
| 释义 |
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) [1] 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 theoremLet 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. 1. ^H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure, 1969. References
| 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. Second Series | volume = 93 | year = 1971 | pages = 390–408 | issn = 0003-486X | doi=10.2307/1970780}} {{MathSciNet|id=0279272}} 2 : Theorems in measure theory|Probability theorems |
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