Structure theorem for Gaussian measures
- 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 1977 by Kallianpur-Sato-Stefan and Dudley-Feldman-le Cam.
tatement 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".
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