In mathematics, the structure theorem for Gaussian measures shows that the

abstract Wiener space The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Camer ...

construction is essentially the only way to obtain a strictly positive

Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named ...

on a separable

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

. It was proved in the 1970s by Kallianpur–Sato–Stefan and

Dudley Dudley is a large market town and administrative centre in the county of West Midlands, England, southeast of Wolverhampton and northwest of Birmingham. Historically an exclave of Worcestershire, the town is the administrative centre of the ...

–

Feldman Feldman is a German and Ashkenazi Jewish surname. Notable people with the surname include: Academics * Arthur Feldman (born 1949), American cardiologist * David B. Feldman, American psychologist * David Feldman (historian), American historian * ...

– le Cam. There is the earlier result due to H. Satô (1969) 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...

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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natura ...

(''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 The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...

Gaussian

cylinder set measure In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder ...

on ''H''.

References

* {{Banach spaces Banach spaces Probability theorems Theorems in measure theory