Structure theorem for Gaussian measures
Encyclopedia
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
.
(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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the structure theorem for Gaussian measures shows that the abstract Wiener space
Abstract Wiener space
An abstract Wiener space is a mathematical object in measure theory, used to construct a "decent" measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener...
construction is essentially the only way to obtain a strictly positive
Strictly positive measure
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that it is zero "only on points".-Definition:...
Gaussian measure
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...
on a separable Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
. It was proved in the 1970s by Kallianpur–Sato–Stefan and Dudley
Richard M. Dudley
Richard Mansfield Dudley is Professor of Mathematics at the Massachusetts Institute of Technology. He received his PhD at Princeton University in 1962 under the supervision of Edward Nelson and Gilbert Hunt. He was a Putnam Fellow in 1958....
–Feldman–le Cam
Lucien le Cam
Lucien Marie Le Cam was a mathematician and statistician. He obtained a Ph.D. in 1952 at the University of California, Berkeley, was appointed Assistant Professor in 1953 and continued working there beyond his retirement in 1991 until his death.Le Cam was the major figure during the period 1950...
.
Statement of the theorem
Let γ be a strictly positive Gaussian measure on a separable Banach space (E, || ||). Then there exists a separable Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
(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
Canonical
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
Gaussian cylinder set measure
Cylinder set measure
In mathematics, cylinder set measure is a kind of prototype for a measure on an infinite-dimensional vector space...
on H.