Structure theorem for Gaussian measures: Difference between revisions

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{{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 measure|strictly positive]] [[Gaussian measure]] on a [[separable space|separable]] [[Banach space]]. It was proved in [[1977]]the 1970s by [[Gopinath Kallianpur|Kallianpur]]-Sato-&ndash;Satô&ndash;Stefan and [[Richard M. Dudley|Dudley]]-&ndash;[[Jacob Feldman-|Feldman]]&ndash;[[Lucien_le_CamLucien le Cam|le Cam]].<!-- I don't see those papers cited here. -->

There is the earlier result due to H. Satô (1969) <ref>[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.</ref> which proves that "any Gaussian measure on a separable Banach space is an [[abstract Wiener space | abstract Wiener measure]] in the sense of [[Leonard Gross | 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 <math>\gamma</math>''γ'' be a [[Strictly positive measure|strictly positive]] Gaussian measure on a separable Banach space <math>(''E</math>'',&nbsp;||&nbsp;||). Then there exists a separable [[separable Hilbert space]] <math>(''H</math>'',&nbsp;&lang;&nbsp;,&nbsp;&rang;) and a map <math>''i ''&nbsp;: &nbsp;''H \to ''&nbsp;→&nbsp;''E</math>'' such that <math>''i ''&nbsp;: &nbsp;''H \to ''&nbsp;→&nbsp;''E</math>'' is an abstract Wiener space with ''γ''&nbsp;=&nbsp;''i''<mathsub>∗</sub>\gamma = i_{*} \left( \gamma^{''γ''<sup>''H} \right)''</mathsup>), where ''γ''<mathsup>\gamma^{''H}''</mathsup> is the [[canonical form|canonical]] Gaussian [[cylinder set measure]] on <math>''H</math>''.

==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&ndash;408

| issn = 0003-486X

| doi=10.2307/1970780

| jstor = 1970780 | mr=0279272}}

{{Analysis in topological vector spaces}}

{{Measure theory}}

{{Banach spaces}}

[[Category:MathematicalBanach theoremsspaces]]

[[Category:MeasureTheorems in probability theory]]

[[Category:Theorems in measure theory]]