View source for Sazonov's theorem

In [[mathematics]], '''Sazonov's theorem''', named after [[Vyacheslav Vasilievich Sazonov]] ({{lang|ru|Вячесла́в Васи́льевич Сазо́нов}}), is a [[theorem]] in [[functional analysis]].

It states that a [[bounded operator|bounded linear operator]] between two [[Hilbert space]]s is [[&gamma;-radonifying|''&gamma;''-radonifying]] if  it is a [[Hilbert–Schmidt operator]]. The result is also important in the study of [[stochastic processes]] and the [[Malliavin calculus]], since results concerning [[probability measure]]s on infinite-dimensional spaces are of central importance in these fields. Sazonov's theorem also has a converse: if the map is not Hilbert–Schmidt, then it is not ''γ''-radonifying.

==Statement of the theorem==

Let ''G'' and ''H'' be two Hilbert spaces and let ''T'' : ''G'' → ''H'' be a [[bounded operator]] from ''G'' to ''H''. Recall that ''T'' is said to be '''''γ''-radonifying''' if the [[Pushforward measure|push forward]] of the [[Cylinder set measure#Cylinder set measures on Hilbert spaces|canonical Gaussian cylinder set measure]] on ''G'' is a ''bona fide'' [[measure (mathematics)|measure]] on ''H''. Recall also that ''T'' is said to be a '''[[Hilbert–Schmidt operator]]''' if there is an [[orthonormal basis]] {{nowrap|{ ''e''<sub>''i''</sub> : ''i'' &isin; ''I'' }}} of ''G'' such that

: <math>\sum_{i \in I} \| T(e_i) \|_H^2 < + \infty.</math>

Then '''Sazonov's theorem''' is that ''T'' is ''&gamma;''-radonifying if it is a Hilbert–Schmidt operator.

The proof uses [[Prokhorov's theorem]].

==Remarks==

The canonical Gaussian [[cylinder set measure]] on an infinite-dimensional Hilbert space can never be a ''bona fide'' measure; equivalently, the [[identity function]] on such a space cannot be ''&gamma;''-radonifying.

==See also==

* {{annotated link|Cameron–Martin theorem}}
* {{annotated link|Girsanov theorem}}
* {{annotated link|Radonifying function}}
* {{annotated link|Minlos–Sazonov theorem}}

==References==

{{reflist}}

* {{citation|mr=0426084
|last=Schwartz|first= Laurent
|title=Radon measures on arbitrary topological spaces and cylindrical  measures. 
|series=Tata Institute of Fundamental Research Studies in Mathematics|issue= 6|publisher=  Oxford University Press |publication-place=London|year=  1973|pages= xii+393}}

{{Measure theory}}
{{Analysis in topological vector spaces}}
{{Functional analysis}}

[[Category:Stochastic processes]]
[[Category:Theorems in functional analysis]]
[[Category:Theorems in measure theory]]