Sazonov's theorem

In mathematics, Sazonov's theorem is a theorem in functional analysis. It states that a bounded linear operator between two Hilbert spaces is "γ"-radonifying if it is Hilbert-Schmidt. The result is also important in the study of stochastic processes and the Malliavin calculus, since results concerning probability measures 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.

tatement 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 push forward of the canonical Gaussian cylinder set measure on "G" is a "bona fide" measure on "H". Recall also that "T" is said to be Hilbert-Schmidt if there is an orthonormal basis { "e"_"i" | "i" ∈ "I" } of "G" such that:

Then Sazonov's theorem is that "T" is "γ"-radonifying if it is Hilbert-Schmidt.

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 "γ"-radonifying.

References

*citation|id=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