In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov (), is a theorem in functional analysis.
It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if it is a HilbertâÂÂSchmidt operator. 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 HilbertâÂÂSchmidt, then it is not ó-radonifying.
Let G and H be two Hilbert spaces and let T : G â H be a bounded operator from G to H. Recall that T is said to be ó-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 a HilbertâÂÂSchmidt operator if there is an orthonormal basis } of G such that
Then Sazonov's theorem is that T is γ-radonifying if it is a HilbertâÂÂSchmidt operator.
The proof uses Prokhorov's theorem.
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.