Sazonov's theorem

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 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 Hilbert–Schmidt operator if there is an orthonormal basis { e**i : i ∈ I} of G such that

: \sum_{i \in I} | T(e_i) |_H^2

Then Sazonov's theorem is that T is γ-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 γ-radonifying.

References

• {{citation|mr=0426084