Feldman–Hájek theorem

Remarks

Here $Q^{-1/2}$ is the inverse of $Q^{1/2}$ on its image, and $Q^{1/2}(H)$ is the Cameron–Martin space of $\mu$. When $Q = R$, the theorem reduces to the Cameron–Martin theorem: $N(a, Q)$ and $N(b, Q)$ are equivalent exactly when $a - b$ lies in the Cameron–Martin space. The proof compares the two measures along an orthonormal basis and applies Kakutani's theorem on infinite products of measures, which also has an all-or-nothing form.

Example: on $\R^n$, two Gaussian measures with invertible covariance matrices are always equivalent, since both have everywhere positive densities. Condition (c) is automatic in finite dimensions.

Example: on an infinite-dimensional $H$ with $Q$ injective, $N(0, Q)$ and $N(0, c Q)$ with $c > 0$, $c \ne 1$, are mutually singular. Indeed the operator in (c) is $(c - 1) I$, which is not Hilbert–Schmidt.

Going deeper

For a student

The dichotomy is a basic fact for statistics of Gaussian processes. If two models give singular laws, one can tell them apart without error from a single infinite-dimensional observation. For example, the variance parameter of Brownian motion can be recovered exactly from one path, which matches the scaling example above.