---
tag: 4TXG13F5N0
kind: statement
role: theorem
name: Feldman–Hájek theorem
msc: [28C20, 60G15, 60G30]
uses: [P0DV2TCH1N, HM2HHE3J51, 0NEGN5CRSC, Z2CP530WDE, P0P21KDV6E, EBGECZNCV5]
about: [P0DV2TCH1N]
sources:
  - url: https://en.wikipedia.org/wiki/Feldman%E2%80%93H%C3%A1jek_theorem
same_as:
  wikidata: Q60794154
  wikipedia: Feldman–Hájek theorem
---

## Statement

Let $H$ be a real separable [[Hilbert space|HM2HHE3J51]].
Let $\mu = N(a, Q)$ and $\nu = N(b, R)$ be [[Gaussian measures|P0DV2TCH1N]] on $H$ with means $a, b \in H$ and covariance operators $Q, R$.
Here $Q$ and $R$ are positive [[self-adjoint|0NEGN5CRSC]] [[trace class|Z2CP530WDE]] operators on $H$.

1. Either $\mu$ and $\nu$ are equivalent, that is, each is [[absolutely continuous|P0P21KDV6E]] with respect to the other, or they are [[mutually singular|EBGECZNCV5]].
2. They are equivalent if and only if the following three conditions hold:
   (a) $Q^{1/2}(H) = R^{1/2}(H)$; call this space $H_0$.
   (b) $a - b \in H_0$.
   (c) The operator $(Q^{-1/2} R^{1/2})(Q^{-1/2} R^{1/2})^* - I$ is a Hilbert–Schmidt operator on the closure of $H_0$.

## 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.

## Explanations

### L1 human

In finite dimensions, two non-degenerate Gaussian laws have the same null sets.
In infinite dimensions this is false, and two Gaussian measures either have exactly the same null sets or live on disjoint sets.
The theorem says precisely when the first case happens.

### L2 human

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.