# Feldman–Hájek theorem

In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures $\displaystyle{ \mu }$ and $\displaystyle{ \nu }$ on a locally convex space $\displaystyle{ X }$ are either equivalent measures or else mutually singular:[1] there is no possibility of an intermediate situation in which, for example, $\displaystyle{ \mu }$ has a density with respect to $\displaystyle{ \nu }$ but not vice versa. In the special case that $\displaystyle{ X }$ is a Hilbert space, it is possible to give an explicit description of the circumstances under which $\displaystyle{ \mu }$ and $\displaystyle{ \nu }$ are equivalent: writing $\displaystyle{ m_{\mu} }$ and $\displaystyle{ m_{\nu} }$ for the means of $\displaystyle{ \mu }$ and $\displaystyle{ \nu }$, and $\displaystyle{ C_\mu }$ and $\displaystyle{ C_\nu }$ for their covariance operators, equivalence of $\displaystyle{ \mu }$ and $\displaystyle{ \nu }$ holds if and only if[2]

• $\displaystyle{ \mu }$ and $\displaystyle{ \nu }$ have the same Cameron–Martin space $\displaystyle{ H = C_{\mu}^{1/2}(X) = C_{\nu}^{1/2}(X) }$;
• the difference in their means lies in this common Cameron–Martin space, i.e. $\displaystyle{ m_{\mu} - m_{\nu} \in H }$; and
• the operator $\displaystyle{ (C_{\mu}^{-1/2} C_{\nu}^{1/2}) (C_{\mu}^{-1/2} C_{\nu}^{1/2})^{\ast} - I }$ is a Hilbert–Schmidt operator on $\displaystyle{ \bar{H} }$.

A simple consequence of the Feldman–Hájek theorem is that dilating a Gaussian measure on an infinite-dimensional Hilbert space $\displaystyle{ X }$ (i.e. taking $\displaystyle{ C_{\nu} = s C_{\mu} }$ for some scale factor $\displaystyle{ s \geq 0 }$) always yields two mutually singular Gaussian measures, except for the trivial dilation with $\displaystyle{ s = 1 }$, since $\displaystyle{ (s^{2} - 1) I }$ is Hilbert–Schmidt only when $\displaystyle{ s = 1 }$.

## References

1. Bogachev, Vladimir I. (1998). Gaussian Measures. Mathematical Surveys and Monographs. 62. Providence, RI: American Mathematical Society. doi:10.1090/surv/062. ISBN 0-8218-1054-5.  (See Theorem 2.7.2)
2. Da Prato, Giuseppe; Zabczyk, Jerzy (2014). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. 152 (Second ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107295513. ISBN 978-1-107-05584-1.  (See Theorem 2.25)