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 $\mu$ and $\nu$ on a locally convex space $X$ are either equivalent measures or else mutually singular: (See Theorem 2.7.2) there is no possibility of an intermediate situation in which, for example, $\mu$ has a density with respect to $\nu$ but not vice versa. In the special case that $X$ is a Hilbert space, it is possible to give an explicit description of the circumstances under which $\mu$ and $\nu$ are equivalent: writing $m_$ and $m_$ for the means of $\mu$ and $\nu,$ and $C_\mu$ and $C_\nu$ for their covariance operators, equivalence of $\mu$ and $\nu$ holds if and only if (See Theorem 2.25)
* $\mu$ and $\nu$ have the same Cameron–Martin space $H = C_\mu^(X) = C_\nu^(X)$;
* the difference in their means lies in this common Cameron–Martin space, i.e. $m_\mu - m_\nu \in H$; and
* the operator $(C_\mu^ C_\nu^) (C_\mu^ C_\nu^)^ - I$ is a Hilbert–Schmidt operator on $\bar.$

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

See also

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References

{{DEFAULTSORT:Feldman-Hájek theorem

Probability theorems
Theorems in measure theory