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

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 $$s = 1.$$