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 and on a locally convex space 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, has a density with respect to but not vice versa. In the special case that is a Hilbert space, it is possible to give an explicit description of the circumstances under which and are equivalent: writing and for the means of and and and for their covariance operators, equivalence of and holds if and only if (See Theorem 2.25) and have the same Cameron–Martin space ; the difference in their means lies in this common Cameron–Martin space, i.e. ; and the operator is a Hilbert–Schmidt operator on
A simple consequence of the Feldman–Hájek theorem is that dilating a Gaussian measure on an infinite-dimensional Hilbert space (i.e. taking for some scale factor ) always yields two mutually singular Gaussian measures, except for the trivial dilation with since is Hilbert–Schmidt only when
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