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 μ {\displaystyle \mu } and ν {\displaystyle \nu } on a locally convex space X {\displaystyle X} are either equivalent measures or else mutually singular: there is no possibility of an intermediate situation in which, for example, μ {\displaystyle \mu } has a density with respect to ν {\displaystyle \nu } but not vice versa. In the special case that X {\displaystyle X} is a Hilbert space, it is possible to give an explicit description of the circumstances under which μ {\displaystyle \mu } and ν {\displaystyle \nu } are equivalent: writing m μ {\displaystyle m_{\mu }} and m ν {\displaystyle m_{\nu }} for the means of μ {\displaystyle \mu } and ν , {\displaystyle \nu ,} and C μ {\displaystyle C_{\mu }} and C ν {\displaystyle C_{\nu }} for their covariance operators, equivalence of μ {\displaystyle \mu } and ν {\displaystyle \nu } holds if and only if
μ {\displaystyle \mu } and ν {\displaystyle \nu } have the same Cameron–Martin space H = C μ 1 / 2 ( X ) = C ν 1 / 2 ( X ) {\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. m μ − m ν ∈ H {\displaystyle m_{\mu }-m_{\nu }\in H} ; and the operator ( C μ − 1 / 2 C ν 1 / 2 ) ( C μ − 1 / 2 C ν 1 / 2 ) ∗ − I {\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 H ¯ . {\displaystyle {\bar {H}}.}
A simple consequence of the Feldman–Hájek theorem is that dilating a Gaussian measure on an infinite-dimensional Hilbert space X {\displaystyle X} (i.e. taking C ν = s C μ {\displaystyle C_{\nu }=sC_{\mu }} for some scale factor s ≥ 0 {\displaystyle s\geq 0} ) always yields two mutually singular Gaussian measures, except for the trivial dilation with s = 1 , {\displaystyle s=1,} since ( s 2 − 1 ) I {\displaystyle (s^{2}-1)I} is Hilbert–Schmidt only when s = 1. {\displaystyle s=1.}
See also Canonical Gaussian cylinder set measure
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