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 ${\displaystyle C_{\nu }=sC_{\mu }}$ 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 ${\displaystyle s=1.}$

See also

• Canonical Gaussian cylinder set measure – way to generate a measure over product spacesPages displaying wikidata descriptions as a fallback