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 and on a locally convex space are either equivalent measures or else mutually singular:[1] 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[2]
- 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
See also
- Canonical Gaussian cylinder set measure – way to generate a measure over product spacesPages displaying wikidata descriptions as a fallback
References
- ^ Bogachev, Vladimir I. (1998). Gaussian Measures. Mathematical Surveys and Monographs. Vol. 62. Providence, RI: American Mathematical Society. doi:10.1090/surv/062. ISBN 0-8218-1054-5. (See Theorem 2.7.2)
- ^ Da Prato, Giuseppe; Zabczyk, Jerzy (2014). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Vol. 152 (Second ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107295513. ISBN 978-1-107-05584-1. (See Theorem 2.25)
- v
- t
- e
- Absolute continuity of measures
- Lebesgue integration
- Lp spaces
- Measure
- Measure space
- Probability space
- Measurable space/function
- Almost everywhere
- Atom
- Baire set
- Borel set
- Borel space
- Carathéodory's criterion
- Cylindrical σ-algebra
- 𝜆-system
- Essential range
- Locally measurable
- π-system
- σ-algebra
- Non-measurable set
- Null set
- Support
- Transverse measure
- Universally measurable
- Atomic
- Baire
- Banach
- Besov
- Borel
- Brown
- Complex
- Complete
- Content
- (Logarithmically) Convex
- Decomposable
- Discrete
- Equivalent
- Finite
- Inner
- (Quasi-) Invariant
- Locally finite
- Maximising
- Metric outer
- Outer
- Perfect
- Pre-measure
- (Sub-) Probability
- Projection-valued
- Radon
- Random
- Regular
- Saturated
- Set function
- σ-finite
- s-finite
- Signed
- Singular
- Spectral
- Strictly positive
- Tight
- Vector
- Carathéodory's extension theorem
- Convergence theorems
- Decomposition theorems
- Egorov's
- Fatou's lemma
- Fubini's
- Hölder's inequality
- Minkowski inequality
- Radon–Nikodym
- Riesz–Markov–Kakutani representation theorem
For Lebesgue measure |
---|