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:^[1] 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^[2]

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

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)

[Bogachev-1] 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)

[DaPratoZabczyk-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)

[1]

[2]

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
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Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

See also

References