probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of

Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...

s. It states that two Gaussian measures

\mu

and

\nu

on a

locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

X

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,

\mu

has a

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

with respect to

\nu

but not vice versa. In the special case that

X

is a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

, it is possible to give an explicit description of the circumstances under which

\mu

and

\nu

are equivalent: writing

m_

and

m_

for the means of

\mu

and

\nu,

and

C_\mu

and

C_\nu

for their

covariance operator In probability theory, for a probability measure P on a Hilbert space ''H'' with inner product \langle \cdot,\cdot\rangle , the covariance of P is the bilinear form Cov: ''H'' × ''H'' → R given by :\mathrm(x, y) = ...

s, equivalence of

\mu

and

\nu

holds if and only if (See Theorem 2.25) *

\mu

and

\nu

have the same Cameron–Martin space

H = C_\mu^(X) = C_\nu^(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^ C_\nu^) (C_\mu^ C_\nu^)^ - I

is a

Hilbert–Schmidt operator In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^ ...

\bar.

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 = s C_\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

{{DEFAULTSORT:Feldman-Hájek theorem Theorems in probability theory Theorems in measure theory

See also

References