By Van Der Merwe A. J., Du Plessis J. L.
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The random variables can be chosen to be inverses of each other; that is, if the two random variables are f and g, then f (g(λ)) = λ and g(f (ω)) = ω. Note that if the two probability spaces (Ω, B, P ) and (Λ, S, Q) are isomorphic and f : Ω → Λ is an isomorphism with inverse g, then the random variable f g defined by f g(λ) = f (g(λ)) is equivalent to the identity random variable i : Λ → Λ defined by i(λ) = λ. With the ideas of isomorphic measurable spaces and isomorphic probability spaces in hand, we now can define isomorphic dynamical systems.
Thus the process distribution indeed agrees with the finite dimensional distributions for all possible index subsets, completing the proof. ✷ By simply observing in the above proof that we only required the probability measure on the coordinate generating fields and that these fields are themselves standard, we can weaken the hypotheses of the theorem somewhat to obtain the following corollary. The details of the proof are left as an exercise. 1 Given standard measurable spaces (Ai , Bi ), i ∈ I, for a countable index set I, let Fi be corresponding generating fields possessing a basis.
Let ||P ||, P ∈ P be a matrix norm such as Pij2 )1/2 , ||P || = ( i,j the Euclidean norm. Then P is a normed linear space. 10: An inner product space A with inner product (·, ·) and distance d(a, b) = ||a−b||, where ||a|| = (a, a)1/2 is a norm. An inner product space (or pre-Hilbert space) is a linear vector space A such that for each pair of vectors a, b ∈ A there is a real number (a, b) called an inner product such that for a, b, c ∈ A, r ∈ (a, b) (a + b, c) = = (b, a) (a, c) + (b, c) (ra, b) (a, a) = ≥ r(a, b) 0 and (a, a) = 0 if and only if a = 0.
A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case by Van Der Merwe A. J., Du Plessis J. L.