Probability Measures on Functional Spaces

Part of the Classics in Mathematics book series


Kolmogorov’s theorem on the construction of a probability space from finite-dimensional distributions of a random process with values in a metric space ℋ shows, in particular, how to construct a measure μ on a measurable space (ℱ,ℬ) — where ℱis the space of all the functions with values in ℋ and ℬ is the minimal σ-algebra containing all the cylinders in ℱ,such that, for any cylinder C, the value μ(C) coincides with the probability that the sample function of the random process belongs to C. This measure is called the measure associated with (or corresponding to) the random process ξ(t)and it can always be constructed irrespective of the probability space on which the process ξ(t) is defined.


Hilbert Space Probability Measure Functional Space Gaussian Measure Sample Function 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsAcademy of Sciences of the Ukrainian SSRDonetskUSSR
  2. 2.Institute of MathematicsAcademy of Sciences of the Ukrainian SSRKievUSSR

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