Probability Measures on Functional Spaces
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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.
KeywordsHilbert Space Probability Measure Functional Space Gaussian Measure Sample Function
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