The Theory of Stochastic Processes I pp 308-361 | Cite as

# Probability Measures on Functional Spaces

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## Abstract

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.

## Keywords

Hilbert Space Probability Measure Functional Space Gaussian Measure Sample Function## Preview

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