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Algebraic Models for Probability Measures Associated with Stochastic Processes

B. M. Schreiber, T.-C. Sun and A. T. Bharucha-Reid
Transactions of the American Mathematical Society
Vol. 158, No. 1 (Jul., 1971), pp. 93-105
DOI: 10.2307/1995773
Stable URL: http://www.jstor.org/stable/1995773
Page Count: 13
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Algebraic Models for Probability Measures Associated with Stochastic Processes
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Abstract

This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foiaʂ notion of algebraic models for probability measures. The main result is a general extension theorem of Kolmogorov type which can be summarized as follows: Let $\{(X,\scr{A}_{i},\mu _{i}),i\in I\}$ be a directed family of probability measure spaces. Then there is an associated directed family of probability measure spaces $\{(G,\scr{B}_{i},\nu _{i}),i\in I\}$ and a probability measure ν on the σ-algebra B generated by the $\scr{B}_{i}$ such that (i) $\nu (B)=\nu _{i}(B),B\in \scr{B}_{i},i\in I$, and (ii) for each i ∊ I the spaces ($X,\scr{A}_{i},\mu _{i}$) and ($G,\scr{B}_{i},\nu _{i}$) are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding ψ: X → G such that the induced measures ν i on G are extendable to ν, although the measures μ i on X may not be extendable. Using the algebraic model formulation, the Kolmogorov extension property and the notion of a representation of a directed family of probability measure spaces are discussed.

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