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Transactions of the American Mathematical Society

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Algebraic models for probability measures associated with stochastic processes


Authors: B. M. Schreiber, T.-C. Sun and A. T. Bharucha-Reid
Journal: Trans. Amer. Math. Soc. 158 (1971), 93-105
MSC: Primary 60.05
DOI: https://doi.org/10.1090/S0002-9947-1971-0279844-1
MathSciNet review: 0279844
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Abstract: This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foiaş 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,{\mathcal{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,{\mathcal{B}_i},{v_i}),i \in I\} $ and a probability measure $ v$ on the $ \sigma $-algebra $ \mathcal{B}$ generated by the $ {\mathcal{B}_i}$ such that (i) $ v(B) = {v_i}(B),B \in {\mathcal{B}_i},i \in I$, and (ii) for each is $ i \in I$ the spaces $ (X,{\mathcal{A}_i},{\mu _i})$ and $ (G,{\mathcal{B}_i},{v_i})$ are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding $ \psi :X \to G$ such that the induced measures $ {v_i}$ on $ G$ are extendable to $ v$, although the measures $ {\mu _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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0279844-1
Keywords: Stochastic processes, probability measures, algebraic models for probability measures, extension of probability measures, directed families of probability measure spaces, positive-definite functions, lifting of measure spaces
Article copyright: © Copyright 1971 American Mathematical Society

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