Algebraic models for probability measures associated with stochastic processes

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.