## Algebraic models for probability measures associated with stochastic processes

HTML articles powered by AMS MathViewer

- by B. M. Schreiber, T.-C. Sun and A. T. Bharucha-Reid
- Trans. Amer. Math. Soc.
**158**(1971), 93-105 - DOI: https://doi.org/10.1090/S0002-9947-1971-0279844-1
- PDF | Request permission

## 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.## References

- Erik Sparre Andersen and Børge Jessen,
*On the introduction of measures in infinite product sets*, Danske Vid. Selsk. Mat.-Fys. Medd.**25**(1948), no. 4, 8. MR**27041** - Patrick Billingsley,
*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396** - Salomon Bochner,
*Harmonic analysis and the theory of probability*, University of California Press, Berkeley-Los Angeles, Calif., 1955. MR**0072370** - J. R. Choksi,
*Inverse limits of measure spaces*, Proc. London Math. Soc. (3)**8**(1958), 321–342. MR**96768**, DOI 10.1112/plms/s3-8.3.321 - N. Dinculeanu and C. Foiaş,
*Algebraic models for measures*, Illinois J. Math.**12**(1968), 340–351. MR**225958** - I. M. Gel′fand and G. E. Shilov,
*Generalized functions. Vol. 1*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR**0435831** - Paul R. Halmos,
*Lectures on Boolean algebras*, Van Nostrand Mathematical Studies, No. 1, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR**0167440** - A. Ionescu Tulcea and C. Ionescu Tulcea,
*Topics in the theory of lifting*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 48, Springer-Verlag New York, Inc., New York, 1969. MR**0276438** - J. L. Kelley and Isaac Namioka,
*Linear topological spaces*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR**0166578** - R. B. Kirk,
*Kolmogorov type consistency theorems for products of locally compact, $B$-compact spaces*, Nederl. Akad. Wetensch. Proc. Ser. A 73 = Indaf. Math.**32**(1970), 77–81. MR**0264019** - A. Kolmogoroff,
*Grundbegriffe der Wahrscheinlichkeitsrechnung*, Springer-Verlag, Berlin-New York, 1977 (German). Reprint of the 1933 original. MR**0494348** - Michel Métivier,
*Limites projectives de mesures. Martingales. Applications*, Ann. Mat. Pura Appl. (4)**63**(1963), 225–352 (French). MR**162908**, DOI 10.1007/BF02412184 - R. A. Minlos,
*Generalized random processes and their extension to a measure*, Selected Transl. Math. Statist. and Prob., Vol. 3, Amer. Math. Soc., Providence, R.I., 1963, pp. 291–313. MR**0154317** - Jacques Neveu,
*Bases mathématiques du calcul des probabilités*, Masson et Cie, Éditeurs, Paris, 1964 (French). MR**0198504** - K. R. Parthasarathy,
*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684** - Jean-Pierre Raoult,
*Limites projectives de mesures $\sigma$-finies et probabilités conditionnelles*, C. R. Acad. Sci. Paris**260**(1965), 4893–4896 (French). MR**182996** - Walter Rudin,
*Fourier analysis on groups*, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR**0152834** - Carel L. Scheffer,
*Limits of directed projective systems of probability spaces*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**13**(1969), 60–80. MR**263126**, DOI 10.1007/BF00535797 - I. E. Segal,
*Abstract probability spaces and a theorem of Kolmogoroff*, Amer. J. Math.**76**(1954), 721–732. MR**63602**, DOI 10.2307/2372714 - G. E. Šilov,
*Measures in linear spaces*, Dokl. Akad. Nauk SSSR**169**(1966), 46–48 (Russian). MR**0210861** - Antonín Špaček,
*Probability measures in infinite Cartesian products*, Illinois J. Math.**4**(1960), 210–220. MR**121817**

## Bibliographic Information

- © Copyright 1971 American Mathematical Society
- 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