Summary
For the two Markov processes associated with the application of a measurable transformation in a probability space in the forward and backward direction respectively, the equivalent descriptions by kernel functions and by Markov operators in L 1, L ∞, and in the space of absolutely continuous finite signed measures are identified. The connections between the conservative parts of the probability space with respect to these processes and the various conservative parts associated with a measurable transformation in the literature are clarified. Finally the inclusion relations between these various conservative parts are established.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Choksi, J. R.: Extension of a theorem of E. Hopf. J. London math. Soc. 36, 81–88 (1961).
Dean, D. W., and L. Sucheston: On invariant measures for operators. Z. Wahrscheinlichkeitstheorie verw. Geb. 6, 1–9 (1966).
Feldman, J.: Subinvariant measures for Markoff operators. Duke math. J. 29, 71–98 (1962).
Halmos, P. R.: Measure theory. New York: Van Nostrand 1956.
—: Lectures on ergodic theory. Publ. Math. Soc. Japan 3, Tokyo 1956. New York: Chelsea 1960.
Helmberg, G.: über die Zerlegung einer me\baren Transformation in konservative und dissipative Bestandteile. Math. Z. 88, 358–367 (1965).
—: über rein dissipative Transformationen. Math. Z. 90, 41–53 (1965).
—: über konservative Transformationen. Math. Ann. 165, 44–61 (1966).
Hopf, E.: Ergodentheorie. Ergebnisse der Mathematik und ihre Grenzgebiete V. Berlin: Springer 1937.
—: The general temporally discrete Markoff process. J. rat. Mech. Analysis 3, 13–54 (1954).
Natanson, J. R.: Theory of functions of a real variable. New York: Frederic Ungar 1955.
Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco: Holden-Day 1965.
Parry, W.: Note on the ergodic theorem of Hurewicz. J. London math. Soc. 39, 202–210 (1964).
Simons, F. H.: Ein weiterer Beweis eines Zerlegungsatzes für me\bare Transformationen. Math. Z. 89, 247–249 (1965).
Tsrurumi, S.: Note on an ergodic theorem. Proc. Japan Acad. 30, 419–423 (1954).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Helmberg, G., Simons, F.H. On the conservative parts of the Markov processes induced by a measurable transformation. Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 165–180 (1969). https://doi.org/10.1007/BF00536378
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00536378