On a theorem of Furstenberg and the structure of topologically ergodic measures

Authors:
Lewis Pakula and Robert Sine

Journal:
Proc. Amer. Math. Soc. **65** (1977), 52-56

MSC:
Primary 28A65; Secondary 60J05

DOI:
https://doi.org/10.1090/S0002-9939-1977-0507575-X

MathSciNet review:
0507575

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An almost everywhere convergence theorem for topologically ergodic measures stated by Furstenberg for homeomorphisms is extended to Markov operators on with compact Hausdörff state space. A structure theorem for topologically ergodic measures is obtained in the compact metric case again in the more general setting of Markov operators.

**[1]**E. M. Alfsen,*Compact convex sets and boundary integrals*, Springer-Verlag, Berlin and New York, 1971. MR**0445271 (56:3615)****[2]**S. R. Foguel,*The ergodic theory of positive operators on continuous functions*, Ann. Scuola Norm. Sup. Pisa.**27**(1973), 19-51. MR**0372154 (51:8370)****[3]**H. Furstenberg,*Stationary processes and prediction theory*, Princeton Univ. Press, Princeton, N.J., 1960. MR**0140151 (25:3573)****[4]**A. Garsia,*Topics in almost everywhere convergence*, Markham, Chicago, Ill., 1970. MR**0261253 (41:5869)****[5]**B. Jamison and R. C. Sine,*Sample path convergence for stable Markov operators*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**28**(1974), 173-177.**[6]**R. R. Phelps,*Lectures on Choquet's theorem*, Van Nostrand, Princeton, N.J., 1966. MR**0193470 (33:1690)****[7]**R. C. Sine,*Geometric theory of a single Markov operator*, Pacific J. Math.**27**(1968), 155-166. MR**0240281 (39:1630)****[8]**-,*Sample path convergence of stable Markov processes*. II, Indiana Univ. Math. J.**25**(1976), 23-44. MR**0391261 (52:12082)****[9]**B. Weiss,*Topological transitivity and ergodic measures*, Math. Systems Theory**5**(1971), 71-75. MR**0296928 (45:5987)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28A65,
60J05

Retrieve articles in all journals with MSC: 28A65, 60J05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0507575-X

Keywords:
Topologically ergodic measures,
Choquet representation

Article copyright:
© Copyright 1977
American Mathematical Society