On ergodic sequences of measures

Authors:
J. R. Blum and R. Cogburn

Journal:
Proc. Amer. Math. Soc. **51** (1975), 359-365

MSC:
Primary 43A05; Secondary 22D40

MathSciNet review:
0372529

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Abstract: Let be the group of integers and its Bohr compactification. A sequence of probability measures defined on is said to be ergodic provided converges weakly to , the Haar measure on . Let and define by where is the cardinality of . Then it is easy to show that if for every , then is ergodic. Let . In this paper we construct (random) sequences which are ergodic, and such that , for every .

**[1]**Julius Blum and Bennett Eisenberg,*Generalized summing sequences and the mean ergodic theorem*, Proc. Amer. Math. Soc.**42**(1974), 423–429. MR**0330412**, 10.1090/S0002-9939-1974-0330412-0**[2]**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. I*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR**551496****[3]**H. Niederreiter,*On a paper of Blum, Eisenberg and Hahn concerning ergodic theory and the distribution of sequences in the Bohr group*, Acta. Sci. Math. (to appear).**[4]**Herbert Robbins,*On the equidistribution of sums of independent random variables*, Proc. Amer. Math. Soc.**4**(1953), 786–799. MR**0056869**, 10.1090/S0002-9939-1953-0056869-7

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1975-0372529-1

Article copyright:
© Copyright 1975
American Mathematical Society