Haar measure for compact right topological groups

Authors:
Paul Milnes and John Pym

Journal:
Proc. Amer. Math. Soc. **114** (1992), 387-393

MSC:
Primary 22C05; Secondary 28C10, 43A05

MathSciNet review:
1065088

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Abstract: Compact right topological groups arise in topological dynamics and in other settings. Following H. Furstenberg's seminal work on distal flows, R. Ellis and I. Namioka have shown that the compact right topological groups of dynamical type always admit a probability measure invariant under the continuous left translations; however, this invariance property is insufficient to identify a unique probability measure (in contrast to the case of compact topological groups). In the present paper, we amplify on the proofs of Ellis and Namioka to show that a right invariant probability measure on the compact right topological group exists provided admits an appropriate system of normal subgroups, that it is uniquely determined and that it is also invariant under the continuous left translations. Using Namioka's work, we show that has such a system of subgroups if its topological centre contains a countable dense subset, or if it is a closed subgroup of such a group.

**[1]**Joseph Auslander,*Minimal flows and their extensions*, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR**956049****[2]**L. Auslander and F. Hahn,*Real functions coming from flows on compact spaces and concepts of almost periodicity*, Trans. Amer. Math. Soc.**106**(1963), 415–426. MR**0144325**, 10.1090/S0002-9947-1963-0144325-8**[3]**J. F. Berglund, H. D. Junghenn, and P. Milnes,*Compact right topological semigroups and generalizations of almost periodicity*, Lecture Notes in Mathematics, vol. 663, Springer, Berlin, 1978. MR**513591****[4]**John F. Berglund, Hugo D. Junghenn, and Paul Milnes,*Analysis on semigroups*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR**999922****[5]**Robert Ellis,*Locally compact transformation groups*, Duke Math. J.**24**(1957), 119–125. MR**0088674****[6]**Robert Ellis,*Lectures on topological dynamics*, W. A. Benjamin, Inc., New York, 1969. MR**0267561****[7]**Robert Ellis,*The Furstenberg structure theorem*, Pacific J. Math.**76**(1978), no. 2, 345–349. MR**0487995****[8]**H. Furstenberg,*The structure of distal flows*, Amer. J. Math.**85**(1963), 477–515. MR**0157368****[9]**I. Namioka,*Right topological groups, distal flows, and a fixed-point theorem*, Math. Systems Theory**6**(1972), 193–209. MR**0316619**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1065088-1

Keywords:
Compact right topological group,
strong normal system of subgroups,
invariant probability measure

Article copyright:
© Copyright 1992
American Mathematical Society