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Some problems on idempotent measures on semigroups

Published online by Cambridge University Press:  17 April 2009

N. A. Tserpes  and
A. Mukherjea
N. A. Tserpes
Affiliation:
University of South Florida, Tampa.
A. Mukherjea
Affiliation:
University of South Florida, Tampa.
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Abstract

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Essentially this paper does the following: In Section 2 it gives necessary and sufficient conditions in order that the support of an idempotent measure on a locally compact semigroup S, be completely simple. In Section 3 it proves that if I is an ideal of S of positive measure μ (= any probability measure), then μn (I) strictly increases to the limit 1. If in addition μ is idempotent, then μ (N-1N) and μ(NN-1) are positive for any open set N. In Section 4 certain compactness conditions are proven equivalent to joint weak*–continuity of the convolution of bounded measures and a limit theorem concerning the convolution powers (Cesarò sums) of μ is proven.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 3 , June 1970 , pp. 299 - 315
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Argabright, L.N., “A note on invariant integrals on locally compact semigroups”, Proc. Amer. Math. Soc. 17 (1966), 377–382.CrossRefGoogle Scholar
[2]Berglund, J.F., Hofmann, K.H., Compact semitopological semigroups and weakly almost periodic functions (Lecture Notes in Mathematics, no. 42, Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[3]Chung, Kai Lai, “The general theory of Markov processes according to Doeblin”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 230–254.CrossRefGoogle Scholar
[4]Harris, T.E., “The existence of stationary measures for certain Markov processes”, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability II (1954–1955), 113–124; (University of California Press, Berkeley, Los Angeles, 1956).Google Scholar
[5]Heble, M. and Rosenblatt, M., “Idempotent measures on a compact topological semigroup”, Proc. Amer. Math. Soc. 14 (1963), 177–184.CrossRefGoogle Scholar
[6]Kartsatos, A. and Tserpes, N.A., “A sufficient condition for the support of a measure to be a left group”, Boll. Un. Mat. Ital. (4) 1 (1968), 538–539.Google Scholar
[7]Pym, J.S., “Idempotent measures on semigroups”, Pacific J. Math. 12 (1962), 685–698.CrossRefGoogle Scholar
[8]Rosenblatt, M., “Limits of convolution sequences of measures on a compact topological semigroup”, J. Math. Mech. 9 (1960), 293–305.Google Scholar
[9]Sun, T.C. and Tserpes, N.A., “Idempotent measures on locally compact semigroups”, (to appear).Google Scholar
[10]Sun, T.C., “Random walks on topological semigroups”, (unpublished note).Google Scholar
[11]Tortrat, A., “Lois tendues μ sur un demi-groupe topologique complètement simple X”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 (1966), 145–160.CrossRefGoogle Scholar
[12]Tserpes, Nicolas et Kartsatos, Athanasius, “Mesures semi-invariantes sur un semi-groupe localment compact”, C.R. Acad. Sci. Paris Sér. A–B 267 (1968), A507–A509.Google Scholar
[13]Williamson, J.H., “Harmonic analysis on semigroups”, J. London Math. Soc. 42 (1967), 1–41.CrossRefGoogle Scholar