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Measure in Semigroups

Published online by Cambridge University Press:  20 November 2018

B. R. Gelbaum  and
G. K. Kalisch
B. R. Gelbaum
Affiliation:
University of Minnesota
G. K. Kalisch
Affiliation:
University of Minnesota
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The major portion of this paper is devoted to an investigation of the conditions which imply that a semigroup (no identity or commutativity assumed) with a bounded invariant measure is a group. We find in §3 that a weakened form of “shearing” is sufficient and a counter-example (§5) shows that “shearing” may not be dispensed with entirely. In §4 we discuss topological measures in locally compact semigroups and find that shearing may be dropped without affecting the results of the earlier sections (Theorem 2). The next two theorems show that under certain circumstances (shearing or commutativity) the topology of the semigroup (already known to be a group by virtue of earlier results) can be weakened so that the structure becomes a separated compact topological group. The last section treats the problem of extending an invariant measure on a commutative semigroup to an invariant measure on its quotient structure.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 396 - 406
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Gelbaum, B. R., Kalisch, G. K., and Olmsted, J.M.H., On the embedding of topological semigroups and integral domains, Proc. Amer. Math. Soc, vol. 2 (1951), 807–821.Google Scholar
2. Halmos, P. R., Measure theory (New York, 1950).Google Scholar
3. Montgomery, D., Continuity in topological groups, Bull. Amer. Math. Soc, vol. 42 (1936), 879–882.Google Scholar
4. Weil, A., L'intégration dans les groupes topologiques et ses applications (Paris, 1938).Google Scholar