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Measures equivalent to the Haar measure

Published online by Cambridge University Press:  18 May 2009

S. Świerczkowski
S. Świerczkowski
Affiliation:
The University Glasgow
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We call two measures equivalent if each is absolutely continuous with respect to the other (cf. [1]). Let G be a locally compact topological group and let μ be a non-negative Baire measure on G (i.e. μ is denned on all Baire sets, finite on compact sets and positive on open sets). We say that μ is stable if μ (E)=0 implies μ(tE)=0 for each t ∈ G. A. M. Macbeath made the conjecture that every stable non-trivial Baire measure is equivalent to the Haar measure. In this paper we prove the following slightly stronger result:

Theorem. Every stable non-trivial measure defined on Baire sets and finite on some open set is equivalent to the Haar measure.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 4 , September 1960 , pp. 157 - 162
Copyright
Copyright © Glasgow Mathematical Journal Trust 1960

References

1.Halmos, P. R., Measure theory (New York, 1951).Google Scholar
2.Macbeath, A. M. and Świerczkowski, S., Measures in homogeneous spaces, Fundamenta Math. 49 (1960), 15–24.Google Scholar