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Invariant Measures on Coset Spaces

Published online by Cambridge University Press:  18 May 2009

S. Świerczkowski
S. Świerczkowski
Affiliation:
The University Glasgow
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In this note we consider measures on a left coset space G/H, where G is a locally compact group and H is a closed subgroup. We assume the natural topology in G/H and we denote the generic element of this space by xH (x∈G). Every element t∈G defines a homeomorphism of G/H given by t(xH) = (tx)H. A. Weil showed that a Baire measure on G/H invariant under all these homeomorphisms can exist only if

Δ(ξ) = δ(ξ) for each ξ ∈ H,

where Δ(x), δ(ξ) denote the modular functions in G, H [6, pp. 42–45]. We shall devote our investigations to inherited measures on G/H (cf. [3] and the definition below) invariant under homeomorphisms belonging to a normal and closed subgroup T ⊂ G.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 2 , July 1961 , pp. 80 - 85
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

1.Halmos, P. R., Measure theory (New York, 1951).Google Scholar
2.Loomis, L. H., An introduction to abstract harmonic analysis(New York, 1953).Google Scholar
3.Macbeath, A. M. and Świerczkowski, S., Inherited measures, Proc. Roy. Soc. Edinburgh A 65 (1961), 237–246.Google Scholar
4.Macbeath, A. M. and Świerczkowski, S., Measures in homogeneous spaces, Fund. Math. 49 (1960), 15–24.CrossRefGoogle Scholar
5.Macbeath, A. M. and Świerczkowski, S., Limits of lattices in a compactly generated group, Canadian J. Math. 12 (1960), 426–436.CrossRefGoogle Scholar
6.Weil, A., L'integration dans les groupes topologiques et ses applications (Paris, 1938).Google Scholar