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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

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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
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6.Weil, A., L'integration dans les groupes topologiques et ses applications (Paris, 1938).Google Scholar