Quasi-invariant Radon measures on groups

Gowrisankaran, Chandra

doi:10.1090/S0002-9939-1972-0318777-5

Quasi-invariant Radon measures on groups
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by Chandra Gowrisankaran

Proc. Amer. Math. Soc. 35 (1972), 503-506

DOI: https://doi.org/10.1090/S0002-9939-1972-0318777-5

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Abstract:

Let G be a Hausdorff topological group which is a Baire space. It is proved that if there is a quasi-invariant Radon measure on G then G is locally compact. Examples of non-Baire groups with and without quasi-invariant measures are considered. In particular, it is shown that there is no $\sigma$-finite measure on the Wiener space which preserves sets of measure zero under translation.

References

R. H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623–627. MR 65033
Jacob Feldman, Nonexistence of quasi-invariant measures on infinite-dimensional linear spaces, Proc. Amer. Math. Soc. 17 (1966), 142–146. MR 190292, DOI 10.1090/S0002-9939-1966-0190292-9
Chandra Gowrisankaran, Radon measures on groups, Proc. Amer. Math. Soc. 25 (1970), 381–384. MR 255724, DOI 10.1090/S0002-9939-1970-0255724-7
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
John C. Oxtoby, Invariant measures in groups which are not locally compact, Trans. Amer. Math. Soc. 60 (1946), 215–237. MR 18188, DOI 10.1090/S0002-9947-1946-0018188-5
Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR 0426084
V. N. Sudakov, Linear sets with quasi-invariant measure, Dokl. Akad. Nauk SSSR 127 (1959), 524–525 (Russian). MR 0107689