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Quasi-invariant Radon measures on groups

Author: Chandra Gowrisankaran
Journal: Proc. Amer. Math. Soc. 35 (1972), 503-506
MSC: Primary 43A05; Secondary 28A40, 46G05
MathSciNet review: 0318777
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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.

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  • [1] R. H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623-627. MR 16, 375. MR 0065033 (16:375b)
  • [2] J. Feldman, Nonexistence of quasi-invariant measures on infinite-dimensional linear spaces, Proc. Amer. Math. Soc. 17 (1966), 142-146. MR 32 #7705. MR 0190292 (32:7705)
  • [3] C. Gowrisankaran, Radon measures on groups, Proc. Amer. Math. Soc. 25 (1970), 381-384. MR 41 #384. MR 0255724 (41:384)
  • [4] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I. Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [5] J. C. Oxtoby, Invariant measures in groups which are not locally compact, Trans. Amer. Math. Soc. 60 (1946), 215-237. MR 8, 253. MR 0018188 (8:253d)
  • [6] L. Schwartz, Radon measures on general topological spaces, Tata Institute of Fundamental Research Monographs (to appear). MR 0426084 (54:14030)
  • [7] V. N. Sudakov, Linear sets with quasi-invariant measure, Dokl. Akad. Nauk SSSR 127 (1959), 524-525. (Russian) MR 21 #6412. MR 0107689 (21:6412)

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Keywords: Topological group, quasi-invariant, Radon measure, locally compact, Baire space, vector space, Wiener algebra
Article copyright: © Copyright 1972 American Mathematical Society

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