Metric invariance of Haar measure

Bandt, Christoph

doi:10.1090/S0002-9939-1983-0677233-2

Metric invariance of Haar measure

Author: Christoph Bandt
Journal: Proc. Amer. Math. Soc. 87 (1983), 65-69
MSC: Primary 43A05; Secondary 28C10
DOI: https://doi.org/10.1090/S0002-9939-1983-0677233-2
MathSciNet review: 677233
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Abstract: Let be a left invariant metric for a locally compact group . We prove that isometric subsets of have equal Haar measure.

[1] Christoph Bandt, Many measures are Hausdorff measures, Bull. Polish Acad. Sci. Math. 31 (1983), no. 3-4, 107–114 (English, with Russian summary). MR 742794
[2] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
[3] James Fickett and Jan Mycielski, A problem of invariance for Lebesgue measure, Colloq. Math. 42 (1979), 123–125. MR 567553, https://doi.org/10.4064/cm-42-1-123-125
[4] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
[5] F. Hausdorff, Dimension und äusseres Mass, Math. Ann. 79 (1919), 157-179.
[6] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. 1, Springer-Verlag, Berlin and New York, 1963.
[7] D. Kahnert, Haar-Mass und Hausdorff-Mass, Measure theory (Proc. Conf., Oberwolfach, 1975) Springer, Berlin, 1976, pp. 13–23. Lecture Notes in Math., Vol. 541 (German). MR 0450523
[8] Jan Mycielski, Remarks on invariant measures in metric spaces, Colloq. Math. 32 (1974), 105–112, 150. MR 361005, https://doi.org/10.4064/cm-32-1-105-112
[9] Jan Mycielski, A conjecture of Ulam on the invariance of measure in Hilbert’s cube, Studia Math. 60 (1977), no. 1, 1–10. MR 430215, https://doi.org/10.4064/sm-60-1-1-10
[10] Leopoldo Nachbin, The Haar integral, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0175995
[11] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
[12] C. A. Rogers, Packing and covering, Cambridge Tracts in Mathematics and Mathematical Physics, No. 54, Cambridge University Press, New York, 1964. MR 0172183
[13] A. Schrijver (ed.), Packing and covering in combinatorics, Mathematical Centre Tracts, vol. 106, Mathematisch Centrum, Amsterdam, 1979. MR 562282
[14] Helmut Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume, J. Reine Angew. Math. 246 (1971), 46–75 (German). MR 273585, https://doi.org/10.1515/crll.1971.246.46