Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the measure theoretic structure
of compact groups


Authors: S. Grekas and S. Mercourakis
Journal: Trans. Amer. Math. Soc. 350 (1998), 2779-2796
MSC (1991): Primary 22C05, 28A35; Secondary 43A05
DOI: https://doi.org/10.1090/S0002-9947-98-02182-5
MathSciNet review: 1473441
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $G$ is a compact group with $w(G)=a\geq \omega$, we show the following results:

(i)
There exist direct products $\displaystyle{\prod _{\xi<a}G_{\xi},\ \prod _{\xi<a}H_{\xi}}$ of compact metric groups and continuous open surjections $\displaystyle{\prod _{\xi<a}G_{\xi} \stackrel{p}{\rightarrow }G \stackrel{q}{\rightarrow }\prod _{\xi<a}H_{\xi}}$ with respect to Haar measure; and
(ii)
the Haar measure on $G$ is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups.
Applications of the above results in measure theory are given.


References [Enhancements On Off] (What's this?)

  • [B] N. Bourbaki, Integration, Ch. 8, Hermann, Paris, 1959-1967. MR 31:3539
  • [Ch] M.M. Choban, Baire isomorphisms and Baire topologies. Solution of a problem of Comfort, Soviet Math. Dokl., 30 (1984), 780-784. MR 87d:54029
  • [C-F] J.R. Choksi and D.H. Fremlin, Completion regular measures on product spaces, Math. Ann. 241 (1979), 113-128. MR 81m:28004
  • [Cl-Mo$_1$] Joan Cleary and Sidney A. Morris, Locally dyadic topological groups, Bull. Austral. Math. Soc., 40 (1989), 417-419. MR 90m:22005
  • [Cl-Mo$2$] Joan Cleary and Sidney A. Morris, Compact groups and products of the unit interval, Math. Proc. Camb. Phil. Soc. 110 (1991), 293-297. MR 92k:22005
  • [Co] W.W. Comfort, Topological groups. In: K. Kunen, J.E. Vaughan (eds.) Handbook of Set-theoretic Topology, Ch. 24, pp. 1143-1263, Amsterdam: North-Holand 1984. MR 86g:22001
  • [Co-Ho-Re] W.W. Comfort, K.H. Hofmann and D. Remus, Topological Groups and Semigroups, Recent Progress in General Topology, M. Husek and J. Van Mill (eds.), Elsevier Sciences Publishers, 1992. CMP 93:15
  • [Co-Re] W.W. Comfort and Dieter Remus, Pseudocompact Refinements of Compact Group Topologies, Math. Z. 215 (1994), 337-346. MR 95f:54035
  • [D] J. Dieudonne, Treatise on Analysis, Vol II, Academic Press, New York, 1976. MR 58:26627
  • [Di] J. Dixmier, Sur certains espaces consideres par M.H. Stone, Sum. Bras. Math. 2 (1951), 151-181. MR 14:69e
  • [E] D.A. Edwards, On independent group characters, Bull. Amer. Math. Soc., 65 (1959), 352-354. MR 21:5692
  • [F-G] D.H. Fremlin and S. Grekas, Products of completion regular measures, Fund. Math. 147 (1995) N. 1, 27-37. MR 96e:28003
  • [G$_1$] S. Grekas, Isomorphic measures on compact groups, Math. Proc. Camb. Phil. Soc., 112 (1992), 349-360 (and a corrigendum in Math. Proc. Camb. Phil. Soc., 1994).
  • [G$_2$] S. Grekas, Structural properties of compact groups with measure theoretic applications, Israel J. Math. 87 (1994), 89-95. MR 95g:28025
  • [G$_3$] S. Grekas, Measure-theoretic problems in topological dynamics, J. d'Analyse Math. 65 (1995), 207-220. MR 96e:54003
  • [Gr] C. Gryllakis, Product of completion regular measures, Proc. Amer. Math. Soc. 103 (1988), 563-568. MR 89d:28010
  • [H-R] Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis I, Spinger-Verlag 1963. MR 28:158
  • [I-T] A. & C. Ionescu Tulcea, Topics in the theory of lifting, Springer-Verlag, Berlin - Heidelberg - New York 1969. MR 43:2185
  • [J] R.A. Johnson, Disintegrating measures on compact group extensions, A. Wahr. Verw. Gebiete, 53 (1980), 271-281. MR 81k:28009
  • [K-N] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, J. Wiley, New York 1974. MR 54:2612
  • [Kuz] V. Kuz'minov, On a hypothesis of P.S. Alexandroff in the theory of topological groups, Dokl. Akad. Nauk, SSSR, 125 (1959), 727-729. MR 21:3506
  • [Ku] J. Kupka, Strong liftings with application to measurable cross sections in locally compact groups, Israel J. Math. Vol. 44, No 3 (1983), 243-261. MR 84g:28006
  • [L$_1$] V. Losert, On the existence of uniformly distributed sequences in compact topological spaces I, Trans. Amer. Soc. Vol. 246 (1978), 463-471. MR 80f:10062a
  • [L$_2$] V. Losert, On the existence of uniformly distributed sequences in compact topological spaces II, Mh. Math. 87 (1979), 247-260. MR 80f:10062b
  • [L$_3$] V. Losert, Strong liftings for certain classes of compact spaces, Measure Theory Oberwolfach 1981, Lecture Notes in Math. 945, Springer Verlag, Berlin - Heildeberg - New York, 170-179. MR 84h:28010
  • [M] S. Mercourakis, Some remarks on countably determined measures and uniform distribution of sequences, Mh. Math., 121 (1996), 79-111. MR 97j:28029
  • [Mon-Zip] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience 1955. MR 17:383b
  • [Mos] Paul S. Mostert, Sections in principal fibre spaces, Duke Math. J. 23 (1956), 57-71. MR 17:771f
  • [P] A. Pelczynski, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. 58 (1968). MR 37:3335
  • [Pr] John F. Price, Lie Groups and Compact Groups, Cambridge University Press 1977. MR 56:8743
  • [V] W.A. Veech, Some questions of uniform distribution, Ann. of Math. (2) 94 (1971), 125-138. MR 44:4187

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 22C05, 28A35, 43A05

Retrieve articles in all journals with MSC (1991): 22C05, 28A35, 43A05


Additional Information

S. Grekas
Affiliation: Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece
Email: sgrekas@eudoxos.dm.uoa.gr

S. Mercourakis
Affiliation: Department of Mathematics, University of Athens, Panepistemiopolis, 157 84 Athens, Greece
Email: smerkour@eudoxos.dm.uoa.gr

DOI: https://doi.org/10.1090/S0002-9947-98-02182-5
Keywords: Compact group, Haar measure, Baire isomorphism, Riemann integrable function, Jordan measurable set.
Received by editor(s): February 1, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society