Uniqueness of invariant means for measurepreserving transformations
Author:
Joseph Rosenblatt
Journal:
Trans. Amer. Math. Soc. 265 (1981), 623636
MSC:
Primary 28D15; Secondary 43A07, 58F11
MathSciNet review:
610970
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Abstract: For some compact abelian groups (e.g. , , and ), the group of topological automorphisms of has the Haar integral as the unique invariant mean on . This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets of , ; it is the unique normalized positive finitelyadditive measure on which is invariant under isometries and the transformation of . Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on , which is invariant by some group of measurepreserving transformations of the probability space , are described.
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 [1]
 S. Banach, Sur le problème de la mesure, Oeuvres, Vol. I, PWN, Warsaw, 1967, pp. 318322.
 [2]
 A. del Junco and J. Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), 185197. MR 553340 (81d:10042)
 [3]
 E. Granirer, Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615624. MR 0340962 (49:5712)
 [4]
 F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, New York, 1969. MR 0251549 (40:4776)
 [5]
 V. Losert and H. Rindler, Almost invariant sets (preprint). MR 608100 (82i:43001)
 [6]
 E. Marczewski (Szprilrajn), Problem , The Scottish Book, 19371938.
 [7]
 J. Mycielski, Equations insoluable on and related problems, Amer. Math. Monthly 84 (1977), 723726; 85 (1978), 263265. MR 0470100 (57:9867)
 [8]
 , Finitelyadditive invariant measures. I, Colloq. Math. 42 (1979), 309318. MR 567569 (82g:43003a)
 [9]
 , Finitelyadditive invariant measures. III, Colloq. Math, (to appear). MR 567575 (82g:43003b)
 [10]
 I. Namioka, Følner's conditions for amenable semigroups, Math. Scand. 15 (1964), 1828. MR 0180832 (31:5062)
 [11]
 J. Rosenblatt, Finitelyadditive invariant measures. II, Colloq. Math. 42 (1979), 361363. MR 567575 (82g:43003b)
 [12]
 , Invariant means for the bounded measurable functions on a nondiscrete locally compact group, Math. Ann. 220 (1976), 219228. MR 0397305 (53:1164)
 [13]
 W. Rudin, Invariant means on , Studia Math. 44 (1972), 219227. MR 0304975 (46:4105)
 [14]
 K. Schmidt, Asymptotically invariant sequences and an action of on the sphere (preprint).
 [15]
 , Amenability, Kazhdan's property , strong ergodicity, and invariant means for ergodic actions (preprint).
 [16]
 D. Sullivan, For there is only one finitelyadditive rotationally invariant measure on the sphere defined on all Lebesgue measurable sets, Bull. Amer. Math. Soc. (N.S.) 1 (1981), 121123. MR 590825 (82b:28035)
 [17]
 S. Trott, A pair of generators for the unimodular group, Canad. Math. Bull. 5 (1962), 245252. MR 0141716 (25:5113)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106109707
PII:
S 00029947(1981)06109707
Article copyright:
© Copyright 1981
American Mathematical Society
