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Uniform distribution, invariant means, and Riemann integrals. (English. Russian original) Zbl 0853.43002

Math. Notes 56, No. 3, 978-983 (1994); translation from Mat. Zametki 56, No. 3, 144-154 (1994).
The author studies actions of semigroups \(S\) satisfying certain amenability conditions. He proves a local version of Dixmier’s criterion, looking for invariant means on closed invariant subspaces of \(m(S)\) (the space of complex valued bounded functions on \(S\) generated by a single \(f \in m(S))\) (existence and uniqueness). He also proves some results on the representation of \(S\) on Banach spaces \(E\), considering also the dual representation \(\sigma\) on the dual space \(E'\).
In the second part a general concept of uniform distribution is studied: A mapping \(\omega : S \to X\), \((X \mu)\) a compact probability space, is called \(M\)-uniformly distributed (this concept is even generalized to not necessarily positive weights), for certain subsets \(M\) of the set of all \(S\)-left invariant means on \(m(S) \), if \(I(f \circ \omega) = \mu (f)\) for all continuous \(f\) on \(X\). For \(S = N\) (and more generally for left amenable semigroups) it is easy to find subsets \(M\), such that this concept coincides with the usual one (of Cesaro convergence of \(f(x_n)\) if \(S = N)\).
Strengthening a result of E. Yu. Terekhina on the substitution of variables in Riemann integrals [Mosc. Univ. Math. Bull. 40, 75-79 (1985); translation from Vestn. Mosk. Univ., Ser. I 1985, No. 3, 78-80 (1985; Zbl 0594.26006)] the author shows that there exist certain “distinctly distributed” sequences. Finally a result on Riemann integrable functions on compact measure spaces is proved which is related to the variable-substitution problem.
Reviewer: H.Rindler (Wien)

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
28A25 Integration with respect to measures and other set functions

Citations:

Zbl 0594.26006
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References:

[1] Alan Paterson, Amenability, American Math. Soc., Providence (1988).
[2] J. Rosenblatt and Z. Yang, ”Functions with a unique mean value,” Illinois J. Math.,34, No. 4, 744–764 (1990). · Zbl 0687.43002
[3] J. Dixmier, ”Les moyennes invariantes dans les semigroupes et leurs applications,” Acta Sci. Math.,12, 213–227 (1950). · Zbl 0037.15501
[4] E. Granirer, ”On finite equivalent invariant measures for semigroups of transformations,” Duke. Math. J.,38, No. 2, 395–408 (1971). · Zbl 0218.43002 · doi:10.1215/S0012-7094-71-03849-X
[5] G. G. Lorentz, ”A contribution to the theory of divergent sequences,” Acta Math.,80, 167–190 (1948). · Zbl 0031.29501 · doi:10.1007/BF02393648
[6] M. Day, ”Amenable semigroups,” Illinois Math. J.,1, No. 4, 509–544 (1957). · Zbl 0078.29402
[7] P. Greenleaf, Invariant Means on Topological Groups and Their Applications [Russian translation], Mir, Moscow (1973). · Zbl 0252.43005
[8] L. Alaoglu and G. Birkhoff, ”General ergodic theorems,” Ann. Math.,41, 293–309 (1940). · Zbl 0024.12402 · doi:10.2307/1969004
[9] A. A. Tempel’man, Ergodic Theorems on Groups [in Russian], Mokslas, Vil’nyus (1986).
[10] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Interscience, New York (1974). · Zbl 0281.10001
[11] E. Yu. Terekhina, ”On subsitution of variables in Riemann integrals,” Vestn. MGU, Ser. Mat., Mekh., No. 3, 78–80 (1985).
[12] C. J. de la Valle Pussen, A Course in the Analysis of the Infinitely Small [Russian translation], Vol. 1, GTTI, Moscow (1933).
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