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Transactions of the American Mathematical Society

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



A constructive ergodic theorem

Author: J. A. Nuber
Journal: Trans. Amer. Math. Soc. 164 (1972), 115-137
MSC: Primary 28A65; Secondary 02E99
Erratum: Trans. Amer. Math. Soc. 216 (1976), 393.
MathSciNet review: 0291411
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Abstract: As discussed by Bishop, Birkhoff's Ergodic Theorem is not constructively valid. In this paper we present an hypothesis which is necessary and sufficient for the constructive almost everywhere convergence of the Césaro averages of the translates of an integrable function by a measure preserving transformation. In addition necessary and sufficient conditions are given for the limit function to be constructively integrable. Also we present a necessary and sufficient condition that the averages converge to a constant function and give an equivalent formulation of this condition for finite measure spaces. Several interesting examples are given which satisfy these conditions.

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

  • [1] Errett Bishop, Foundations of constructive analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221878
  • [2] -, Mathematics as a numerical language (to appear).
  • [3] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523

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Keywords: Birkhoff's Ergodic Theorem, measure preserving transformation, constructive mathematics, almost everywhere convergence, Césaro averages, integrable function
Article copyright: © Copyright 1972 American Mathematical Society

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