A constructive ergodic theorem

Nuber, J. A.

doi:10.1090/S0002-9947-1972-0291411-3

A constructive ergodic theorem
HTML articles powered by AMS MathViewer

by J. A. Nuber

Trans. Amer. Math. Soc. 164 (1972), 115-137

DOI: https://doi.org/10.1090/S0002-9947-1972-0291411-3

PDF | Request permission

Erratum: Trans. Amer. Math. Soc. 216 (1976), 393.

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

Errett Bishop, Foundations of constructive analysis, McGraw-Hill Book Co., New York-Toronto-London, 1967. MR 0221878

Mathematics as a numerical language

Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523