A constructive ergodic theorem
J. A. Nuber
Trans. Amer. Math. Soc. 164 (1972), 115-137
Primary 28A65; Secondary 02E99
Trans. Amer. Math. Soc. 216 (1976), 393.
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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.
Bishop, Foundations of constructive analysis, McGraw-Hill Book
Co., New York-Toronto, Ont.-London, 1967. MR 0221878
-, Mathematics as a numerical language (to appear).
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
- E. Bishop, Foundations of constructive analysis, McGraw-Hill, New York, 1967, pp. 153-242. MR 36 #4930. MR 0221878 (36:4930)
- -, Mathematics as a numerical language (to appear).
- N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958, pp. 675-676. MR 22 #8302. MR 0117523 (22:8302)
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Birkhoff's Ergodic Theorem,
measure preserving transformation,
almost everywhere convergence,
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