A note on the ergodic theorem

Author:
Takeshi Yoshimoto

Journal:
Proc. Amer. Math. Soc. **90** (1984), 401-404

MSC:
Primary 47A35; Secondary 40C05

DOI:
https://doi.org/10.1090/S0002-9939-1984-0728356-1

MathSciNet review:
728356

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a positive regular shift-invariant method of summability and let be a one-to-one transformation which maps onto and which is -bimeasurable, i.e., if and only if , where is a measurable space. Then it is proved that if for a finite measure on the sequence is -summable for each , then for any real the sequence is -summable -almost everywhere for every bounded -measurable function defined on . The result includes the Blum-Hanson theorem.

**[1]**J. R. Blum and D. L. Hanson,*A note on the ergodic theorem*, Proc. Amer. Math. Soc.**16**(1965), 413-414. MR**0209437 (35:335)****[2]**L. W. Cohen,*On the mean ergodic theorem*, Ann. of Math. (2)**41**(1940), 505-509. MR**0002027 (1:339b)****[3]**P. R. Halmos,*Measure theory*, Van Nostrand, New York, 1950. MR**0033869 (11:504d)****[4]**T. Yoshimoto,*Generalized ergodic inequalities and ergodic theorems*, Probability Theory and Mathematical Statistics (ed. by K. Ito and J. V. Prokhorov): Proceedings of the 4th USSR-Japan Symposium, held at Tbilisi, USSR. August 23-29, 1982, Lectures Notes in Math., vol. 1021, Springer-Verlag (to appear). MR**736038 (85h:28024)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0728356-1

Keywords:
Regular summation method,
-summable,
-summable,
Blum-Hanson theorem

Article copyright:
© Copyright 1984
American Mathematical Society