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Hopf’s ratio ergodic theorem by inducing. (English) Zbl 1065.28010
The ratio ergodic theorem of E. Hopf [“Ergodentheorie” (1937; Zbl 0017.28301)] and W. Stepanoff [Compos. Math. 3, 239-253 (1936; Zbl 0014.41804)] is the appropriate extension of Birkhoff’s pointwise ergodic theorem [G. D. Birkhoff, Proc. Natl. Acad. Sci. USA 17, 656-660 (1931; Zbl 0003.25602)] to the setting of infinite measure. A simple proof based on combinatorial ideas has been given by T. Kamae and M. Keane [Osaka J. Math. 34, No.3, 653-657 (1997; Zbl 0890.28010)]. Here a very simple proof is given using induced transformations and the pointwise ergodic theorem.

28D05 Measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
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