Abstract
LetP be a Markov operator recurrent in the sense of Harris, withσ-finite invariant measureμ. (1) Ifμ is finite andP aperiodic, then forf ∈L 1(μ),P nf →f fdμ a.e. (2) Ifμ is infinite,P nf → 0 a.e. for everyf ∈L p (μ), 1≦p <∞.
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Dr. Shlomo Horowitz died before completing the manuscript. He left a draft, to which I added the last paragraph, references, title and abstract. I made only slight modifications in a place or two in the proofs, for clarity (M. Lin).
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Horowitz, S. Pointwise convergence of the iterates of a harris-recurrent Markov operator. Israel J. Math. 33, 177–180 (1979). https://doi.org/10.1007/BF02762159
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DOI: https://doi.org/10.1007/BF02762159