Abstract: D. S. Ornstein has shown that if the ergodic maximal function of a nonnegative function is in then the function is in . This paper gives a new simple proof of this fact.
-  Roger L. Jones, New proofs for the maximal ergodic theorem and the Hardy-Littlewood maximal theorem, Proc. Amer. Math. Soc. 87 (1983), no. 4, 681–684. MR 687641, https://doi.org/10.1090/S0002-9939-1983-0687641-1
-  Donald Ornstein, A remark on the Birkhoff ergodic theorem, Illinois J. Math. 15 (1971), 77–79. MR 0274719
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