We study the integrability of the supremum of ergodic ratios defined by means of a measure-preserving conservative ergodic transformation of a $\sigma$-finite measure space. Our result implies Ornstein's recent result for the supremum of ergodic averages.
"On the Integrability of the Supremum of Ergodic Ratios." Ann. Probab. 1 (2) 338 - 340, April, 1973. https://doi.org/10.1214/aop/1176996988