In this paper, we consider multiple ergodic means generated by a finite collection of measure-preserving transformations. We prove an almost sure convergence of such means for the functions of the logarithmic Orlicz space associated with the rank of the transformations. It is also shown the optimality of this function class in this convergence property. Our result extends a theorem by Dunford and Zygmund, where the convergence was claimed in the logarithmic space associated with the number of transformations. We prove our result by establishing a weak type inequality for certain maximal functions in Euclidean space.

Research of Lacey was supported in part by grant from the US National Science Foundation, DMS-1949206, and the research of Karagulyan was supported by the Science Committee of RA, in the frames of the research project 21AG-1A045.