Let G = R d G = {{\mathbf {R}}^d} or Z d {{\mathbf {Z}}^d} and consider an ergodic measure-preserving action of G on a probability space ( X , A , P ) (X,\mathfrak {A},P) , let f ∈ L 1 ( X , P ) f \in {L^1}(X,P) and Mf be its maximal ergodic function. Our purpose is to prove the converse of the following theorem of N. Wiener: if | f | log + | f | |f|{\log ^ + }|f| is integrable then Mf is integrable. For the particular case G = Z G = {\mathbf {Z}} this result was already obtained by D. Ornstein whose proof is based on induced transformations and seems to be specific to Z, our proof is based on a result of E. M. Stein on the Hardy-Littlewood maximal function on R d {{\mathbf {R}}^d} and its analogue on Z d {{\mathbf {Z}}^d} .