Uniqueness of the maximal function in the ratio ergodic theorem (Q1766838)

\textit{L. Ephremidze} [Fundam. Math. 174, No.3, 217-228 (2002; Zbl 1004.28010)] has shown that the ergodic maximal function associated to a finite measure-preserving transformation is injective on \(L^1\) functions, and also considered the same property for the \(\sigma\)-finite setting, where the conclusion does not always hold. In this paper it is argued that a more natural maximal function to consider in the infinite measure case is one associated to \textit{E. Hopf's} ratio ergodic theorem [``Ergodentheorie'' (1937; Zbl 0017.28301)]. The main result is that if \(T\) is a conservative measure-preserving transformation on a \(\sigma\)-finite measure space, and \(p>0\), \(f\) are \(L^1\) functions, then \[ f\mapsto\sup_{n\geq1}\left(\sum_{k=0}^{n-1}f\circ T^k\right)\left/ \left(\sum_{k=0}^{n-1}p\circ T^k\right)\right. \] is injective. Even in the case of finite measure, this gives a new result.