It is shown that if two functions share the same uncentered (two-sided) ergodic maximal function, then they are equal almost everywhere.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-1-5, author = {Paul Alton Hagelstein}, title = {On the uniqueness of the uncentered ergodic maximal function}, journal = {Fundamenta Mathematicae}, volume = {184}, year = {2004}, pages = {81-90}, zbl = {1069.28010}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-1-5} }
Paul Alton Hagelstein. On the uniqueness of the uncentered ergodic maximal function. Fundamenta Mathematicae, Tome 184 (2004) pp. 81-90. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm183-1-5/