The uniqueness theorem for the ergodic maximal operator is proved in the continous case.
@article{urn:eudml:doc:44422, title = {On the uniqueness of maximal operators for ergodic flows.}, journal = {Revista Matem\'atica de la Universidad Complutense de Madrid}, volume = {15}, year = {2002}, pages = {75-84}, zbl = {1063.37001}, mrnumber = {MR1915216}, language = {en}, url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44422} }
Ephremidze, Lasha. On the uniqueness of maximal operators for ergodic flows.. Revista Matemática de la Universidad Complutense de Madrid, Tome 15 (2002) pp. 75-84. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44422/