The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of RdRd on Lp(X)Lp(X)-spaces are convergent for d⩾3d⩾3 and p>d/(d-1)p>d/(d-1).This is done by adapting the proof of the spherical maximal theorem by Rubio de Francia so as to obtain directly the ergodic theorem.