We have a new and elementary proof of the pointwise ergodic theorem for amenable groups that uses random walk techniques. This is considerably simpler than Lindenstrauss' original proof (2001), and allows us to prove the classical (Lindenstrauss, 2001) and noncommutative (Cadilhac and Wang, 2022) versions in the same setting. The technique is to use a suitably chosen random walk to dominate the process of averaging along certain Folner sets by the average of an integer action (affected by a majorisation inequality of measures on the group), so that effectively one can reduce any amenable group action to an integer action. The talk will be based on a joint paper with Joachim Zacharias and Runlian Xia (draft in preparation).
