Skew products and ergodic theorems for group actions

We obtain new ergodic theorems for an action of the free semi-group on a probability space by measure-preserving maps. Our method consists in associating with the original semi-group action a skew product over the shift on the space of infinite one-sided sequences of generators of the semi-group, and then integrating Birkhoff–Khinchin ergodic theorems along the base of the skew product.