On pointwise ergodic theorems for infinite measure

TL;DR

This paper proves pointwise convergence of ergodic averages for Dunford-Schwartz operators in various function spaces, extending classical results and confirming Bourgain's Return Times theorem in these contexts.

Contribution

It establishes new pointwise ergodic theorems for infinite measure spaces and fully symmetric spaces with non-trivial Boyd indices, including Bourgain's theorem.

Findings

Pointwise convergence of ergodic averages for Dunford-Schwartz operators.

Extension of Bourgain's Return Times theorem to symmetric spaces.

Validation of classical ergodic results in broader function spaces.

Abstract

For a Dunford-Schwartz operator in the $L^p-$space, $1\leq p< \infty$ , of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of ergodic averages in fully symmetric spaces of measurable functions with non-trivial Boyd indices is studied. In particular, it is shown that for such spaces Bourgain's Return Times theorem is valid.