A note on Furstenberg's filtering problem

TL;DR

This paper confirms that Furstenberg's filtering theorem is valid without requiring integrability, resolving an old question in elementary probability theory related to ergodic theory.

Contribution

It provides a positive answer to a longstanding question, extending Furstenberg's filtering theorem to cases without integrability assumptions.

Findings

Furstenberg's filtering theorem holds without integrability.

Resolved an old question in probability theory.

Extended applicability of ergodic theory results.

Abstract

This short note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg's seminal article "Disjointness in Ergodic Theory." As a consequence, Furstenberg's filtering theorem holds without any integrability assumption.