Slow convergence of Birkhoff ergodic averages

Key Points

• Birkhoff ergodic averages converge slowly for ergodic actions of groups on probability spaces.
• The result holds as measures approach integrals, with a specific sequence defined for convergence.
• An analytic approach using specific functions in L1 demonstrates the behavior of averages in this context.
• These findings enhance understanding of convergence in ergodic theory for probabilistic frameworks.