A General Correspondence between Averages and Integrals

TL;DR

This paper presents a unified theorem that generalizes the Furstenberg correspondence principle, connecting averages and integrals across various mathematical structures such as graphs and functions.

Contribution

It provides a comprehensive framework that encompasses previous generalizations of the Furstenberg correspondence, unifying them into a single theorem.

Findings

Unified correspondence theorem for sequences of graphs and functions

Enables transfer of combinatorial properties to dynamical systems

Simplifies understanding of averages and integrals in diverse contexts

Abstract

Recent work has generalized the Furstenberg correspondence between sets of integers and dynamical systems to versions which involve sequences of finite graphs or sequences of $L^\infty$ functions. We give a unified version of the theorem subsuming all these generalizations.