Axiomatic theory of divergent series and cohomological equations

TL;DR

This paper develops an axiomatic framework for summing divergent series using ergodic theory and cohomological equations, leading to the construction of nonmeasurable functions as conjectured by Kolmogorov.

Contribution

It introduces a general axiomatic approach to divergent series and links it with ergodic theory and cohomological equations, including the construction of nonmeasurable functions.

Findings

Established a summation theory for divergent series based on Hardy-Kolmogorov axioms.

Connected the solvability of cohomological equations with the measurability of solutions.

Constructed a nonmeasurable function as conjectured by Kolmogorov.

Abstract

A general theory of summation of divergent series based on the Hardy-Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some cohomological equations, all solutions to which are nonmeasurable. In particular, this realizes a construction of a nonmeasurable function as first conjectured by Kolmogorov.