Ultrafilter limits of asymptotic density are not universally measurable

TL;DR

This paper demonstrates that for any nonprincipal ultrafilter on positive integers, the measure derived from the ultrafilter limit of asymptotic density is not universally measurable, revealing limitations in measure theory.

Contribution

It establishes that ultrafilter limits of asymptotic density do not produce universally measurable functions, highlighting a fundamental limitation in their measure-theoretic properties.

Findings

Ultrafilter limits of asymptotic density are not universally measurable.

The result applies to any nonprincipal ultrafilter on positive integers.

This reveals a fundamental measure-theoretic limitation in ultrafilter-based limits.

Abstract

We show that for any nonprincipal ultrafilter $U$ on the positive integers, then probability measure induced by the $U$-limit of asymptotic density is not a universally measurable function.