The Filter Dichotomy and medial limits

TL;DR

This paper demonstrates that the Filter Dichotomy principle implies the nonexistence of medial limits, connecting filter theory with measure-theoretic properties under certain set-theoretic assumptions.

Contribution

It proves that assuming the Filter Dichotomy rules out the existence of medial limits, establishing a new link between filter properties and measure theory.

Findings

Filter Dichotomy implies no medial limits

Medial limits exist under the Continuum Hypothesis

The result connects filter theory with measure-theoretic concepts

Abstract

The \emph{Filter Dichotomy} says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A function between topological spaces is \emph{universally measurable} if the preimage of %every open subset of the codomain is measured by every Borel measure on the domain. A \emph{medial limit} is a universally measurable function from $\mathcal{P}(\omega)$ to the unit interval [0,1] which is finitely additive for disjoint sets, and maps singletons to 0and $\omega$ to 1. Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits. We show that the Filter Dichotomy implies that there are no medial limits.