Paradoxical decompositions and finitary rules

TL;DR

This paper investigates paradoxical coloring rules in probability spaces, exploring conditions under which such rules lead to paradoxical partitions, extending classical results like the Hausdorff paradox.

Contribution

It introduces a framework for analyzing paradoxical coloring rules and characterizes when they are paradoxical, generalizing classical measure paradoxes.

Findings

Identification of conditions for paradoxical coloring rules

Extension of Hausdorff paradox to new settings

Characterization of measure-preserving transformations in paradoxical contexts

Abstract

We colour every point x of a probability space X according to the colours of a finite list x_1, ...., x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule (1) does there exist a finitely additive extension of the probability measure for which the x_i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? if the answer to the first question is no and to the second question yes, we say that the colouring rule is paradoxical. A paradoxical colouring rule not only allows for a paradoxical partition of the space, it requires one. We pay special attention to generalizations of the Hausdorff paradox.