A Continuous Paradoxical Colouring Rule Using Group Action

TL;DR

This paper introduces a paradoxical colouring rule in measure spaces with group actions, demonstrating that certain colourings satisfying the rule are impossible to measure finitely additively, revealing a paradox in measure theory.

Contribution

It establishes the existence of paradoxical colourings under group actions with convex, finite-dimensional colour sets, extending the understanding of measure-theoretic paradoxes.

Findings

Paradoxical colourings cannot be finitely measured.

Approximate solutions to the colouring rule are not finitely measurable.

Injective measure-preserving shifts illustrate the paradox.

Abstract

Given a probability space $(X, {\cal B}, m)$, measure preserving transformations $g_1, \dots , g_k$ of $X$, and a colour set $C$, a colouring rule is a way to colour the space with $C$ such that the colours allowed for a point $x$ are determined by that point's location and the colours of the finitely $g_1 (x), \dots , g_k(x)$ with $g_i(x) \not= x$ for all $i$ and almost all $x$. We represent a colouring rule as a correspondence $F$ defined on $X\times C^k$ with values in $C$. A function $f: X\rightarrow C$ satisfies the rule at $x$ if $f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$. A colouring rule is paradoxical if it can be satisfied in some way almost everywhere with respect to $m$, but not in {\bf any} way that is measurable with respect to a finitely additive measure that extends the probability measure $m$ and for which the finitely many transformations $g_1, \dots , g_k$ remain measure preserving. We show that a colouring rule can be paradoxical when the $g_1, \dots, g_k$ are members of a group $G$, the probability space $X$ and the colour set $C$ are compact sets, $C$ is convex and finite dimensional, and the colouring rule says if $c: X\rightarrow C$ is the colouring function then the colour $c(x)$ must lie ($m$ a.e.) in $F(x, c(g_1(x) ), \dots , c(g_k(x)))$ for a non-empty upper-semi-continuous convex-valued correspondence $F$ defined on $X\times C^k$. We show that any colouring that approximates the correspondence by $\epsilon$ for small enough positive $\epsilon$ cannot be measurable in the same finitely additive way. Furthermore any function satisfying the colouring rule illustrates a paradox through finitely many measure preserving shifts defining injective maps from the whole space to subsets of measure summing up to less than one.