A Measure Theoretic Paradox from a continuous colouring rule

TL;DR

This paper constructs a measure-theoretic paradox involving continuous convex colourings on a probability space, showing such paradoxes exist even with convex, compact colour sets and continuous rules.

Contribution

It demonstrates the existence of paradoxical colouring rules with convex, compact colour sets and continuous defining functions, extending measure-theoretic paradoxes to a broader setting.

Findings

Existence of paradoxical colourings with convex, compact sets.

Robustness of paradoxes under approximation by continuous functions.

Paradoxical rules can be defined by continuous functions on Euclidean space.

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 apoint $x$ 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 $m$ defined on ${\cal B}$ and for which the finitely many transformations $g_1, \dots , g_k$ remain measure preserving. Can a colouring rule be paradoxical if both $X$ and the colour set $C$ are convex and compact sets 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 $X\times C^k$? The answer is yes, and we present such an example. We show that this result is robust, including that any colouring that approximates the correspondence by $\epsilon$ for small enough positive $\epsilon$ also cannot be measurable in the same finitely additive way. Because non-empty upper-semi-continuous convex-valued correspondences on Euclidean space can be approximated by continuous functions, there are paradoxical colouring rules that are defined by continuous functions.