{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/optimisation-and-paradoxical-decompositions","title":"A Continuous Paradoxical Colouring Rule Using Group Action","arxiv_id":"2106.02084","date":"2021-05-28","proceeding":null,"authors":["Tugkan Batu","Robert Samuel Simon","Grzegorz Tomkowicz"],"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.","url_abs":"https://arxiv.org/abs/2106.02084v2","url_pdf":"https://arxiv.org/pdf/2106.02084v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"optimisation-and-paradoxical-decompositions","repo_url":"https://github.com/tugkanbatu/paradoxicalcolouring","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}