Answer to a question of Kolmogorov

TL;DR

This paper answers Kolmogorov's long-standing question by constructing a counterexample, showing that not all measurable sets can be approximated by polygons through contractions while preserving measure.

Contribution

It provides the first counterexample to Kolmogorov's question, demonstrating limitations of measure-preserving polygonal approximations via contractions.

Findings

Counterexample set is bounded and simply connected.

Counterexample can be extended to higher dimensions.

Shows negative answer to Kolmogorov's question.

Abstract

More than 80 years ago Kolmogorov asked the following question. Let $E\subseteq \mathbb{R}^{2}$ be a measurable set with $\lambda^{2}(E)<\infty$, where $\lambda^2$ denotes the two-dimensional Lebesgue measure. Does there exist for every $\varepsilon>0$ a contraction $f\colon E\to \mathbb{R}^{2}$ such that $\lambda^{2}(f(E))\geq \lambda^{2}(E)-\varepsilon$ and $f(E)$ is a polygon? We answer this question in the negative by constructing a bounded, simply connected open counterexample. Our construction can easily be modified to yield the analogous result in higher dimensions.