The Measure Game

TL;DR

This paper introduces a game related to measure theory, using it to prove key results like Fubini's theorem and the Borel-Cantelli lemma through combinatorial methods, offering new constructive proofs.

Contribution

It presents a novel measure-theoretic game and employs it to derive fundamental theorems and results with direct combinatorial proofs, avoiding traditional measure-theoretic arguments.

Findings

Proves measure-theoretic results using the game approach

Provides a constructive proof of the Rényi-Lamperti lemma

Establishes the measurability of _1 sets via the game

Abstract

We study a game first introduced by Martin (actually we use a slight variation of this game) which plays a role for measure analogous to the Banach-Mazur game for category. We first present proofs for the basic connections between this game and measure, and then use the game to prove fundamental measure theoretic results such as Fubini's theorem, the Borel-Cantelli lemma, and a general unfolding result for the game which gives, for example, the measurability of $\boldsymbol{\Sigma}^1_1$ sets. We also use the game to give a new, more constructive, proof of a strong form of the R\'{e}nyi-Lamperti lemma, an important result in probability theory with many applications to number theory. The proofs we give are all direct combinatorial arguments using the game, and do not depend on known measure theoretic arguments.