The Measure Game

David Samuel Simmons; Logan Crone; Lior Fishman; Stephen Jackson; Houston Schuerger

The Measure Game

David Samuel Simmons, Logan Crone, Lior Fishman, Stephen Jackson, Houston Schuerger

Mathematics

Research output: Working paper › Preprint

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 Σ11 sets. We also use the game to give a new, more constructive, proof of a strong form of the Rényi-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.

Original language	English
Publisher	arXiv
Publication status	Submitted - 24 Oct 2019

Access to Document

https://arxiv.org/abs/1910.11181

Cite this

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title = "The Measure 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 Σ11 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.",

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T1 - The Measure Game

AU - Simmons, David Samuel

AU - Crone, Logan

AU - Fishman, Lior

AU - Jackson, Stephen

AU - Schuerger, Houston

PY - 2019/10/24

Y1 - 2019/10/24

N2 - 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 Σ11 sets. We also use the game to give a new, more constructive, proof of a strong form of the Rényi-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.

AB - 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 Σ11 sets. We also use the game to give a new, more constructive, proof of a strong form of the Rényi-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.

M3 - Preprint

BT - The Measure Game

PB - arXiv

ER -