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Influence in product spaces

Part of: Foundations of probability theory Classical measure theory

Published online by Cambridge University Press:  25 July 2016

Geoffrey R. Grimmett ,
Svante Janson  and
James R. Norris
Geoffrey R. Grimmett*
Affiliation:
University of Cambridge
Svante Janson*
Affiliation:
Uppsala University
James R. Norris*
Affiliation:
University of Cambridge
*
Statistical Laboratory, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: [email protected]
Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden. Email address: [email protected]
Statistical Laboratory, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: [email protected]
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Abstract

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

Keywords

Influencesharp thresholdproduct spaceseparable spacemeasure-space isomorphism

MSC classification

Primary: 60A10: Probabilistic measure theory
Secondary: 28A35: Measures and integrals in product spaces
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 145 - 152
Copyright
Copyright © Applied Probability Trust 2016 

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