Influence in product spaces

TL;DR

This paper extends the influence inequality to general product spaces, connecting it to the Lebesgue cube and resolving a key assertion in the theory of influence and sharp thresholds.

Contribution

It establishes a generalized influence inequality for arbitrary product spaces, broadening the applicability of influence theory in probabilistic combinatorics.

Findings

Derived influence inequality for general product spaces

Linked influence inequality to Lebesgue cube analysis

Resolved a key assertion of BKKKL in influence theory

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.