A structure theorem for Boolean functions with small total influences

TL;DR

This paper proves a structure theorem for Boolean functions with small total influences, showing they are nearly measurable with respect to certain sub-sigma algebras, which helps understand properties without sharp thresholds.

Contribution

It generalizes Friedgut's core results to arbitrary product spaces and improves Bourgain's related findings, providing a broader structural understanding.

Findings

Boolean functions with small influences are almost measurable with respect to natural sub-sigma algebras

The theorem extends Friedgut's results to general product probability spaces

It offers insights into the structure of monotone properties lacking sharp thresholds

Abstract

We show that on every product probability space, Boolean functions with small total influences are essentially the ones that are almost measurable with respect to certain natural sub-sigma algebras. This theorem in particular describes the structure of monotone set properties that do not exhibit sharp thresholds. Our result generalizes the core of Friedgut's seminal work [Ehud Friedgut. Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc., 12(4):1017-1054, 1999.] on properties of random graphs to the setting of arbitrary Boolean functions on general product probability spaces, and improves the result of Bourgain in his appendix to Friedgut's paper.