A structure theorem for Boolean functions with small total influences

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.