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We show that on every product probability space, Boolean functions with smalltotal influences are essentially the ones that are almost measurable withrespect to certain natural sub-sigma algebras. This theorem in particulardescribes the structure of monotone set properties that do not exhibit sharpthresholds.  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 ofarbitrary Boolean functions on general product probability spaces, and improvesthe result of Bourgain in his appendix to Friedgut's paper.

A structure theorem for Boolean functions with small total influences