On Russo's Approximate Zero-One Law

Abstract

Consider the product measure mu(p) on {0, 1}(n), when 0 (resp. 1) is given weight 1 - p (resp, p). Consider a monotone subset A of {0, 1}(n). We give a precise quantitative form to the following statement: if A does not depend much on any given coordinate, d mu(p)(A)/dp is large. Thus, in that case, there is a threshold effect and mu(p)(A) jumps from near 0 to near 1 in a small interval.