THE INFLUENCE OF VARIABLES IN PRODUCT-SPACES

Let X be a probability space and let f : X(n) --> {0, 1} be a measurable map. Define the influence of the k-th variable on f, denoted by I(f)(k), as follows: For u = (u1, u2, ..., u(n-1)) is-an-element-of X(n-1) consider the set l(k)(u) = {(u1, U2, ..., u(k-1), t, u(k), ..., u(n-1)): t is-an-element-of X}. I(f)(k) = Pr(u is-an-element-of X(n-1) : f is not constant on l(k)(u)). More generally, for S a subset of [n] = {1, ..., n} let the influence of S on f, denoted by I(f)(S), be the probability that assigning values to the variables not in S at random, the value of f is undetermined. THEOREM 1: There is an absolute constant c1 so that for every function f: X(n) --> {0, 1}, with Pr(f-1(1)) = p less-than-or-equal-to 1/2, there is a variable k so that I(f)(k) greater-than-or-equal-to c1p log n/n. THEOREM 2: For every f : X(n) --> {0, 1}, with Prob(f = 1) = 1/2, and every epsilon > 0, there is S subset-of [n], \S\ = c2 (epsilon)n/log n so that I(f)(S) greater-than-or-equal-to 1 - epsilon. These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the case X = {0, 1}.