## Abstract

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=(u_{ 1}, u_{ 2},..., u_{ n-1}) ∈X^{ n-1} consider the set l_{ k} (u)={(u_{ 1}, u_{ 2},..., u_{ k-1}, t, u_{ k},..., u_{ n-1}):t ∈X}. {Mathematical expression} 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 c_{ 1} so that for every function f: X^{ n} → {0, 1}, with Pr(f^{ -1}(1))=p≤1/2, there is a variable k so that {Mathematical expression} Theorem 2:For every f: X^{ n} → {0, 1}, with Prob(f=1)=1/2, and every ε>0, there is S ⊂ [n], |S|=c_{ 2}(ε)n/log n so that I_{ f} (S)≥1-ε. These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the case X={0, 1}.

Original language | American English |
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Pages (from-to) | 55-64 |

Number of pages | 10 |

Journal | Israel Journal of Mathematics |

Volume | 77 |

Issue number | 1-2 |

DOIs | |

State | Published - Feb 1992 |