Influences in Product Spaces: KKL and BKKKL Revisited
Combinatorics, Probability and Computing
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Decision trees and influences of variables over product probability spaces
Combinatorics, Probability and Computing
A simple reduction from a biased measure on the discrete cube to the uniform measure
European Journal of Combinatorics
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In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube, where there is a clear definition of influence, in the general case several definitions have been presented in different papers. We propose a family of definitions for the influence that contains all the known definitions, as well as other natural definitions, as special cases. We show that the proofs of the BKKKL theorem and of other results can be adapted to our new definition. The adaptation leads to generalizations of these theorems, which are tight in terms of the definition of influence used in the assertion.