  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 at least three
definitions were 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 prove a generalization of
the BKKKL theorem, which is tight in terms of the definition of influence used
in the assertion, and use it to generalize several known results on influences
in general product spaces.
