  The theory of influences in product measures has profound applications in
theoretical computer science, combinatorics, and discrete probability. This
deep theory is intimately connected to functional inequalities and to the
Fourier analysis of discrete groups. Originally, influences of functions were
motivated by the study of social choice theory, wherein a Boolean function
represents a voting scheme, its inputs represent the votes, and its output
represents the outcome of the elections. Thus, product measures represent a
scenario in which the votes of the parties are randomly and independently
distributed, which is often far from the truth in real-life scenarios.
  We begin to develop the theory of influences for more general measures under
mixing or correlation decay conditions. More specifically, we prove analogues
of the KKL and Talagrand influence theorems for Markov Random Fields on bounded
degree graphs with correlation decay. We show how some of the original
applications of the theory of in terms of voting and coalitions extend to
general measures with correlation decay. Our results thus shed light both on
voting with correlated voters and on the behavior of general functions of
Markov Random Fields (also called ``spin-systems") with correlation decay.
