A short proof and a generalization of the BKR-inequality

The BKR-inequality (a.k.a. BK-conjecture) asserts that the probability of two events occurring on disjoint subsets of a finite family of independent variables is dominated by the product of the probabilities of the two events. This correlation inequality has been conjectured and partially established by van den Berg and Kesten (1985) and has been proved by Reimer (2000). We give a short proof of the BKR-inequality and generalize it to arbitrary families of independent variables.