Proof of the Van den Berg–Kesten Conjecture

We prove the following conjecture of J. van den Berg and H. Kesten. For any events Ascr and Bscr in a product probability space, Prob(Ascr □Bscr ) les Prob(Ascr )Prob(Bscr ), where Ascr □Bscr is the event that Ascr and Bscr occur ‘disjointly’.