A Katona-type proof of an Erdős-Ko-Rado-type theorem

Let p⩽1/2 and let μp be the product measure on {0,1}n, where μp(x)=p∑xi(1-p)n-∑xi. Let A⊂{0,1}n be an intersecting family, i.e. for every x,y∈A there exists 1⩽i⩽n such that xi=yi=1. Then μp(A)⩽p. Our proof uses a probabilistic trick first applied by Katona to prove the Erdős-Ko-Rado theorem.