Proof of an intersection theorem via graph homomorphisms

Let 0 <= p <= 1/2 and let {0,1}(n) be enclosed with the product measure mu(p) defined by mu(p)(x) = p(vertical bar x vertical bar) (1-p)(n-vertical bar x vertical bar), where vertical bar x vertical bar = Sigma x(i). Let I subset of {0,1}(n) be an intersecting family i.e. for every x,y epsilon I there exists a coordinate 1 <= i <= n such that x(i) = y(i) = 1. Then mu(p) (I) <= p. Our proof uses measure preserving homorphisms between graphs.