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

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