A localization inequality for set functions

We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f1,f2 defined on the subsets of a finite set S, satisfying for i∈{1,2}, there exists a positive multiplicative set function μ over S and two subsets A,B⊆S such that for i∈{1,2} μ(A)fi(A)+μ(B)fi(B)+μ(A∪B)fi(A∪B)+μ(A∩B)fi(A∩B)⩾0. The Ahlswede–Daykin four function theorem can be deduced easily from this.