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 f 1 , f 2 defined on the subsets of a finite set S, satisfying ∑ X ⊆ S f i ( X ) ⩾ 0 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 ) f i ( A ) + μ ( B ) f i ( B ) + μ ( A ∪ B ) f i ( A ∪ B ) + μ ( A ∩ B ) f i ( A ∩ B ) ⩾ 0 . The Ahlswede–Daykin four function theorem can be deduced easily from this.