A characterization of functions with vanishing averages over products of disjoint sets

Given α1,…,αm∈(0,1)α1,…,αm∈(0,1), we characterize all integrable functions f:[0,1]m→Cf:[0,1]m→C satisfying ∫A1×⋯×Amf=0∫A1×⋯×Amf=0 for any collection of disjoint measurable sets A1,…,Am⊆[0,1]A1,…,Am⊆[0,1] of respective measures α1,…,αmα1,…,αm. We use this characterization to settle some of the conjectures in Janson and Sós (2015) about the relation between subgraph counts and quasi-randomness.