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

Given ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_m\in (0,1)$, we characterize all integrable functions $f:[0,1{]}^m\to \mathbb{C}$ satisfying $\int_{A_1 \times …\times A_m} f =0$foranycollectionofdisjointsets$A_1,…,A_m \subseteq [0,1]$ of respective measures ${\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_m$. We use this characterization to settle some of the conjectures in [S. Janson and V. S'os, More on quasi-random graphs, subgraph counts and graph limits, arXiv:1405.6808].