 Balister–Bollobás entropy inequality

Let $\mathcal{A}$ be a finite multiset of subsets and let $r_i$ count its occurrences of coordinate $i$. Its nested compression consists of $L_j=\{i:r_i\ge j\}$. Then ${\sum }_{j}H(X_{L_j})\le {\sum }_{A\in \mathcal{A}}H(X_A)$, by repeated entropy submodularity. This is the compression formulation in Balister and Bollobás, "Projections, Entropy and Sumsets".