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Take information entropy in bits and put $H(X_{\varnothing })=0$. Write $C=A\cap B$, $U=A\setminus B$, and $V=B\setminus A$. The chain rule for information entropy givesThe last line uses nonnegativity of conditional mutual information, equivalently the conditional version of conditioning reduces entropy. All random variables are finite-valued, so every conditional entropy here is finite. Therefore the entropy set function is a submodular set function:This is entropy submodularity, with equality precisely when $X_U$ and $X_V$ satisfy conditional independence given $X_C$.

A real-valued set function is supermodular if it satisfies the displayed inequality. Its negative is a submodular set function. Equivalently, the gain from adding an element cannot decrease as the set grows. This is the defining property of a convex cooperative game, and makes coalition marginal allocations lie in the core of a cooperative game.

An elementary union-intersection compression of a finite multiset of subsets replaces occurrences of $A,B$ by $A\cup B,A\cap B$. A compression is a finite sequence of these moves. The move preserves every coordinate multiplicity. On incomparable sets it increases ${\sum }_S|S{|}^2$ by $2|A\setminus B||B\setminus A|$, so repeated compression reaches a chain under inclusion. For any submodular set function, the sum of its values cannot increase.