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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 set function is submodular if $f(A)+f(B)\ge f(A\cup B)+f(A\cap B)$ for every pair of subsets. Entropy submodularity is a fundamental example.

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.