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On a finite ground set $E$, a submodular rank function is a set function $r:2^E\to \mathbb{R}$ with the following properties:These are normalization, monotonicity, submodularity, and unit integer increments. In particular its values are nonnegative integers with $r(A)\le |A|$. Equivalently, one can require an integer-valued normalized monotone submodular function with $r(\{e\})\le 1$: diminishing returns then bounds every marginal increment by one. A general real-valued submodular function need not be a matroid rank.