 Modular set function

On a finite ground set, a modular set function is a set function satisfying the displayed equality for every pair of subsets. Equivalently,To prove this representation, apply the equality to disjoint $S$ and $\{x\}$: $h(S+x)-h(S)=h(\{x\})-h(\varnothing )$. Add these fixed marginal increments along any ordering of $S$. Conversely, the contributions of each element and of the constant cancel in the displayed identity. Thus a modular set function need not be a normalised set function; its normalised instances are precisely the real-valued set functions with finite additivity on a finite power set. Adding or subtracting a modular set function preserves submodularity.