Monotone set function ($A\subseteq B\Rightarrow f(A)\le f(B)$)

A real-valued set function is monotone, in the nondecreasing convention, when enlarging its input subset cannot decrease its value. Equivalently, each marginal increment ${\Delta }_{x}f(S)$ is nonnegative. This follows by adding elements individually; the converse follows by considering $S\subseteq S\cup \{x\}$. A normalised set function with this property is a nonnegative function.