Coverage function
= Coverage function
{title2=$F(S)=|\bigcup_{v\in S}C_v|$}
A <coverage function> assigns a value to the union of objects covered by a selection. Let $V$ index <sets> $C_v$ in a finite universe $U$. The unweighted version is
$$
F(S)=\left|\bigcup_{v\in S}C_v\right|,\qquad S\subseteq V.
$$
A <weighted coverage function> gives each $u\in U$ a nonnegative weight $w_u$ and sums those weights over the same <set union>. Its value is zero on the empty selection and cannot decrease when more sets are selected.<Learning with Submodular Functions: A Convex Optimization Perspective by Francis Bach>[Section 6.3, Set covers, printed pp. 59–61; reinterpretation as the nonnegative measure of a union]{ref}
The <submodular set function> property follows from an explicit marginal calculation:
$$
F(S\cup\{v\})-F(S)=\sum_{u\in C_v\setminus\bigcup_{s\in S}C_s}w_u.
$$
If $S\subseteq T$, the objects still uncovered after $T$ form a subset of those uncovered after $S$. Nonnegative weights therefore make the gain no larger after $T$: precisely the <diminishing returns> property.
For an original example, take $C_1=\{a,b\}$, $C_2=\{b,c\}$ and weights $(w_a,w_b,w_c)=(2,1,3)$. Then $F(\{1\})=3$, $F(\{2\})=4$ and $F(\{1,2\})=6$. The second set adds only three after the first, because their shared object $b$ is counted once. This models the value of distinct coverage, rather than the sum of overlapping collection sizes.