Definition:Extended Weight Function

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Let $S$ be a set.

Let $\mathscr F$ be the set of all finite subsets of $S$.

Let $w: S \to \R$ be a weight function.

The extended weight function of $w$ is the function $w^+: \mathscr F \to \R$ defined by:

$\forall A \in \mathscr F : \map {w^+} A = \ds \sum_{a \mathop \in A} \map w a$