Mapping Defines Additive Function of Subalgebra of Power Set
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Theorem
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $S$ be a finite set.
Let $f: S \to \mathbb A$ be a mapping.
Let $B$ be an algebra of sets over $S$.
Define $\Sigma: B \to \mathbb A$ using summation as:
- $\ds \map \Sigma T = \sum_{t \mathop \in T} \map f t$
for $T\subseteq S$.
Then $\Sigma$ is an additive function on $B$.
Proof
Note that by Subset of Finite Set is Finite, $B$ consists of finite sets.
The result now follows from Sum over Disjoint Union of Finite Sets.
$\blacksquare$