Mapping Defines Additive Function of Subalgebra of Power Set

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.

Also see

 * Inclusion-Exclusion Principle
 * Sum over Union of Finite Sets