Definition:Strongly Additive Function

Definition
Let $\mathcal S$ be an algebra of sets.

Let $f: \mathcal S \to \overline{\R}$ be a function, where $\overline{\R}$ denotes the extended set of real numbers.

Then $f$ is defined to be strongly additive iff, for all $S, T \in \mathcal S$:


 * $f \left({S \cup T}\right) + f \left({S \cap T}\right) = f \left({S}\right) + f \left({T}\right)$

Examples

 * An additive function is strongly additive (proof)
 * Thus, a measure is also strongly additive (proof)