Definition:Strongly Additive Function

Definition
Let $\Sigma$ be a $\sigma$-algebra.

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

Then $f$ is defined to be strongly additive iff, for all $E, F \in \Sigma$:


 * $f \left({E \cup F}\right) + f \left({E \cap F}\right) = f \left({E}\right) + f \left({F}\right)$

Examples

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