Measure is Strongly Additive

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Then $\mu$ is strongly additive, that is:

$\forall E, F \in \Sigma: \map \mu {E \cap F} + \map \mu {E \cup F} = \map \mu E + \map \mu F$

Proof

Combine Measure is Finitely Additive Function with Additive Function is Strongly Additive.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.3 \ \text{(iv)}$