Measure is Strongly Additive

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

$\forall E, F \in \Sigma: \mu \left({E \cap F}\right) + \mu \left({E \cup F}\right) = \mu \left({E}\right) + \mu \left({F}\right)$

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): $4.3 \ \text{(iv)}$