Measure is Strongly Additive
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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): $4.3 \ \text{(iv)}$