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