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): $4.3 \ \text{(iv)}$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.