Measure is Monotone

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Theorem

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


Then $\mu$ is monotone, that is:

$\forall E, F \in \Sigma: E \subseteq F \implies \mu \left({E}\right) \le \mu \left({F}\right)$


Proof

A direct corollary of Non-Negative Additive Function is Monotone.

$\blacksquare$


Sources