Measure is Monotone

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

• René L. Schilling: Measures, Integrals and Martingales (2005)... (previous)... (next) $4.3 \ \text{(ii)}$