Measure is Monotone
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Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Then $\mu$ is monotone, that is:
- $\forall E, F \in \Sigma: E \subseteq F \implies \map \mu E \le \map \mu F$
Proof
A direct corollary of Non-Negative Additive Function is Monotone.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.3 \ \text{(ii)}$
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $1.2$: Measures