Measure is Monotone

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Then $\mu$ is monotone, that is:

$\forall E, F \in \Sigma: E \subseteq F \implies \map \mu E \le \map \mu F$

Resolution of the Identity

Let $X$ be a topological space.

Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\map B \HH$ be the space of bounded linear transformations on $\HH$.

Let $\le_{\map B \HH}$ be the canonical preordering of $\map B \HH$.

Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.

Let $E, F \in \map \BB X$ be such that $E \subseteq F$.

Then:

$\map \EE E \le_{\map B \HH} \map \EE F$

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