Null Sets Closed under Subset

Theorem

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

Let $N \in \Sigma$ be a $\mu$-null set, and let $M \in \Sigma$ be a subset of $N$.

Then $M$ is also a $\mu$-null set.

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

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

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

Let $E, E' \in \map \BB X$ be such that:

$E \subseteq E'$ and $\map \EE {E'} = {\mathbf 0}_{\map B \HH}$

Then $\map \EE E = {\mathbf 0}_{\map B \HH}$.

As $\mu$ is a measure, $\map \mu M \ge 0$.

Also, by Measure is Monotone, $\map \mu M \le \map \mu N = 0$.

Hence $\map \mu M = 0$, and $M$ is a $\mu$-null set.

$\blacksquare$