Null Sets and Properties Holding Almost Everywhere

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $(X,\mathcal{F},\mu)$ be a measure space, so that $(X,\mathcal{F})$ is a measurable space and $\mu$ is a measure on it, with values in the set $[0,\infty]$ of The Extended Real Line, Its Order, and [0,∞] (ii). Let $\mathcal{P}(X)$ be the power set of $X$, let unions, intersections and differences of classes be those of The Boolean Operations on Classes, Disjointness, and the Universal Class (i), let $\subseteq$ be the subclass relation, and, for a sequence $(A_{m})$, let $\bigcup_{m\in\mathbb{N}}A_{m}$ be the union of $(A_{m})$ as a family indexed by $\mathbb{N}$. A subset of $X$ is $\mu$-null as in Null Set of a Measure (i), and a class holds $\mu$-almost everywhere ($\mu$-a.e.) as in A Property Holding Almost Everywhere (i).

(i) Let $B\in\mathcal{F}$. Then $B$ is $\mu$-null if and only if $\mu(B)=0$.

(ii) Let $N$ and $M$ be sets with $N\subseteq M\subseteq X$. If $M$ is $\mu$-null, then $N$ is $\mu$-null.

(iii) If $N$ and $M$ are $\mu$-null subsets of $X$, then $N\cup M$ is $\mu$-null.

(iv) Let $(B_{m})$ be a sequence in $\mathcal{F}$ with $\mu(B_{m})=0$ for every $m\in\mathbb{N}$. Then $\mu\bigl(\bigcup_{m\in\mathbb{N}}B_{m}\bigr)=0$, where $\bigcup_{m\in\mathbb{N}}B_{m}\in\mathcal{F}$ by σ-Algebra and Measurable Space (i).

(v) Let $(B_{m})$ be a sequence in $\mathcal{F}$ with $\mu(B_{m})=0$ for every $m\in\mathbb{N}$, and let $(N_{m})$ be a sequence in $\mathcal{P}(X)$ with $N_{m}\subseteq B_{m}$ for every $m\in\mathbb{N}$. Then $\bigcup_{m\in\mathbb{N}}N_{m}$ is $\mu$-null.

Let $Q$ and $R$ be classes.

(vi) If $Q$ holds $\mu$-a.e. and $X\cap Q\subseteq R$, then $R$ holds $\mu$-a.e.

(vii) If $Q$ and $R$ hold $\mu$-a.e., then $Q\cap R$ holds $\mu$-a.e.