σ-Algebra and Measurable Space

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $X$ be a set, let $\mathcal{P}(X)$ be its power set, let $X\setminus A$ be the difference of $X$ and a set $A$, and, for a sequence $(A_{n})$, let $\bigcup_{n\in\mathbb{N}}A_{n}$ be the union of $(A_{n})$ as a family indexed by $\mathbb{N}$, a set by The Union and Product of a Family of Sets Indexed by a Set, and the Intersection of a Family with an Inhabited Index Class, Are Sets (i).

(i) A $\sigma$-algebra on $X$ is a set $\mathcal{F}$ that is a subset of $\mathcal{P}(X)$ and satisfies: $X\in\mathcal{F}$; $X\setminus A\in\mathcal{F}$ for every $A\in\mathcal{F}$; and $\bigcup_{n\in\mathbb{N}}A_{n}\in\mathcal{F}$ for every sequence $(A_{n})$ in $\mathcal{F}$.

(ii) A measurable space is an ordered pair $(X,\mathcal{F})$ of a set $X$ and a $\sigma$-algebra $\mathcal{F}$ on $X$. The elements of $\mathcal{F}$ are the measurable sets of $(X,\mathcal{F})$.