The σ-Algebra Generated by a Set of Subsets

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, and let $\mathcal{C}$ be a subset of $\mathcal{P}(X)$.

(i) Let $\mathcal{S}_{\mathcal{C}}$ be the set of those $\mathcal{G}\in\mathcal{P}(\mathcal{P}(X))$ that are $\sigma$-algebras on $X$ with $\mathcal{C}\subseteq\mathcal{G}$ (this set exists by Predicative Formulas for σ-Algebras, π-Systems and λ-Systems, and the Sets of Them Containing a Given Family (i)). Every $\sigma$-algebra $\mathcal{G}$ on $X$ with $\mathcal{C}\subseteq\mathcal{G}$ is an element of $\mathcal{S}_{\mathcal{C}}$, since $\mathcal{G}\subseteq\mathcal{P}(X)$ by σ-Algebra and Measurable Space (i). Moreover $\mathcal{P}(X)\in\mathcal{S}_{\mathcal{C}}$: $\mathcal{C}\subseteq\mathcal{P}(X)$; $X\in\mathcal{P}(X)$ and $X\setminus A\in\mathcal{P}(X)$ for every set $A$, by The Union Set and the Power Set of a Set (ii), the difference $X\setminus A$ being a subset of $X$; and for a sequence $(A_{n})$ in $\mathcal{P}(X)$, the union $\bigcup_{n\in\mathbb{N}}A_{n}$ is 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), each of its elements lies in some $A_{n}$ by Indexed Families of Sets and Their Union, Intersection and Product (ii) and hence in $X$, so it is an element of $\mathcal{P}(X)$ by The Union Set and the Power Set of a Set (ii). So $\mathcal{S}_{\mathcal{C}}$ has an element, and the intersection $\bigcap_{\mathcal{G}\in\mathcal{S}_{\mathcal{C}}}\mathcal{G}$ of the family $\mathcal{G}\mapsto\mathcal{G}$ indexed by $\mathcal{S}_{\mathcal{C}}$ is 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 (ii). The $\sigma$-algebra generated by $\mathcal{C}$ is the set $\sigma(\mathcal{C})=\bigcap_{\mathcal{G}\in\mathcal{S}_{\mathcal{C}}}\mathcal{G}$.

(ii) For every set $A$: $A\in\sigma(\mathcal{C})$ if and only if $A\subseteq X$ and $A\in\mathcal{G}$ for every $\sigma$-algebra $\mathcal{G}$ on $X$ with $\mathcal{C}\subseteq\mathcal{G}$. Indeed, by Indexed Families of Sets and Their Union, Intersection and Product (iii), $A\in\sigma(\mathcal{C})$ if and only if $A\in\mathcal{G}$ for every $\mathcal{G}\in\mathcal{S}_{\mathcal{C}}$; the elements of $\mathcal{S}_{\mathcal{C}}$ are exactly the $\sigma$-algebras on $X$ containing $\mathcal{C}$, as noted in (i); and $A\in\mathcal{P}(X)$, which holds when $A\in\sigma(\mathcal{C})$ because $\mathcal{P}(X)\in\mathcal{S}_{\mathcal{C}}$, means $A\subseteq X$ by The Union Set and the Power Set of a Set (ii).