Proof of The σ-Algebra Generated by Measurable Functions into Metric Spaces

Throughout, preimages $X^{-1}(B)$ are subsets of $\Omega$ (Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (i)), $\mathcal{C}_{X}$ is the image of $\mathcal{B}(S)$ under the preimage map of Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (iii), and unions of families of sets are sets (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)), so $\mathcal{C}_{X}$, $\mathcal{C}$ and the union in (ii) are sets of subsets of $\Omega$, and $\sigma(\cdot)$ of each is a $\sigma$-algebra on $\Omega$ containing it (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)).

Measurability as an inclusion. Let $\mathcal{G}$ be a $\sigma$-algebra on $\Omega$. By Measurable Map, and the Set of Measurable Maps (i), a function $X:\Omega\to S$ is $\mathcal{G}/\mathcal{B}(S)$-measurable if and only if $X^{-1}(B)\in\mathcal{G}$ for every $B\in\mathcal{B}(S)$, that is, if and only if

$$\mathcal{C}_{X}\subseteq\mathcal{G}.$$

In the situation of (ii), by the definition of a union (Indexed Families of Sets and Their Union, Intersection and Product (ii)), every $X_{j}$ is $\mathcal{G}/\mathcal{B}(S_{j})$-measurable if and only if $\mathcal{C}\subseteq\mathcal{G}$.

Proof of (i). Since $\mathcal{C}_{X}\subseteq\sigma(\mathcal{C}_{X})$, $X$ is $\sigma(\mathcal{C}_{X})/\mathcal{B}(S)$-measurable. If $X$ is $\mathcal{G}/\mathcal{B}(S)$-measurable, then $\mathcal{C}_{X}\subseteq\mathcal{G}$, hence $\sigma(\mathcal{C}_{X})\subseteq\mathcal{G}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv)). As $X$ is $\mathcal{F}/\mathcal{B}(S)$-measurable, the case $\mathcal{G}=\mathcal{F}$ gives $\sigma(\mathcal{C}_{X})\subseteq\mathcal{F}$.

Proof of (ii). The same argument with $\mathcal{C}$ in place of $\mathcal{C}_{X}$ gives the first three assertions. For the last, let $\mathcal{D}=\bigcup_{j\in J}\sigma(\mathcal{C}_{X_{j}})$, 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). Since $\mathcal{C}_{X_{j}}\subseteq\sigma(\mathcal{C}_{X_{j}})$ for every $j$, we have $\mathcal{C}\subseteq\mathcal{D}$, so $\sigma(\mathcal{C})\subseteq\sigma(\mathcal{D})$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (v)). Conversely, every $X_{j}$ is $\sigma(\mathcal{C})/\mathcal{B}(S_{j})$-measurable, so $\sigma(\mathcal{C}_{X_{j}})\subseteq\sigma(\mathcal{C})$ by (i); hence $\mathcal{D}\subseteq\sigma(\mathcal{C})$, and $\sigma(\mathcal{D})\subseteq\sigma(\mathcal{C})$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv).