The σ-Algebra Generated by Measurable Functions into Metric Spaces

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(\Omega,\mathcal{F})$ be a measurable space. For a metric space $(S,d)$ and a function $X:\Omega\to S$, let

$$\mathcal{C}_{X}=\{X^{-1}(B):B\in\mathcal{B}(S)\}$$

(i) One function. Let $(S,d)$ be a metric space and let $X:\Omega\to S$ be $\mathcal{F}/\mathcal{B}(S)$-measurable. Then $\sigma(\mathcal{C}_{X})$ is a $\sigma$-algebra on $\Omega$ with $\sigma(\mathcal{C}_{X})\subseteq\mathcal{F}$, $X$ is $\sigma(\mathcal{C}_{X})/\mathcal{B}(S)$-measurable, and $\sigma(\mathcal{C}_{X})\subseteq\mathcal{G}$ for every $\sigma$-algebra $\mathcal{G}$ on $\Omega$ for which $X$ is $\mathcal{G}/\mathcal{B}(S)$-measurable.

(ii) A family of functions. Let $J$ be a set, let $((S_{j},d_{j}))_{j\in J}$ be a family of metric spaces, and let $(X_{j})_{j\in J}$ be a family of functions $X_{j}:\Omega\to S_{j}$, each $\mathcal{F}/\mathcal{B}(S_{j})$-measurable. Let

$$\mathcal{C}=\bigcup_{j\in J}\mathcal{C}_{X_{j}}$$

Then $\sigma(\mathcal{C})$ is a $\sigma$-algebra on $\Omega$ with $\sigma(\mathcal{C})\subseteq\mathcal{F}$, every $X_{j}$ is $\sigma(\mathcal{C})/\mathcal{B}(S_{j})$-measurable, and $\sigma(\mathcal{C})\subseteq\mathcal{G}$ for every $\sigma$-algebra $\mathcal{G}$ on $\Omega$ for which every $X_{j}$ is $\mathcal{G}/\mathcal{B}(S_{j})$-measurable. Moreover,

$$\sigma(\mathcal{C})=\sigma\Bigl(\bigcup_{j\in J}\sigma(\mathcal{C}_{X_{j}})\Bigr).$$