σ-Algebras Generated by Random Elements

In the setting of Probability Spaces: Notation, let $(S,d)$ be a metric space, let $J$ be a set and let $((S_{j},d_{j}))_{j\in J}$ be a family of metric spaces.

(i) The $\sigma$-algebra $\sigma(X)$. For a random element $X$ of $S$, the $\sigma$-algebra generated by $X$ is

$$\sigma(X)=\sigma\bigl(\{X^{-1}(B):B\in\mathcal{B}(S)\}\bigr)$$

(Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (iii)), a $\sigma$-algebra on $\Omega$ contained in $\mathcal{F}$ (The σ-Algebra Generated by Measurable Functions into Metric Spaces (i)).

(ii) The $\sigma$-algebra $\sigma(X_{j}:j\in J)$. For a family $(X_{j})_{j\in J}$ in which each $X_{j}$ is a random element of $S_{j}$, the $\sigma$-algebra generated by $(X_{j})_{j\in J}$ is

$$\sigma(X_{j}:j\in J)=\sigma\Bigl(\bigcup_{j\in J}\{X_{j}^{-1}(B):B\in\mathcal{B}(S_{j})\}\Bigr),$$

a $\sigma$-algebra on $\Omega$ contained in $\mathcal{F}$ (The σ-Algebra Generated by Measurable Functions into Metric Spaces (ii)).