Independence of Classes of Events and of σ-Algebras

In the setting of Probability Spaces: Notation, let $J$ be a set.

(i) Classes of events. A family $(\mathcal{C}_{j})_{j\in J}$ of subsets of $\mathcal{F}$ is independent if for every $n\in\mathbb{N}$, all distinct indices $j_{1},\dots,j_{n}\in J$ and all events $A_{1}\in\mathcal{C}_{j_{1}},\dots,A_{n}\in\mathcal{C}_{j_{n}}$,

$$P\Bigl(\bigcap_{k=1}^{n}A_{k}\Bigr)=\prod_{k=1}^{n}P(A_{k}).$$

(ii) $\sigma$-Algebras. A family $(\mathcal{G}_{j})_{j\in J}$ of $\sigma$-algebras on $\Omega$ contained in $\mathcal{F}$ is independent if it is independent as a family of subsets of $\mathcal{F}$, as in (i).

(iii) Two independent $\sigma$-algebras. Two $\sigma$-algebras $\mathcal{G}$ and $\mathcal{H}$ on $\Omega$ contained in $\mathcal{F}$ are independent if

$$P(A\cap B)=P(A)\,P(B)\qquad(A\in\mathcal{G},\ B\in\mathcal{H}).$$