Independence of σ-Algebras: π-Systems, Grouping and Pairs

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

(i) Subfamilies. Let $(\mathcal{C}_{j})_{j\in J}$ be an independent family of subsets of $\mathcal{F}$, let $J'\subseteq J$, and for $j\in J'$ let $\mathcal{D}_{j}\subseteq\mathcal{C}_{j}$. Then the family $(\mathcal{D}_{j})_{j\in J'}$ is independent.

(ii) Independent $\pi$-systems generate independent $\sigma$-algebras. Let $(\mathcal{C}_{j})_{j\in J}$ be an independent family of $\pi$-systems on $\Omega$ contained in $\mathcal{F}$. Then the family $(\sigma(\mathcal{C}_{j}))_{j\in J}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv)) is independent.

(iii) Grouping. Let $(\mathcal{G}_{j})_{j\in J}$ be an independent family of $\sigma$-algebras on $\Omega$ contained in $\mathcal{F}$, let $K$ be a set and let $(J_{k})_{k\in K}$ be a family of pairwise disjoint subsets of $J$. For $k\in K$ let

$$\mathcal{H}_{k}=\sigma\Bigl(\bigcup_{j\in J_{k}}\mathcal{G}_{j}\Bigr).$$

Then the family $(\mathcal{H}_{k})_{k\in K}$ is independent.

(iv) Two $\sigma$-algebras as a family. Let $\mathcal{G}$ and $\mathcal{H}$ be $\sigma$-algebras on $\Omega$ contained in $\mathcal{F}$. Then $\mathcal{G}$ and $\mathcal{H}$ are independent if and only if the family $(\mathcal{G}_{k})_{k\in[2]}$ with $\mathcal{G}_{1}=\mathcal{G}$ and $\mathcal{G}_{2}=\mathcal{H}$ is independent.

(v) Two independent $\pi$-systems. Let $\mathcal{C}$ and $\mathcal{D}$ be $\pi$-systems on $\Omega$ contained in $\mathcal{F}$ with

$$P(C\cap D)=P(C)\,P(D)\qquad(C\in\mathcal{C},\ D\in\mathcal{D}).$$

Then $\sigma(\mathcal{C})$ and $\sigma(\mathcal{D})$ are independent.