Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{R}$ be the set of real numbers, and let $\mathcal{B}(\mathbb{R})$ be the Borel $\sigma$-algebra.

Generated $\sigma$-algebra of a family of random variables. Let $J$ be a nonempty set and let $(X_j)_{j\in J}$ be a family of random variables on $(\Omega,\mathcal{F},P)$. The $\sigma$-algebra generated by $(X_j)_{j\in J}$, written $\sigma(X_j: j\in J)$, is the generated $\sigma$-algebra $\sigma(\mathcal{C})$ on $\Omega$ of the family

Since each $X_j$ is a random variable, every member of $\mathcal{C}$ lies in $\mathcal{F}$, and $\mathcal{F}$ is a $\sigma$-algebra containing $\mathcal{C}$; hence $\sigma(X_j:j\in J)\subseteq\mathcal{F}$, i.e., it is a sub-$\sigma$-algebra of $\mathcal{F}$.

For a single random variable $X$ we write $\sigma(X)=\sigma(X:j\in\{1\})$. In this case the family $\{X^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ is itself a $\sigma$-algebra: $X^{-1}(\mathbb{R})=\Omega$, $\Omega\setminus X^{-1}(B)=X^{-1}(\mathbb{R}\setminus B)$, and $\bigcup_m X^{-1}(B_m)=X^{-1}(\bigcup_m B_m)$, with $\mathbb{R}\setminus B$ and $\bigcup_m B_m$ Borel because $\mathcal{B}(\mathbb{R})$ is a $\sigma$-algebra. Hence $\sigma(X)=\{X^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$.

Independence of $\sigma$-algebras. Let $B$ be a nonempty set and for each $b\in B$ let $\mathcal{G}_b\subseteq\mathcal{F}$ be a sub-$\sigma$-algebra. The family $(\mathcal{G}_b)_{b\in B}$ is independent if for every finite nonempty set of distinct indices $b_1,\dots,b_p\in B$ and every choice of events $A_l\in\mathcal{G}_{b_l}$ ($1\le l\le p$),

with the finite product notation.

Because $\Omega\in\mathcal{G}_b$ for every $b$ and $P(\Omega)=1$, the family $(\mathcal{G}_b)_{b\in B}$ is independent if and only if every finite subfamily is independent, and events $A_1,\dots,A_r$ are independent if and only if the product identity above holds for every choice of indices after inserting $\Omega$ for omitted factors; in particular, random variables $X_1,\dots,X_r$ are independent in the sense of Independence of Events and of Random Variables if and only if the $\sigma$-algebras $\sigma(X_1),\dots,\sigma(X_r)$ are independent, since $\sigma(X_i)=\{X_i^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ as shown above.