Exercise 2.31 (Independence of $\sigma$-algebras)

Question

If $(X_i)_{i \in A}$ are a collection of random variables, show that $(X_i)_{i \in A}$ are jointly independent random variables if and only if $(\sigma(X_i))_{i \in A}$ are jointly independent $\sigma$-algebras.

Elu Thingol · Accepted Answer

\begin{itemize}
\item[$\implies$] Suppose that $(X_i)_{i\in A}$ are independent. We use the equivalent definition of the independence of $\sigma$-algebras, i.e., we verify whether
$$\displaystyle {\bf P}\left( \bigcap_{i \in B} E_i \right) = \prod_{i \in B} {\bf P}( E_i )$$
whenever $B$ is a finite subset of $A$ and $E_i \in {\mathcal F}_i$ for $i \in B$. Hence, pick an arbitrary collection $(E_i)_{i\in B}$ satisfying these conditions. By definition, there is a collection $(S_i)_{i\in B}$ of measurable sets in the range of $X$ such that $X^{-1}(S_i) = E_i$ for each $i$. We then have
$$\displaystyle {\bf P}\left( \bigcap_{i \in B} E_i \right) = {\bf P}\left( \bigwedge_{i \in B} X_i \in S_i \right) = \prod_{i \in B} {\bf P}\left(X_i \in S_i \right) = \prod_{i \in B}  {\bf P}( E_i ).$$
\item[$\impliedby$] Suppose that $\sigma(X_i)_{i \in A}$ are independent, i.e., whenever $Y_i$ is a random variable measurable with respect to $\sigma(X_i)$ for $i \in A$, the tuple $(Y_i)_{i \in A}$ is jointly independent. Since $(X_i)_{i \in A}$ is one of such tuples, the assertion holds trivially.
\end{itemize}

Exercise 2.31 (Independence of $\sigma$-algebras)

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Exercise 2.31 (Independence of $\sigma$-algebras)

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