\begin{definition}[Definition:Independent Random Variables/General Definition]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. 
Let $\struct {S, \Sigma'}$ be a measurable space.
Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of random variables on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.
For each $i \in \N$, let $\map \sigma {X_i}$ be the $\sigma$-algebra generated by $X_i$. 
We say that $\sequence {X_n}_{n \mathop \in \N}$ is a '''sequence of independent random variable''' {{iff}}:
:$\sequence {\map \sigma {X_n} }_{n \mathop \in \N}$ is a sequence of independent $\sigma$-algebras.
\end{definition}