I have the following problem: We have to sigma algebras $\sigma (u)$ and $\sigma (X_{i})$ which are not independent. $u$ and $X_{i}$ are random variables.
Now my question: Why is it possible, in some cases, to find $X_{k}, X_{k+1} .....$ so that $\sigma (u)$ is independent from $\sigma (X_{i}, X_{k}, X_{k+1} .....)$.
The only thing I know is that the sigma Algebra $\sigma (X_{i}, X_{k}, X_{k+1} .....)$ would be finer than $\sigma (X_{i})$.