Thank you for the reply!
I was wondering, the condition of the lemma that $A_1, A_2,...,A_n$ are independent pi-systems is too strong, stronger than pairwise independent, not sure how to apply it in case of pairwise independent systems. In any case I will read the sections in Billingsley's...
I am trying to establish whether the following is true (my intuition tells me it is), more importantly if it is true, I need to establish a proof.
If $X_1, X_2$ and $X_3$ are pairwise independent random variables, then if $Y=X_2+X_3$, is $X_1$ independent to $Y$? (One can think of an...