$\displaystyle \Omega=[0,1]$ with its borelian subset and P the Lebesgue measure over [0,1].

Let $\displaystyle A_{n}=\bigcup_k]\frac{2(k-1)}{2^n},\frac{2k-1}{2^n}]$ , k from 1 to 2^(n-1).

I have showed that the $\displaystyle (A_{n})$ family is mutually independent.

Now,

Let $\displaystyle X_{n}=I_{A_{n}}=\sum_k I_{]\frac{2(k-1)}{2^n},\frac{2k-1}{2^n}]}, n \in N$ from [0,1] in {0,1} (k from 1 to 2^(n-1))

I have to show that the random variables are mutually independent too, but I can't get there...

Thank you for helping me