Random Variables Assignments-

1) Let f be a random variable. Assume that f ≥ 0. Assume that f and f² are independent. Prove that f is almost surely a constant.

2) Prove that two random variables are independent if and only if for every pair of Borel sets B₁ and B₂ the following equality holds

E(χ(B₁) · χ(B₂)) = E(χ(B₁)) · E(χ(B₂))

3) Let (?, A, µ) be a probability space. Prove that the collection of all self-independent sets is a σ-algebra. Prove that this σ-algebra is independent with A.