Mathematical statistics

Question 4 Suppose {12 3,3,! is the uniform probability space; that is {[yl], B, ,1) Where zit-is the uniform probability distribution. Define X{m) = w [a] Does there exist a bounded random variable that is both independent of X and not constant almost surely? [b] Define '1" = 1 X}. Construct a random 1Lurariable Z which is not almost surely constant and such that Z and 'r' are independent