Exercise1.Let = [0,1] andFbe the-algebra generated by the dyadic intervals

Exercise 1. Let 1 = [0, 1] and F be the o-algebra generated by the dyadic intervals k k+1 2' 2 k = 0, 1 . (a) List all events in F. (b) Give an example of a function X : 0 - R such that X takes only finite number of values and is F-measurable. (c) Give an example of a function Z : 0 - R such that Z takes only finite number of values and is not F-measurable. (d) Let P be the uniform distribution on [0, 1] and let Y(w) = w for w E [0, 1]. Compute E(Y X) where X is the example you have given in (b) (e) Let P be discrete, that is P 4 k = 0, 1, 2, 3. For Y defined in (d) find again E(Y| X)