1. a) Let X be a random variable defined on (, F,Pr). The collection of subsets of of the form { : X () B} for B B (x) is called the -field generated by X. Show that this collection of sets is a -field.

b). Show that B (x) := {B x : B B} is a -field if x is a Borel set (i.e., x B), where B is the Borel -field on R.