Please do both parts

Problem 1. (5 points) Let (S, B) be a measurable space and X a r.v. taking values in R. Let o(X) be the o-field generated by X andIB(R) the Borel o-field on R. (1) Show that if f : R - R is a Borel measurable function, then Z = foX : 0 - Ris o(X)/ B(R) measurable. (2) Show that, conversely, for any Z : 0 - R which is o(X) / B(R) measurable, there exists a Borel measurable function f : R - R such that Z = fo X. (Hint: start with indicator functions and approximate an o(X) / 8(R) function by simple functions.)