Exercise 3. Assume that X is an infinite set and let S = {{x}, x = X}. Find the σ-algebra generated by S (hint: distinguish the countable and uncountable case). Exercise 4. (Characterization of measurable functions.) Let (X,A) be a measurable space and f: X-R 1. Show that = {BR: f¹(B) EA} is a σ-algebra. 2. Let be a generating set of the Borel algebra (R). Show that the following assertions are equivalent: (a) f is measurable, (b) f¹(C) EA, for all C = C. Exercise 5. Let (X,A) be a measurable space and suppose that f: XR is increasing. Prove that f is measurable.