Let X be a set and A a σ-algebra of subsets of X. (a) What doesit mean for a function f : X → R to be measurable? [2%] (b) If fand g are measurable and α, β ∈ R show that the function αf + βg isalso measurable. [7%] (c) (i) Suppose that f is a measurablefunction. Is |f| measurable? (Give a proof or a counterexample.)[3%] (ii) Suppose that |f| is a measurable function. Is fmeasurable? (Give a proof or a counterexample.) [3%] (iii) Let X =R and let f(x) = 3 if x is rational and f(x) = 1 if x is not. Whatis the smallest σ-algebra of subsets of R with respect to which fis measurable?
