... thus, µ ∼ ν.(b) Show that ifdνdµ>0, µ-a.e. onX and ifµ and νareσ-finite, thendµdνexists and dµdν=dνdµ−1, µ − a.e. and ν − a.e. on X.Solution(a) For every E ∈ A, by definition, ... =n∈NCAnthen E is countable and E ⊂ C and An∈ σ(E) for all n ∈ N.By definition of σ-algebra,n∈NAn∈ σ(E), and son∈NAn∈ B.Thus, B is a σ-algebra of subsets of X and E ⊂ B. Hence,σ(E) ... that {µ, ν} is a Jordan de-composition of λ, and E and F are two measurable subsets of X such thatE ∩ F = ∅, E ∪ F = X, E is a null set for ν and F is a null set for ν.Showthat {E, F } is...