(Convergence: Examples and counterexamples). A sequence {ξn, n ∈ N} converges setwise to ξ , if for any A ∈ X , limn→∞ ξn(A) = ξ (A). (a) Show that total variation convergence implies setwise...

(Convergence: Examples and counterexamples). A sequence {ξn, n ∈ N} converges setwise to ξ , if for any A ∈ X , limn→∞ ξn(A) = ξ (A).

(a) Show that total variation convergence implies setwise convergence

(b) Set X = R. Show that the sequence {δ1/n ,n ∈ N ∗} converges weakly to δ0 but does not converge setwise.

(c) Let X = [0,1] and let ξn denote the probability measure with density x 7→ 1 + sin(2πnx) with respect to Lebesgue measure on [0,1]. Show that the sequence {ξn, n ∈ N} converges setwise to the uniform distribution on [0,1] but does not converge in the total variation distance.