Give an example of a sequence P n converging weakly to P and a function f that is continuous but not bounded such that  f dP n does not converge to  f dP. Give an example of a sequence P n converging...

Give an example of a sequence P_n
converging weakly to P and a function f that is continuous but not bounded such that  f dP_n
does not converge to  f dP.

Give an example of a sequence P_n
converging weakly to P and a function f that is bounded but not continuous such that  f dP_n
does not converge to  f dP.

Show that if X_n
converges weakly to X and Y_n
converges in probability to 0, then X_nY_n
converges in probability to 0.