(1) Give an example of a uniformly integrable family of random variables cfw_Xn nN for which the dominated convergence theorem does not apply - that is, there is no integrable X 0 with |Xn | X for
(2) Show that if Fn
F and Yn Y in L1 , t
(1) Give an example of a martingale Xn with Xn a.s.
(2) Suppose Xn and Xn are supermartingales with respect to Fn , and N is a stopping time with
XN XN . Show that Yn = Xn 1(N > n) + Xn 1(N n) is a supermartingale
(3) Show that any
(1) Show that if X = Y on B G , then E [X |G ] = E [Y |G ] a.s. on B .
(2) Suppose X 0 and E [X ] = . Show that there is a unique Y G with 0 Y so that
Y dP for all A G .
(Hint: Consider Xn = X n.)
(3) Prove the following conditional l