1.*8. For any two sets Al and A2 in qT, define then p is a pseudo-metric in the space of sets in :F; call the resulting metric space M(0T, g"». PIOve that fOl each integrable r .v. X the mapping of MC~ , go)

to ll?l given by A -+ .~ X dq? is continuous. Similarly, the mappmgs on M(::f , ;:7J1) x M(~ , q?) to M(~ , ,9?) given by are all continuous. If (see Sec. 4.2 below)

lim sup A'l -lIm mf A'l II n modulo a null set, we denote the common equivalence class of these two sets by limll All' Prove that in this case {All} converges to limll All in the metric p. Deduce Exercise 2 above as a special case.