5.2 No Free Lunch 83(iv) C∩ball∞is closed with respect to the topology of convergence in measure.
Proof.(ii)⇒(i): This is the crucial implication. It is a direct consequence of
Krein-ˇSmulian theorem ([S 94, Theorem IV 6.4]).
(i)⇒(ii) trivial.
(i)⇔(i’) and (ii)⇔(ii’): this is the assertion of the Mackey-Arens theorem
([S 94, Theorem IV 3.2]).
(ii)⇒(iv): Let (fn)∞n=1be a Cauchy sequence inC∩ball∞with respect
to convergence in measure. By passing to a subsequence we may assume that
(fn)∞n=1converges a.s. to somef 0 ∈ball∞. We have to show thatf 0 ∈C.
This follows from Lebesgue’s theorem on dominated convergence, which
implies that, for eachg∈L^1 (Ω,F,P), we have limn→∞E[fng]=E[f 0 g].
Hence (fn)∞n=1converges tof 0 in theσ(L∞,L^1 )-topology so that by hypoth-
esis (ii) we have thatf 0 ∈C.
(iv)⇒(iii’)⇒(iii)⇒(ii’): trivial, noting that the Mackey topology
τ(L∞,L^1 ) is finer than the topology induced by‖.‖p, for any 0<p<∞.