14.5 Duality Results and Maximal Elements 317Proof.Clearly (2) implies (1) by Theorem 14.5.12.
For the reverse implication take nowQas in (1), then the duality result
(Theorem 14.5.9) givesα∈Ras well as aw-admissible integrandHsuch that
f≤α+(H·S)∞,whereα=supR∈Me;ER[w]<∞[f]. Here we use explicitly
that the infimum in the duality theorem is a minimum. But then it follows
fromEQ[w]<∞and from the equalityEQ[f]=αthatf=α+(H·S)∞
and thatH·Sis aQ-uniformly integrable martingale.