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I think that if we use Girsanov's theorem there are cases in which the local martingale stochastic exponential is not uniformly integrable (Delbaen and Schachermayer constructed such a stochastic process). My question is that supposed we know that the optimal equivalent martingale measure is the variance optimal ( i.e. the one which minimizes the variance of the martingale measure), so its existence is granted, we want to prove that this measure is unique.