In an incomplete market the price of a claim f in general can not be uniquely identified by no arbitrage arguments. However, the "classical" super-replication price is a sensible indicator of the (maximum selling) value of the claim. When f satisfies certain pointwise conditions (e.g. f is bounded from below), the super-replication price is equal to sup_{Q}E_{Q}[f], where Q varies on the whole set of pricing measures. Unfortunately, this price is often too high: a typical situation is here discussed in the examples. We thus define the less expensive weak super-replication price and we relax the requirements on f by asking just for "enough" integrability conditions. By building up a proper duality theory, we show its economic meaning and its relation with the investor's preferences. Indeed, it turns out that the weak super-replication price of f coincides with sup_{Q∈M_{Φ}}E_{Q}[f], where M_{Φ} is the class of pricing measures with finite generalized entropy (i.e. E[Φ(((dQ)/(dP)))]<∞) and where Φ is the convex conjugate of the utility function of the investor.