  Consider a financial market in which an agent trades with utility-induced
restrictions on wealth. For a utility function which satisfies the condition of
reasonable asymptotic elasticity at $-\infty$ we prove that the utility-based
super-replication price of an unbounded (but sufficiently integrable)
contingent claim is equal to the supremum of its discounted expectations under
pricing measures with finite {\it loss-entropy}. For an agent whose utility
function is unbounded from above, the set of pricing measures with finite
loss-entropy can be slightly larger than the set of pricing measures with
finite entropy. Indeed, the former set is the closure of the latter under a
suitable weak topology.
  Central to our proof is the representation of a cone $C_U$ of utility-based
super-replicable contingent claims as the polar cone to the set of finite
loss-entropy pricing measures. The cone $C_U$ is defined as the closure, under
a relevant weak topology, of the cone of all (sufficiently integrable)
contingent claims that can be dominated by a zero-financed terminal wealth.
  We investigate also the natural dual of this result and show that the polar
cone to $C_U$ is generated by those separating measures with finite
loss-entropy. The full two-sided polarity we achieve between measures and
contingent claims yields an economic justification for the use of the cone
$C_U$, and an open question.
