Essential supremum and essential maximum with respect to random preference relations

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X$X$ endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X$X$ endowed by a random preference relation to the space L⁰(X)$L^0\left(X\right)$ of X$X$-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.