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We introduce the concepts of max-closedness and numeraires of convex subsetsin the nonnegative orthant of the topological vector space of all randomvariables built over a probability space, equipped with a topology consistentwith convergence in probability. Max-closedness asks that maximal elements ofthe closure of a set already lie on the set. We discuss how numeraires arisenaturally as strictly positive optimisers of certain concave monotonemaximisation problems. It is further shown that the set of numeraires of aconvex, max-closed and bounded set of of nonnegative random variables thatcontains at least one strictly positive element is dense in the set of itsmaximal elements.

Maximality and numeraires in convex sets of nonnegative random variables