Maximality and numéraires in convex sets of nonnegative random variables

We introduce the concepts of max-closedness and numéraires of convex subsets of L+0, the nonnegative orthant of the topological vector space L0 of all random variables built over a probability space, equipped with a topology consistent with convergence in probability. Max-closedness asks that maximal elements of the closure of a set already lie on the set. We discuss how numéraires arise naturally as strictly positive optimisers of certain concave monotone maximisation problems. It is further shown that the set of numéraires of a convex, max-closed and bounded set of L+0 that contains at least one strictly positive element is dense in the set of its maximal elements.