Mathematics – Functional Analysis
Scientific paper
2012-02-21
Mathematics
Functional Analysis
11 pages
Scientific paper
We introduce the concepts of max-closedness and outer support points of convex sets in the nonnegative orthant of the topological vector space of all random variables built over a probability space, equipped with a topology consistent with convergence of sequences in probability. Max-closedness asks that maximal elements of the closure of a set already lie on the set. We show that outer support points arise naturally as optimizers of concave monotone maximization problems. It is further shown that the set of outer support points of a convex, max-closed and bounded set of nonnegative random variables is dense in the set of its maximal elements, which can be regarded as a version of the celebrated Bishop-Phelps theorem in a space that even fails to be locally convex.
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