Abstract

Properties of increasing ∗-weakly lower semicontinuous (LSC) sublinear functionals on the positive cone L+∞L^\infty_+ are of importance for study miscellaneous non-controlled factors from a unified viewpoint based on the notion of sublinear expectation [3, 4 and 5]. For every such functional N there are defined the class AN\mathcal{A}_N of closed convex subsets A ⊂ L¹₊ satisfying the condition N(φ) = sup{⟨φ, f⟩: f ∈ A} ∀φ ∈ L+∞L^\infty_+ and the class GN\mathcal{G}_N of increasing ∗-weakly LSC sublinear extentions of N from L+∞L^\infty_+ to L∞L^\infty. AN\mathcal{A}_N is ordered for inclusion and GN\mathcal{G}_N is ordered in a natural way: Q₁ ≤ Q₂ ⇔ Q₁ (φ) ≤ Q₂ (φ) ∀φ ∈ L∞L^\infty, where Q₁ , Q₂ ∈ GN\mathcal{G}_N. The existence of the minimal elements in AN\mathcal{A}_N and in GN\mathcal{G}_N is proved and their description is given. The orders induced in L∞∗L^{\infty*} by convex cones conjugate to KεK_\varepsilon = {φ ∈ L∞L^\infty: φ ≥ ε‖φ‖}, ε > 0, are of substantial use in proving the theorem.

Author information

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A.A. Lebedev

Faculty of Cosmonautics, Moscow Aviation Institute, Moscow, 125871, Volokolamskoe shosse 4, Russia.

V.A. Lebedev

Faculty of Machanics and Mathematics, Moscow State University, Moscow, 119899, Vorobjevy gory, Russia.

Keywords

  • sublinear functionals
  • closed in measure convex sets in L¹
  • lower semicontinuity
  • sublinear expectation
  • quasi-extremal elements of convex sets

References

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Suggested citation

A. A. Lebedev, V. A. Lebedev. “On the Minimal Extension of Increasing *-Weakly Semicontinuous Sublinear Functionals from L_+^∞.” Journal of Convex Analysis 4 (1997), No. 1, 177–187. https://doi.org/10.68381/jca04009

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Published by Heldermann Verlag, 1997. Rights now held by Banach Press.