On the Minimal Extension of Increasing *-Weakly Semicontinuous Sublinear Functionals from L₊^∞

Properties of increasing ∗-weakly lower semicontinuous (LSC) sublinear functionals on the positive cone $L^\infty_+$ 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 $\mathcal{A}_N$ of closed convex subsets A ⊂ L¹₊ satisfying the condition N(φ) = sup{⟨φ, f⟩: f ∈ A} ∀φ ∈ $L^\infty_+$ and the class $\mathcal{G}_N$ of increasing ∗-weakly LSC sublinear extentions of N from $L^\infty_+$ to $L^\infty$. $\mathcal{A}_N$ is ordered for inclusion and $\mathcal{G}_N$ is ordered in a natural way: Q₁ ≤ Q₂ ⇔ Q₁ (φ) ≤ Q₂ (φ) ∀φ ∈ $L^\infty$, where Q₁ , Q₂ ∈ $\mathcal{G}_N$. The existence of the minimal elements in $\mathcal{A}_N$ and in $\mathcal{G}_N$ is proved and their description is given. The orders induced in $L^{\infty*}$ by convex cones conjugate to $K_\varepsilon$ = {φ ∈ $L^\infty$: φ ≥ ε‖φ‖}, ε > 0, are of substantial use in proving the theorem.