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

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 A N

of closed convex subsets A ⊂ L¹₊ satisfying the condition N(φ) = sup{⟨φ, f⟩: f ∈ A} ∀φ ∈

L^\infty_+ L + ∞

and the class

\mathcal{G}_N G N

of increasing ∗-weakly LSC sublinear extentions of N from

L^\infty_+ L + ∞

to

L^\infty L ∞

.

\mathcal{A}_N A N

is ordered for inclusion and

\mathcal{G}_N G N

is ordered in a natural way: Q₁ ≤ Q₂ ⇔ Q₁ (φ) ≤ Q₂ (φ) ∀φ ∈

L^\infty L ∞

, where Q₁ , Q₂ ∈

\mathcal{G}_N G N

. The existence of the minimal elements in

\mathcal{A}_N A N

and in

\mathcal{G}_N G N

is proved and their description is given. The orders induced in

L^{\infty*} L ∞ ∗

by convex cones conjugate to

K_\varepsilon K ε

= {φ ∈

L^\infty L ∞

: φ ≥ ε‖φ‖}, ε > 0, are of substantial use in proving the theorem.