Maximum Lebesgue extension of monotone convex functions

Abstract

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a "nice" dual representation of the function. arXiv:1304.7934v2 [math.FA] 19 Jan 2014 2 K. OwariAs a first (trivial) example, we briefly see what happens when ϕ 0 is linear.Example 1.2. Let ϕ 0 be a positive (monotone) linear functional on L ∞ . Then it is finite-valued and identified with a finitely additive measure ν 0 (A) := ϕ 0 (1 A ) as ϕ 0 (X) = Ω Xdν 0 , while (1.2) is equivalent to saying that ν 0 is σ-additive. If the latter is the case, the "usual" integralφ(X) := Ω Xdν 0 defines a Lebesgue-preserving extension of ϕ 0 to L 1 (ν 0 ) := {X ∈ L 0 : Ω |X|dν 0 < ∞}. On the other hand, if ϕ is a monotone convex function on a solid space X ⊂ L 0 with (1.1) and ϕ| L ∞ = ϕ 0 , it is easy that ϕ must be positive, linear and finite on X . Then |X|dν 0 = lim nφ (|X| ∧ n) = lim n ϕ 0 (|X| ∧ n) = lim n ϕ(|X| ∧ n) = ϕ(|X|) < ∞ if X ∈ X , hence X ⊂ L 1 (ν 0 ), where the first equality follows from the monotone convergence theorem, and the fourth from the Lebesgue property of ϕ on X . Similarly, but with X1 {|X|≤n} instead of |X| ∧ n, we see also that ϕ =φ| X . Namely, (φ, L 1 (ν 0 )) is the maximum Lebesgue-preserving extension of ϕ 0 . ♦

A non-exponential extension of Sanov’s theorem via convex duality

“…The form of our strong Cramér condition (3.1) was heavily inspired by the work of Owari [44] on continuous extensions of monotone convex functionals. In several cases of interest (namely, Propositions 4.1 and 5.3 below), it turns out that a converse to Lemma 3.1 is true, that is, the strong Cramér condition (3.1) is equivalent to the statement that ρ(λψ) < ∞ for all λ > 0.…”

Model spaces for risk measures

“…(iii) follows from [27,Proposition 4.10], and (iv) is an immediate consequence of the monotonicity of ρ(| · |).…”

“…The latter means X c,R ≤ 1 2 < 1 for some c > 0, and thus H R ⊆ B. Finally, as (H R , · R ) is a Banach lattice and both η and ρR are convex, monotone and finite-valued on (H R , · R ), ρR| H R and η| H R are continuous according to Remark 2.4(v).From Proposition 4.9 we can derive the following characterisation of M R , a result which can also be found as[27, Lemma 3.3].…”

On the Lebesgue property of monotone convex functions

“…If the converse was also true, Theorem 1.1 would provide us an even nicer interpretation of the Lebesgue property as the Fatou property (easy to check) plus "something extra", with the "extra" being precisely specified. See [16,Theorem 3.9] for more discussion, where a characterization of Lebesgue property in the form of Theorem 1.1 is obtained under solely the Fatou property as the a priori assumption, but with the conjugate (ϕ| L ∞ ) * of the restriction to L ∞ of ϕ instead of ϕ * . The implication (1.8) ⇒ (1.3) for proper convex functions is indeed true for some good spaces X , but its validity in the generality of this paper is still open (to us).…”