Qualification Conditions-Free Characterizations of the $$\varepsilon $$-Subdifferential of Convex Integral Functions

Characterizations of the subdifferential of convex integral functions under qualification conditions

“…The same holds true if the space of pintegrable functions is replaced by any decomposable space (see e.g. [8,Definition 3,§VII], or [19,Definition 3.2] for this definition).…”

“…Proof. According to [19,Theorem 5.1], we only have to prove that for every L ∈ F (x) and ε ≥ 0, N ε dom I f ∩L (x) = A ε L (x) = B ε L (x). Indeed, it suffices to apply Proposition 9 with the measurable space ( T , Σ, μ), where T := T ∪ {ω 0 } for an element ω 0 / ∈ T , Σ is the σ-Algebra generated by (Σ ∪ {ω 0 }), and μ is defined by…”

“…General formulae have been established in [19] using a finite-dimentional reduction approach, without additional assumptions on the data represented by the integrand f . In this paper, we use natural qualifications condition, involving appropriate continuity assumption on the integrand, to give more explicit characterization of the subdifferential and the ε-subdifferential of the function I f .…”

“…The conclusion when X is reflexive and separable comes from the fact that we can take sequences instead of nets used above (using [35,Theorem 2.3]) and countable family of finite-dimensional subspaces (see [19,Remark 4.3]). It remains to verify the sufficiency implication.…”

Random multifunctions as the set minimizers of infinitely many differentiable random functions

“…where cl w denotes the closure with respect to the weak topology on L p (Ω, X). Moreover, in both cases, it is possible to choose ϕ satisfying the estimations (9).…”