
Journal of Convex Analysis 20 (2013), No. 2, 355376 Copyright Heldermann Verlag 2013 An Upper Bound for the Convergence of Integral Functionals Emmanuel Giner Institut de Mathématiques, Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse 04, France giner@math.univtoulouse.fr [Abstractpdf] Given a $\sigma$finite measure space $(\Omega, \mathcal{T}, \mu)$ endowed with a $\mu$complete tribe, a separable Banach space $E$, we consider a topological vector space $(X,T)$, $X$ being a decomposable subspace of measurable $E$valued functions defined on $\Omega$. Under a reasonable assumption on the vector topology $T$, we show that if ${(f_{n})}_{n}$ is a sequence of extended realvalued measurable integrands defined on the product $\Omega\times E$, with upper epilimit (or upper $\Gamma$limit) $f=ls_{e} f_{n}$, then $I_f$ is in many cases an upper bound for the $T$upper epilimit of the sequence $(I_{f_{n}})_n$, where $I_f$, $I_{f_{n}}$ are the integral functionals defined on $X$ associated to the integrands $f$, $f_n$. The cases of Lebesgue spaces endowed with its strong, weak, or Mackey topologies are reached. We discuss also the necessity of the given conditions. Keywords: Integral functional, upper epilimits, $\Gamma$convergence, finally equiintegrable sets. MSC: 26A16, 26A24, 26E15, 28B20, 49J52, 54C35 [ Fulltextpdf (206 KB)] for subscribers only. 