
Journal of Convex Analysis 30 (2023), No. 3, 10531072 Copyright Heldermann Verlag 2023 Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus Abderrahim Hantoute Departamento de Matemáticas, Universidad de Alicante, Spain, Spain and: Universidad de Chile, Santiago, Chile hantoute@ua.es Abderrahim Jourani Université de Bourgogne, Institut de Mathématiques de Bourgogne, Dijon, France, Dijon, France abderrahim.jourani@ubourgogne.fr José VicentePérez Departamento de Matemáticas, Universidad de Alicante, Spain jose.vicente@ua.es [Abstractpdf] We present a subdifferential analysis for a general concept of infinite sum $f:=\sum_{i\in I}f_{i}$ of arbitrary collections of convex functions $f_{i}$, called Lebesgue infinite sum. Since this problem cannot be addressed, at least directly, through classical arguments from the theory of normal convex integrands, we perform a reduction analysis showing that the $\varepsilon$subdifferential of $f$ reduces to that of countable/finite subsums via appropriate lower limit and closure processes. Then, the usual calculus rules of (countable) integral functions give rise to characterizations of the $\varepsilon$subdifferential of $f$, which are written exclusively by means of $\varepsilon$subdifferentials of the data $f_{i}$. The resulting characterizations do not assume any qualification or boundedness condition. Keywords: Lebesgue infinite sum, convex functions, subdifferential calculus. MSC: 26B05, 26J25, 49H05. [ Fulltextpdf (169 KB)] for subscribers only. 