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Journal of Convex Analysis 30 (2023), No. 3, 1053--1072
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

Abderrahim Jourani
Université de Bourgogne, Institut de Mathématiques de Bourgogne, Dijon, France, Dijon, France

José Vicente-Pérez
Departamento de Matemáticas, Universidad de Alicante, Spain


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.

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