Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus
Résumé
We present a subdifferential analysis for a general concept of infinite sum f := Sigma(i is an element of 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 epsilon-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 epsilon-subdifferential of f, which are written exclusively by means of epsilon-subdifferentials of the data f(i). The resulting characterizations do not assume any qualification or boundedness condition.