Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus

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.