Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus

We present a subdifferential analysis for a general concept of infinite sum

of arbitrary collections of convex functions

f_{i} 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 ε

-subdifferential of

f 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 ε

-subdifferential of

f f

, which are written exclusively by means of

\varepsilon ε

-subdifferentials of the data

f_{i} f i

. The resulting characterizations do not assume any qualification or boundedness condition.