Abstract

We provide formulae for the ε-subdifferential of the integral function If(x) : = ∫ Tf(t, x) dμ(t) , where the integrand f: T× X→ R¯ is measurable in (t, x) and convex in x. The state variable lies in a locally convex space, possibly non-separable, while T is given a structure of a nonnegative complete σ-finite measure space (T, A, μ). The resulting characterizations are given in terms of the ε-subdifferential of the data functions involved in the integrand, f, without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.