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We provide formulae for the $$\varepsilon $$ ε -subdifferential of the integral function $$I_f(x):=\int _T f(t,x) d\mu (t)$$ I f ( x ) : = ∫ T f ( t , x ) d μ ( t ) , where the integrand $$f:T\times X \rightarrow \overline{\mathbb {R}}$$ 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 $$\sigma $$ σ -finite measure space $$(T,\mathcal {A},\mu )$$ ( T , A , μ ) . The resulting characterizations are given in terms of the $$\varepsilon $$ 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.

Qualification Conditions-Free Characterizations of the $$\varepsilon $$-Subdifferential of Convex Integral Functions