  The paper concerns the investigation of nonconvex and nondifferentiable
integral functionals on general Banach spaces, which may not be reflexive
and/or separable. Considering two major subdifferentials of variational
analysis, we derive nonsmooth versions of the Leibniz rule on
subdifferentiation under the integral sign, where the integral of the
subdifferential set-valued mappings generated by Lipschitzian integrands is
understood in the Gelfand sense. Besides examining integration over complete
measure spaces and also over those with nonatomic measures, our special
attention is drawn to a stronger version of measure nonatomicity, known as
saturation, to invoke the recent results of the Lyapunov convexity theorem type
for the Gelfand integral of the subdifferential mappings. The main results are
applied to the subdifferential study of the optimal value functions and
deriving the corresponding necessary optimality conditions in nonconvex
problems of stochastic dynamic programming with discrete time on the infinite
horizon.
