Convex integral functionals

Abstract

We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their ε \varepsilon -subdifferential ( ε ≥ 0 ) (\varepsilon \ge 0) and their ε \varepsilon -directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.