Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities

Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. Minimizers and generalized minimizers are explicitly described not assuming the primal constraint qualification. A generalized Pythagorean identity is presented using Bregman distance and a correction term for lack of essential smoothness. Results are applied to minimization of Bregman distances. Existence of generalized dual solutions is established. The main new tool is a modification of the concept of convex core of a Borel measure, suitable to describe the effective domain of the value function.