Epiconvergence of relaxed stochastic optimization problems

We consider relaxation of almost sure constraint in dynamic stochastic optimization problems and their convergence. We show an epiconvergence result relying on the Kudo convergence of $\sigma$-algebras and continuity of the objective and constraint operators. We present classical constraints and objective functions with conditions ensuring their continuity. We are motivated by a Lagrangian decomposition algorithm, known as Dual Approximate Dynamic Programming, that relies on relaxation, and can also be understood as a decision rule approach in the dual.