The optional sampling theorem for submartingales in the sequentially planned context

The optional sampling theorem is considered in the sequentially planned context. We prove the optional sampling theorem for direct successors and for sampling plans with a finite number of stages. Also, the theorem is studied in the general case under a uniform integrability condition; we obtain it for submartingales with a last element, and for submartingales that verify a bounded condition based on uniform integrability.