Set valued Aumann–Pettis integrable martingale representation theorem and convergence

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It is known in the literature that in the RNP Banach space the set valued uniformly integrable martingale is a regular martingale. In this paper by using a selector approach we provide a weaker condition than uniform integrability of a set valued Aumann–Pettis integrable martingale to be a set valued Aumann–Pettis integrable regular martingale. The Converse is also established. As an application of the aforementioned results, a new convergence result of set valued Pettis integrable martingales in Slice topology is provided.