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.

中文翻译：

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集合值 Aumann-Pettis 可积鞅表示定理和收敛

文献中已知，在 RNP Banach 空间中，集合值一致可积鞅是正则鞅。在本文中，通过使用选择器方法，我们提供了一个比集合值 Aumann-Pettis 可积鞅的均匀可积性更弱的条件为集合值 Aumann-Pettis 可积正则鞅。匡威也成立了。作为上述结果的应用，提供了切片拓扑中集合值 Pettis 可积鞅的新收敛结果。