We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.

中文翻译：

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弱紧值可积多功能的分解

我们简要概述了可积多功能的分解特性，即，当“某种意义上可积”多功能可以表示为其可积选择之一和较窄意义上的多功能可积之和。分解定理是多值积分理论的重要工具，因为它们使我们能够将可积多功能视为具有更好属性的多功能的转换。因此，它们可以更好地表征所考虑的可集成多功能。关于该问题的大量文献始于2006年作者的开创性论文，其中证明了在可分离Banach空间X中具有紧凑凸值的Henstock可积多功能的性质。。在本文中，我们总结了较早的结果，我们证明了进一步的结果，并提供了表，这些表显示了该主题的最新状态。