Abstract In this paper we discuss integral inequalities for collection integrals that are a special subclass of decomposition integrals introduced as a general framework for many non-linear integrals, including the Choquet integral, the Shilkret integral, the PAN integral, and the concave integral. We give a full characterization of collection integrals that are comonotone additive and for which Chebyshev's, Jensen's, Cauchy's, and Holder's integral inequalities hold. Interestingly, all these classes of collection integrals coincide and thus we introduce a special subclass of collection integrals, called PCC integrals. The paper is complemented with some examples and remarks for collection and decomposition integrals.

中文翻译：

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集合积分与 Choquet 积分

摘要 在本文中，我们讨论了集合积分的积分不等式，集合积分是分解积分的一个特殊子类，作为许多非线性积分的一般框架引入，包括 Choquet 积分、Shilkret 积分、PAN 积分和凹积分。我们给出了集合积分的完整特征，这些积分是共单调可加的，并且切比雪夫、詹森、柯西和霍尔德的积分不等式成立。有趣的是，所有这些集合积分类都是重合的，因此我们引入了一个特殊的集合积分子类，称为 PCC 积分。本文补充了一些关于收集和分解积分的例子和评论。