In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin-systems turn out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin-systems. Again, coherence and extendability are equivalent. A related, but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities.

翻译：在非精确概率文献中，少有研究关注非精确概率在一组事件上具有精确性这一事实。我们将这类集合称为"精确性系统"。我们证明，在温和假设下，下概率与上概率的精确性系统构成所谓的（前）Dynkin系统。有趣的是，从基于部分数据的机器学习、频率概率理论，到量子概率理论和不确定性决策，若干场景中概率先验地仅需在特定底层集系上保持精确。在此类场景中，（前）Dynkin系统同样被用作精确性系统。我们证明，在可扩展性条件下，配备概率的这些前Dynkin系统可嵌入集合代数。令人惊讶的是，量子概率研究中发展出的可扩展性条件等价于非精确概率文献中的相干性概念。基于此，我们阐明了一种将精确性系统与概率信度集相关联的格对偶性。最后，我们将该框架推广至非精确概率的期望型对应物，其前Dynkin系统的类比物表现为有界实值函数空间中的（集合族）线性子空间。我们引入部分期望——定义在前Dynkin系统上的概率的自然推广。再次证明，相干性与可扩展性等价。一个相关但更一般的格对偶性保留了精确性系统与概率信度集之间的关联。