  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.
