Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces

In the 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 turns 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.


Introduction
Scholarship in imprecise probability largely focuses on the imprecision of probabilities.However, imprecise probability models often lead to precise probabilistic statements on certain events or gambles, i.e. bounded, real-valued functions.In this work, we follow a hitherto not taken route investigating the system of precision, i.e. the set structure on which an imprecise probability is precise. 1 It turns out that (pre-)Dynkin-systems 2 describe the set of events with precise probabilities In all of these scenarios, there does not exist a precise probability over all attributes and events.Or, there is no such precise probability accessible.Two attributes might each on their own exhibit a precise probabilistic description, while a joint precise probabilistic description does not exist.On a more fundamental level, no intersectability is provided.A precise probabilistic description of two events does not imply that the intersection of those events possesses a precise probability.The set system for the description of the events with precise probabilities which independently turned up in the various, previously mentioned fields of research is, again, the (pre-)Dynkin-system.
The question of intersectability (or "intersectionality") is of considerable interest in the social sciences where it is used as a label to describe the problem of the joint effect of various individual attributes on social outcomes [Cole, 2009, Shields, 2008, Weldon, 2008].That this notion of intersectionality has something to do with set systems is clear already from the fact that the Venn diagram pictured on the right 4 is used as an illustration both for the Wikipedia articles on Hypergraphs [Anonymous, 2023a] (another name for a set system [Berge, 1989]) and Intersectionality [Anonymous, 2023b].Needless to say, the concept as used in the social sciences is rich, complex, and somewhat vague, which is not necessarily held to be a weakness: "at least part of its success has been attributed to its vagueness" [Hancock, 2013, page 260].Our interest is in under what circumstances precise probabilities can be ascribed to events; we speculate that such formal results may well contribute to a deeper empirical understanding of social intersectionality, without resorting to fuzzy logic [Hancock, 2007] with its renowned lack of operational definition [Cooke, 2004].
By rethinking the domain of probability measures one might wonder about the origins of Kolmogorov's σ-algebra as the set system for events which possess probabilities.This links back 3 Following Kolmorogov's classical setup, Ω is the base set, F σ a σ-algebra and µ σ a countably additive probability on F σ .Approaches to imprecise probability often do not even presuppose an underlying measure space (e.g.[Walley, 1991]).However, they are often linked to finitely additive measure spaces (Ω, F , µ), where µ is a finitely additive probability and F is an algebra of sets (sometimes called a field). 4By RupertMillard, CC BY-SA 3.0.
to the old problem of measurability [Elstrodt, 2018, page 1-5].The measurability problem is the mathematical problem to assign a uniform measure to all subsets of a continuum.Giuseppe Vitali showed 1905 that this problem is not solvable for countably additive measures [Elstrodt, 2018, page 5] (from [Vitali, 1905]).Hence, more restricted set systems such as the σ-algebra arose.Isaacs et al. [2022] reconsidered this century-old discussion to argue for rationality of imprecise probabilities.We take their argument even further.Inside the borders of mathematical measurability, the set of events which ought to be assigned probabilities is a modelling choice.Measurability is a modelling tool.We show that it is naturally parametrized by the set of (pre-)Dynkin-systems.
All of the preceding considerations bring us to the main question of this paper: What is the system of precision and how does it relate to an imprecise probability on "all" events?We approach this question from three perspectives.
1. First, we show that, under mild assumptions, a pair of lower and upper probabilities assign precise probabilities, i.e. lower and upper probability coincide, to events which form a pre-Dynkin-system or even a Dynkin-system.
2. Second, we define probabilities on pre-Dynkin-systems in accordance with the literature on quantum probability, in particular [Gudder, 1969].We argue that probabilities on pre-Dynkinsystems, as well as their inner and outer extension, exhibit few desirable properties, e.g.subadditivity cannot be guaranteed.Hence, extendability, the ability to extend a probability from a pre-Dynkin-system to a larger set structure, turns out to be crucial, as it implies coherence of the probability defined on the pre-Dynkin-system.This observation links together the research from probabilities defined on weak set structures [Gudder, 1969, Zhang, 2002, Schurz and Leitgeb, 2008] to imprecise probabilities [Walley, 1991, Augustin et al., 2014].Furthermore, extendability guarantees the existence of a nicely behaving, so-called coherent extension.We finally show that the inner and outer extension of a probability defined on a pre-Dynkin-system is always more pessimistic than its corresponding lower and upper coherent extension.
3. Last, we develop a duality theory between pre-Dynkin-systems on a predefined base measure space and their respective credal sets of probabilities.The credal sets consist of all probabilities which coincide with the pre-defined measure on a pre-Dynkin-system.A so-called Galois connection links together the containment structure on the set of set systems with the containment structure on the set of credal sets.
We conclude our presentation with a generalization to expectation-type counterparts of imprecise probabilities in Section 6.These are often called previsions, e.g. in [Walley, 1991].Our main question thus generalizes to: What is the system of precision and how does it relate to an imprecise expectation on "all" gambles?In this case, by "system of precision" we mean the set of gambles with on which a lower and upper expectation coincide.
1. First, we propose a generalization of a finitely additive probability defined on a pre-Dynkinsystem.More concretely, we define partial expectations which correspond to expectation functionals which are only defined on a set of linear subspaces of the space of all gambles.However, on those linear subspaces they behave like "classical" (finitely additive) expectations.
2. Second, we show that under some properties, imprecise expectations are precise on a linear subspace of the linear space of gambles.(cf.Section 3) 3. Third, we present a natural generalization of extendability for partial expectations, which again turns out to be equivalent to coherence of the partial expectation.
4. Last, analogous to the lattice duality5 described in Section 5, we present a lattice duality for linear subsets of the space of gambles and credal sets which define coherent lower and upper previsions.
In summary, our work makes contributions in-between the research field of imprecise probabilities, probabilities defined on general set structures, and partially defined expectation functionals.Part of this work has been presented on the International Symposium on Imprecise Probabilities: Theories and Applications under the title "The Set Structure of Precision" [Derr and Williamson, 2023].The following version is a more complete and exhaustive presentation of this conference version.We included all omitted proofs of the conference version.We elaborated the content of Section 5. We added an entire section about the generalization to expectation-type counterparts of probabilities on pre-Dynkin-systems (Section 6).We presented relations to different research areas in more detail.We shortly discussed countably additive probabilities on Dynkin-systems in the appendix, as we put emphasis on finitely additive probabilities in the main text.Before we begin the structural investigation of pre-Dynkin-systems, we first introduce the used notation and fix the mathematical framework.

Notation and Technical Details
As we deal with a lot of sets, sets of sets, and rarely even sets of sets of sets in this paper, we agree on the following notation: sets are written with capital latin or greek letter, e.g.A or Ω. Sets of sets are denoted A .Sets of sets of sets obtain the notation A. As usual, R is reserved for the set of real numbers, N for the natural numbers.The power set of a set A is written as 2 A .
In the course of this work, we require the notions of σ-algebras and algebras (of sets).An algebra is a subset of 2 Ω which contains the empty set and is closed under complement and finite union.A σ-algebra is an algebra which is closed under countable union [Williams, 1991, Definition 1.1]. 6Probability) measures are denoted by lowercase greek letters, e.g.µ, ν and ψ, except for σ.Generally, we use "σ" to emphasize the countable nature of a mathematical object.This becomes clear when we define Dynkin-systems (Definition 2.1).Other functions are denoted by lowercase latin letters, e.g.f and g .
Regarding the technical setting, we roughly follow the setup of [Walley, 1991, §3.6 and Appendix D].For a summary, see Table 1.Let Ω be an arbitrary set.In several examples Ω = [n], where [n] denotes the set {1, . . ., n}.The set B(Ω) is defined as the set of all real-valued, bounded functions on Ω.We call those functions gambles.For instance, χ A , the indicator function of A ⊆ Ω, is in B(Ω).The supremum norm, ∥ f ∥ sup := sup ω∈Ω | f (ω)| makes B(Ω) a topological linear vector space [Hildebrandt, 1934].With ba(Ω) we denote the set of all bounded, signed, finitely additive measures on 2 Ω .In fact, ba(Ω) is the topological dual space of B(Ω).So in particular, every continuous linear functional φ ∈ B(Ω) * can be identified with a bounded, signed, finitely additive measure [Hildebrandt, 1934]. 7For this reason, we use, with minor abuse of notation, the same notation for bounded, signed, finitely additive measures and for continuous linear functionals in the dual space of B(Ω), i.e. we write ν( f ) = f d ν.The dual space ba(Ω) gets equipped with the weak ⋆ topology, Ω, 2 Ω Base set and its power set [n] Set {1, . . ., n} D Pre-Dynkin-system on Ω (Definition 2.1) D σ Dynkin-system on Ω (Definition 2.1) D(A ) Pre-Dynkin hull of a set system A ⊆ 2 Ω (Definition 2.1) µ Finitely additive probability defined on D (Definition 2.9) µ * , µ * Inner respectively outer extension (Proposition 4. E Partial Expectation (Definition 6.1) S(Ω, A ) Linear space of simple gambles on the set system A B(Ω, F σ ) Linear space of bounded, F σ -measurable functions E Coherent lower prevision (Definition 6.5) E Coherent upper prevision (Definition 6.5) ν Linear prevision defined on B(Ω) (equivalent to ν above) ψ Fixed, linear prevision defined on B(Ω) (equivalent to ψ above) m : 2 B(Ω) → 2 ∆ Generalized credal set function (Definition 6.9) m • : 2 ∆ → 2 B(Ω) Generalized dual credal set function (Definition 6.10) i.e. the weakest topology which makes all evaluation functionals of the form f * ∈ ba(Ω) * such that f * (ν) := f d ν for some f ∈ B(Ω) continuous.With ∆ ⊆ ba(Ω) we denote the convex, weak ⋆ -closed subset of finitely additive probability measures.The set ∆ plays a major role in Walley's theory of previsions, as the measures in ∆ are in one-to-one correspondence to his linear previsions [Walley, 1991, Theorem 3.2.2].The operator co is the convex, weak ⋆ closure on the space ba(Ω).We further introduce the following two notations: let F ⊆ 2 Ω be an algebra.Then S(Ω, F ) ⊆ B(Ω) denotes the linear subspace of simple functions on F , i.e. scaled and added indicator functions of a finite number of disjoint sets (cf. [Rao and Rao, 1983, Definition 4.2.12]).Let F σ ⊆ 2 Ω be a σ-algebra.Then B(Ω, F σ ) ⊆ B(Ω) denotes the linear subspace of all bounded, real-valued, F σ -measurable functions.Equipped with these notions and tools we are ready for a first preliminary question.

What Is a (Pre-)Dynkin-System?
In this work, the main objects under consideration are pre-Dynkin-systems and Dynkin-systems.A (pre-)Dynkin-system is a set system on Ω.It contains the empty set, is closed under complement and (countable) disjoint union.More formally: Definition 2.1 ((Pre-)Dynkin-system).We say D ⊆ 2 Ω is a pre-Dynkin-system on some set Ω if and only if all of the following conditions hold: We call D σ ⊆ 2 Ω a Dynkin-system if and only if the conditions (a), (b) and Observe that every Dynkin-system is a pre-Dynkin-system.We will denote pre-Dynkin-systems by the use of D, in contrast to D σ for Dynkin-systems.This should not be confused with D(A ) for A ⊆ 2 Ω , which is the intersection of all pre-Dynkin-systems which contain A , i.e. the smallest pre-Dynkin-system containing A . 8 In other words, D(A ) is the pre-Dynkin-hull generated by A .The following short lemma will be helpful in later proofs.

Lemma 2.2 (Closedness under Set Difference
In classical probability theory, Dynkin-systems appear as a technical object required for the measure-theoretic link between cumulative distribution functions and probability measures (cf.[Williams, 1991, Proof of Lemma 1.6]).In particular, every σ-algebra, the well-known domain of probability measures, is a Dynkin-system.Thus, all statements within this work are generalizations of classical probability theoretical results.We give a short example of a pre-Dynkin-system, which is not an algebra in the following.This example gets reused to illustrate forthcoming statements.
Pre-Dynkin-and Dynkin-systems naturally arise in probability theory.For instance, the set of all subsets A ⊆ N, such that the natural density µ [Schurz and Leitgeb, 2008]) is a pre-Dynkin-system D N , but not an algebra9 .It is sometimes called the density logic [Pták, 2000] and constitutes the foundation of von Mises' century-old frequential theory of probability [von Mises, 1919] (refined andsummarized in [von Mises andGeiringer, 1964]).
Another class of Dynkin-systems occurs in so-called marginal scenarios [Cuadras et al., 2002].Marginal scenarios are settings in which marginal probability distributions for a subset of a set of random variables are given, but not the entire joint distribution.This restricted "joint measurability" of the involved random variables can be expressed via Dynkin-systems [Gudder, 1984, Example 4.2] [Vorob'ev, 1962].
Pre-Dynkin-systems are so helpful because they structurally align with finitely additive probability measures.The same statement holds for Dynkin-systems and countably additive probabilities.If we know the probability of an event, then we know the probability of the complement, i.e. the event does not happen.If we know the probability of several events which are disjoint, then we know the probability of the union, which is just the sum.Probabilities following their standard definition go hand in hand with Dynkin-systems.We see this observation manifested in many following statements.
Remarkably, (pre-)Dynkin-systems appeared under a variety of names (cf.Appendix B).Fundamental to all its regular, independent occurences in many research areas is the need for a set structure which does not allow for arbitrary intersections.

Compatibility
(Pre-)Dynkin-systems are not necessarily closed under intersections.However, when the intersection of two sets (events) is contained in the (pre-)Dynkin-system, we call the two events compatible.

Definition 2.4 (Compatibility). Let A, B be elements in a pre-Dynkin-system D, then A and B are compatible if and only if
This definition follows the definitions given in e.g.[Gudder, 1969[Gudder, , 1973[Gudder, , 1984]]. 10Compatibility in pre-Dynkin-systems is a symmetric relation, but it is not necessarily transitive.Furthermore, it is complement inherited, i.e. if A, B are compatible in a pre-Dynkin-system then so are A, B c [Gudder, 1979, Lemma 3.6].Lastly, compatibility, even though expressed as intersectability, i.e. "closed under intersection", can be equivalently expressed as unifiability, i.e. "closed under union".Lemma 2.5 (Cup gives Cap gives Cup).Let D be a pre-Dynkin-system and A, B ∈ D.
Proof.Using Lemma 2.2 for pre-Dynkin-systems we can quickly see that the following two decompositions give the desired equivalence: For the "⇒"-direction: (A related result for Dynkin-systems is given in [Gudder, 1969, 5.The term "compatibility" underlines that closedness under intersection gets loaded with further meaning in the context of probability theory.As we define in the next section, D is the set of "measurable" events, i.e. events which get assigned a probability.Hence, two events A, B are called compatible if and only if a precise joint probabilistic description, i.e. a precise probability of A ∩ B , exists. 11ompatibility is not only a property of elements in a pre-Dynkin-systems.One can take compatibility as a primary notion, i.e. one requires the statements of Lemma 2.5 and [Gudder, 1979, Lemma 3.6] to hold.Then, a set structure which contains the empty set and the entire base set and is equipped with this notion of compatibility is a pre-Dynkin-system [Khrennikov, 2009b, Definition 5.1]. 12nterestingly, the assumption of arbitrary compatibility is fundamental to most parts of probability theory.σ-algebras, the domain of probability measures, are exactly those Dynkin-systems in which all events are compatible with all others [Gudder, 1973, Theorem 2.1].Algebras are exactly those pre-Dynkin-systems in which all events are compatible with all others.Surprisingly, it turns out that, as well, all pre-Dynkin-systems can be dissected into such "blocks" of full compatibility.Every pre-Dynkin-system consists of a set of maximal algebras which we call blocks.In particular, maximality here stands for: there is no algebra contained in D such that some A i is a strict sub-algebra of this algebra.13Theorem 2.7 (Pre-Dynkin-Systems Are Made Out of Algebras).Let D be a pre-Dynkin-system on Ω.Then there is a unique family of maximal algebras {A i } i ∈I such that D = i ∈I A i .We call these algebras the blocks of D.
Proof.For the proof we require the definition of a compatible subset of D. A subset A ⊆ D is compatible, if all elements are completely compatible, i.e. any finite intersection of elements in A is contained in D. This is indeed a stronger requirement than pairwise compatibility (cf.Definition 2.4).Certainly, every subset A ⊆ D is compatible if and only if every finite subset of A is compatible.Hence, compatibility is a property of so-called finite character [Schechter, 1997, Definition 3.46].Then, Tuckey's lemma (e.g.[Schechter, 1997, Theorem 6.20.AC5]) guarantees that any compatible subset of D is contained in a maximal compatible subset.Since every element D ∈ D is in at least one compatible subset, e.g.{ , D, D c , Ω} ⊆ D, the (unique) set of maximal compatible subsets {A i } i ∈I covers the entire pre-Dynkin-system D. It remains to show that the maximal compatible subsets are algebras.Consider a maximal compatible subset A i .First, ∈ A i as is compatible to all sets in 2 Ω .Second, A i is closed under finite intersection, otherwise there would exist a finite combination of elements A 1 , . . ., A n ⊆ A i such that A ∩ := n j =1 A j ∈ D, but A ∩ ∉ A i .Then, one can easily see that A i ∪ {A ∩ } would be a compatible subset which strictly contained A i .This is impossible, since A i is maximal.Finally, A i is closed under complement.Consider A ∈ A i , we show that A i ∪ {A c } is again a compatible subset.Let A 1 , . . ., A n ⊆ A i be an arbitrary finite collection of subsets, then Gudder, 1979, Lemma 3.6].By maximality of A i we then know A c ∈ A i .
Theorem 2.7 simplifies several follow-up observations.Instead of pre-Dynkin-systems we can equivalently consider a set of algebras.However, not every union of algebras is a pre-Dynkin-system.If these algebras form a compatibility structure, i.e. a set of maximal π-systems14 , then their union is a pre-Dynkin-system (Definition A.3 and Theorem A.4 in Appendix).Analogous results for Dynkinsystems and σ-algebras exist and are given in Appendix D.1.In summary, pre-Dynkin-systems are set structures which do not allow for arbitrary intersections, but can be split into maximal intersectable subsets, their blocks.

Probabilities on Pre-Dynkin-Systems
We require a notion of probability on a pre-Dynkin-system to elaborate the relationship of imprecise probability and the system of precision in the following.Probabilities are classically defined on σ-algebras.We generalize this definition as e.g.stated in [Williams, 1991, page 18f ] to pre-Dynkinsystems.
Definition 2.9 (Probability Measure on a Pre-Dynkin-System).Let D be a pre-Dynkin-system.We call a function µ : D → [0, 1] a countably additive probability measure on D if and only if it fulfills the following two conditions: If condition (b) holds at least for finite I , we say that µ is a finitely additive probability measure.
For the sake of readability, we use "probability" and "probability measure" exchangeably.Probabilities on pre-Dynkin-systems are monotone, i.e. for A, B ∈ D, if A ⊆ B , then µ(A) ≤ µ(B ). 15But, in contrast to a probability defined on a σ-algebra, a probability on a pre-Dynkin-system is not necessarily modular, i.e. for A, B ∈ D, [Denneberg, 1994, page 16]. 16It is that sophisticated interplay of set structure and probability function which leads us through this paper.In particular, why should we consider pre-Dynkin-systems?

Imprecise Probabilities Are Precise on a Pre-Dynkin-System
As we now demonstrate, pre-Dynkin-systems are, under mild assumptions, the systems of precision.To make this formal, we solely require a normed, conjugate pair of lower and upper probability which fulfill super (resp.sub)-additivity and possibly a continuity assumption.Theorem 3.1 (Imprecise Probability Induces a (Pre-)Dynkin-System). Let ℓ : 2 Ω → [0, 1] and u : 2 Ω → [0, 1] be two set functions, for which all the following properties hold: then u and ℓ define a countably additive probability measure µ σ := u| D σ = ℓ| D σ on a Dynkin-system Proof.We start proving the first part of the theorem.Let (1) We show that D is a pre-Dynkin-system.First, ∈ D by assumption (a).Second, let For (⋆), observe that ℓ(A) ≤ u(A) for all A ∈ 2 Ω , since and thus, Concluding, we define µ := ℓ| D = u| D for which it is trivial to show that it is a finitely additive probability on D.
For the second part, we first notice that continuity from below and from above are equivalent for conjugate set functions on set systems which are closed under complement [Denneberg, 1994, Proposition 2.3].Next, we show that subadditivity of u and continuity from below (of u) imply σ-subadditivity of u: for {A i } i ∈I ⊆ 2 Ω such that I ⊆ N and A i ∩ A j = for all i ̸ = j with i , j ∈ I then u ( i ∈I A i ) ≤ i ∈I u(A i ).In case that I is finite, subadditivity of u is provided by assumption.For infinite I we can construct an increasing sequence of sets, namely The same argument holds analogously for superadditivity and continuity from above of ℓ which is implied by continuity from below and the conjugacy relationship [Denneberg, 1994, Proposition 2.3].In summary, the proof of the first part can then be applied again, now without the restriction that I ⊆ N is finite.Instead it potentially is countable.
Example 3.2.Remember, Ω 4 = {1, 2, 3, 4}.We define ℓ : 2 In summary, imprecise probabilities are, under mild assumptions, precise on a pre-Dynkinsystem or even a Dynkin-system.This, importantly, is also the case if the system of precision is strictly larger than the trivial pre-Dynkin-systems { , Ω}. Exemplarily, a pair of conjugate, coherent lower and upper probability (e.g.[Walley, 1991, §2.7.4]) fulfills the conditions (a) -(d).However, in several cases (e.g.distorted probability distributions) imprecise probabilities are just precise on the system of certainty, i.e. the events which possess 0 or 1 probability (Proposition C.3). Concluding, the system of precision is a pre-Dynkin-system D ⊆ 2 Ω .What if we first define precise, finitely additive probabilities on a pre-Dynkin-system, i.e. we fix a system of precision?We can then ask for "imprecise probabilities" deduced from this probability which are defined on a larger set structure, e.g. an algebra in which the pre-Dynkin-system is contained.

Extending Probabilities on Pre-Dynkin-Systems
Precise probabilities on pre-Dynkin-system naturally arise in many, distinct, applied scenarios as we argued in the Introduction ( § 1).However, we acknowledge that the definition of probabilities on pre-Dynkin-systems is mathematically cumbersome.The possibilities to prove standard theorems is very limited as the approaches by Gudder [1973Gudder [ , 1979]], Gudder and Zerbe [1981] demonstrate.However, if we consider a probability defined on a pre-Dynkin-system as an imprecise probability on a larger set system with a fixed system of precision, we possibly obtain a richer, mathematical toolkit to work with.In this case the larger set system preferably is an algebra in which the pre-Dynkin-system is contained.It remains to clarify how we construct the imprecise probability from the precise probability on the pre-Dynkin-sytem.

Inner and Outer Extension
A simple but, as we show, unsatisfying solution is the use of an inner and outer measure extension.
It does not rely on imposing any conditions on the probability defined on the pre-Dynkin-system.We pay for this generality with the few properties that we can derive for the obtained extension.
Proposition 4.1 (Inner and Outer Extension).[Zhang, 2002, Lemma 2.2] Let D be a pre-Dynkinsystem on Ω and µ a finitely additive probability measure on D. The inner probability measure and outer probability measure e. all of the following conditions are fulfilled: The inner and outer extension are not coherent (Definition 4.7).In particular, the outer extension is not subadditive: In conclusion, the inner and outer extension provides an imprecise probability, which is not necessarily coherent (cf.Definition 4.7) and it does not fulfill the conditions required for Theorem 3.1 to post-hoc guarantee that the set of precision is a pre-Dynkin-system.We remark that there exist normalized, conjugate, monotone superadditive but not subadditive pairs of probabilities, hence possibly inner and outer probabilities as defined here, whose system of precision is not a pre-Dynkin-system (see Example 4.3).For this reason we now explore another, more powerful extension method.

Extendability and Its Equivalence to Coherence
In the following, we try to entirely embed pre-Dynkin systems equipped with a probability into larger algebras.Then, we extend the probability defined on the pre-Dynkin-system in all possible ways to probabilities on the algebra.It turns out that this embedding is only possible under certain conditions on the probability defined on the pre-Dynkin-system.We call this condition extendability.For the sake of generality, we focus on the extension of finitely additive probabilities from pre-Dynkin-systems to algebras here.We treat countably additive probabilities, Dynkin-systems and σ-algebras in Appendix D. In addition, all results until Subsection 4.3 can be formulated in more general terms for non-structured set systems.For the sake of simplicity, we remain within the setting of probabilities defined on pre-Dynkin-systems in this work.
Extendability is the property that a probability measure defined on a pre-Dynkin-system can be extended to a probability measure on an algebra containing the pre-Dynkin-system.Formally: Definition 4.4 (Extendability).Let D be a pre-Dynkin-system on Ω.We call a finitely additive probability measure µ on D extendable to 2 Ω if and only if there is a finitely additive probability measure ν : We defined extendability with respect to the power set 2 Ω .In fact, any relativization to an arbitrary sub-algebra of 2 Ω is equivalent.A finitely additive probability defined on D is extendable to any sub-algebra of 2 Ω which contains D if and only if it is extendable to 2 Ω [Rao and Rao, 1983, Theorem 3.4.4].
The definition is non-vacuous [Gudder, 1984, De Simone et al., 2007].For instance, a probability measure on a pre-Dynkin-System is not generally extendable to a measure on the generated algebra (e.g.Example 3.1 in [Gudder, 1984]).If a probability is extendable, its extension is in general non-unique.
Extendability of probabilities on (pre-)Dynkin-systems has already been part of discussions in quantum probability since 1969 [Gudder, 1969] up to more current times [De Simone and Pták, 2010].Several necessary and/or sufficient conditions on the structure of D and/or the values of µ are known [Gudder, 1984, De Simone et al., 2007, De Simone and Pták, 2010].We present here a sufficient and necessary condition discovered by Horn and Tarski [1948] and restated in [Rao and Rao, 1983, Theorem 3.2.10].17[ Rao and Rao, 1983, Theorem 3.2.10]Let D be a pre-Dynkinsystem on Ω.A finitely additive probability measure for all finite families of sets in The probability µ on D 4 meets the extendability condition.
Extendability proves to be more than a helpful mathematical property for embedding pre-Dynkin-systems and their respective probabilities into algebras.Whether a probability defined on D can be extended to a probability on 2 Ω is directly connected to the question whether the probability measure on D is coherent in the sense of [Walley, 1991, page 68, page 84] or not.Coherence is a minimal consistency requirement for probabilistic descriptions which has been introduced in the fundamental work of De Finetti [1974/2017] and developed by Walley [1991].Shortly summarizing, an incoherent imprecise probability is tantamount to an irrational betting behavior, thus the name.Thus, extendability is, besides its mathematical convenience, a desirable property of probabilities in pre-Dynkin settings.
We adapt here the definition of coherence of previsions in [Walley, 1991, Definition 2.5.1] to probabilities.

Definition 4.7 (Coherent Probability). Let A ⊆ 2 Ω be an arbitrary collection of subsets. A set function
At first sight, the Horn-Tarski condition given in Theorem 4.5 and the coherence condition presented here already appear similar.This becomes even more apparent in Walley's reformulation of coherence for additive probabilities [Walley, 1991, Theorem 2.8.7].In the following, we show that this superficial similarity is indeed based on a rigorous link.Surprisingly, Walley did not mention Horn and Tarski's work in his foundational book.
Theorem 4.8 (Extendability Equals Coherence).Let D be a pre-Dynkin-system on Ω.A finitely additive probability measure µ on D is extendable to 2 Ω if and only if it is a coherent additive probability on D.
For the converse direction, we observe that if µ possess an extension following Definition 4.4, then such an extension is a finitely additive probability on 2 Ω following Definition 2.9.Hence, Walley [1991, Theorem 2.8.9] guarantees that the extension is a coherent additive probability (Definition 4.7).Any restriction to a subdomain D ⊆ 2 Ω keeps the probability coherent and additive.
The linear extension theorem in Walley [Walley, 1991, Theorem 3.4.2]used here is a generalization of de Finetti's fundamental theorem of probability [De Finetti, 1974/2017, Theorems 3.10.1 and3.10.7].De Finetti's theorem is furthermore interesting, as he explicitly states that a coherent additive probability defined on an arbitrary collection of sets can be extended in a precise way (so lower and upper probability coincide) to some sets.De Finetti does not characterize this collection.Our Theorem 3.1, however, gives an answer to this question: the collection forms a pre-Dynkin-system.
Theorem 4.8 provides a missing link between two strands of work: on the one hand, probabilities on pre-Dynkin-systems and related weak set structures have been closely investigated in foundational quantum probability theory [Gudder, 1969[Gudder, , 1979] ] and decision theory [Epstein andZhang, 2001, Zhang, 2002].On the other hand, coherent probabilities are central to imprecise probability, in particular, the more general formulations of coherent previsions and risk measures [Walley, 1991, Delbaen, 2002, Pelessoni and Vicig, 2003].
The reader familiar with the literature on imprecise probability might well not be surprised by the equivalence of extendability and coherence.We still think that this link is indeed valuable to be spelled out explicitly here.The concept of extendability and coherence have been developed separately in two communities with different goals in focus.Coherence tries to capture "rational" betting behavior [De Finetti, 1974/2017, Walley, 1991].Extendability links to what is sometimes called "quantum weirdness".

Extendability, Compatibility and Contextuality
Extendability in quantum theory tightly interacts with a series of properties and concepts which pervade discussions about the "specialness" of quantum theory in comparison to other classical physical theories: compatibility, contextuality, hidden variables and more.To be concrete, two measurements are compatible if, for any initial state18 , there exists a joint measurement such that a fixed joint distribution for both measurement outcomes exists, whose marginals are the distribution of the single measurement [Busch et al., 2012, Xu andCabello, 2018].If measurements are incompatible, then there are potentially still states such that a joint distribution of measurements exist.Only in the cases that no joint distribution of measurements exists, i.e. extendability is not provided, a measuring observer observes contextual behavior [Xu and Cabello, 2018].Translated to the language of imprecise probability, contextuality amounts to non-coherence of a probabilistic description.Compatibility, in contrast, is a structural notion.If any finitely additive probability on a pre-Dynkin-system is extendable, then the pre-Dynkin-system, very roughly, resembles compatible measurements.We are indeed not the first to notice intriguing links between imprecise probability and concepts therein to quantum mechanics.Benavoli and collaborators recovered the four postulates of quantum mechanics with desirability as a starting point [Benavoli et al., 2016].Desirability is a very general framework for imprecise probability [Walley, 2000].

Extendability and Marginal Scenarios
Not far from the relation between extendability and coherence, Miranda and Zaffalon [2018] and Casanova et al. [2022] bridged desirability to marginal scenarios.Marginal scenarios can equivalently be expressed in terms of probabilities on pre-Dynkin-systems [Vorob'ev, 1962, Kellerer, 1964][Gudder, 1984, Example 4.2].In rough terms, the marginal problem for marginal scenarios asks whether for a given set of marginal probability distributions (not necessarily disjoint) there exists a joint distribution. 19This question has been, some while ago, asked for probabilities on finite spaces [Vorob'ev, 1962], countably additive probabilities [Kellerer, 1964], finitely additive probabilities (cf.[Maharam, 1972]) and recently for even more general probability models -sets of desirable gambles [Miranda and Zaffalon, 2018].A recurring theme in all those studies is the so-called running intersection property which characterizes all those marginal structures for which a joint probabilistic description can always be guaranteed.To bring extendability to this picture, one should think of it as a more fine-grained concept: solutions to the marginal problem show under which circumstances every marginal distribution of a certain structure is extendable.But there exist marginal problems for which only specific instantiations of the marginal distributions allow for extendability.The running intersection property is a property of a structure.Extendability is a property of a structure and a probability on this structure.

Coherent Extension
A probability on a pre-Dynkin-system D, even when extendable, only allows for probabilistic statements on D itself.However, extendability guarantees that a "nice" embedding into a larger system of measurable sets exists.More specifically, extendability expressed in terms of credal sets provides a well-known tool for the worst-case extension of a probability from a pre-Dynkin-system to a larger algebra.
If a finitely additive probability on a pre-Dynkin-system is extendable, then we can obtain lower and upper probabilities of events which are not in the pre-Dynkin-system but on a larger algebra.We follow the idea of natural extensions, e.g. as described by [Walley, 1991, page 136].In particular, [Walley, 1991, Theorem 3.3.4 (b)] directly applies as long as a probability on a pre-Dynkin-system is extendable.
Corollary 4.9 (Coherent Extension of Probability).Let D be a pre-Dynkin-system on Ω.For a finitely additive probability measure µ on D we define the credal set where, ℓ and u are defined as in Example 3.2 (cf.[Walley, 1991, page 122]).Figure 1 illustrates the coherent extensions.Even though coherent, µ D 4 is neither supermodular nor submodular: This implies that as well µ D 4 is neither supermodular nor submodular [Denneberg, 1994, Proposition 2.3].
These lower and upper probabilities allow for at least two interpretations: We can assume that a precise probability on a pre-Dynkin-systems D ⊆ 2 Ω just reveals its values on D, but is actually defined over 2 Ω .Then the lower and upper probability constitute lower and upper bounds of the precise "hidden probability" on 2 Ω , which is solely accessible on D. On the other hand, we can even reject the existence of such precise "hidden probability".Then lower and upper probability are the inherently imprecise probability of an event in 2 Ω but not in D. 20The obtained lower and upper probabilities represent the imprecise interdependencies between all events of precise probabilities.We illustrate this statement: in the variety of updating methods in imprecise probability we pick the generalized Bayes' rule [Walley, 1991, §6.4] to exemplarily compute the conditional probability of two events for the coherent extension of a probability from a pre-Dynkin-system.For A, B ∈ D such that µ(B ) > 0 the generalized Bayes' rule gives [Walley, 1991, Theorem 6.4.2]: We can easily rearrange the above as µ D (A ∩ B ) = µ D (A|B )µ(B ).In this case the imprecision of the probability of the intersected event is purely controlled by the conditional probability µ(A|B ) and not by the marginal, which is precise.So, the imprecision captured by the lower and upper probabilities locates solely in the interdependency of the events.We remark that Dempster's rule gives the same conditional probability here [Dempster, 1967].

Inner and Outer Extension Is More Pessimistic Than Coherent Extension
We have presented two extension methods for probabilities defined on pre-Dynkin-systems.We relate the methods in the following.In the case of an extendable probability we can guarantee the following inequalities to hold.
Theorem 4.11 (Extension Theorem -Finitely Additive Case).Let D be a pre-Dynkin-system on Ω and µ a finitely additive probability on D which is extendable to 2 Ω .Then Proof.Since D ⊆ 2 Ω , we easily obtain for all A ∈ 2 Ω .The other inequalities follow by the conjugacy of inner and outer measure, and lower and upper coherent extension.
In words, Theorem 4.11 states that the inner and outer extension is more "pessimistic" than the coherent extension.We use "pessimistic" in the sense of giving a looser bound for the probabilities assigned to elements not in the pre-Dynkin-system D but in 2 Ω .In Appendix D we demonstrate an analogous result for countably additive probabilities on Dynkin-systems.

The Credal Set and its Relation to Pre-Dynkin-System Structure
In the earlier parts of the paper, we derived pre-Dynkin-systems as the system of precision for relatively general imprecise probabilities.Then, we showed that, under extendability conditions, a precise probability on a pre-Dynkin-system gives rise to a coherent imprecise probability on an encompassing algebra.In other words, imprecise probabilities can be "mapped" to pre-Dynkinsystems and vice-versa.We concretize these mappings in the following.This manifestation then reveals structure in the interplay between the systems of precision, i.e. pre-Dynkin-systems, and coherent imprecise probabilities.In particular, we argue that the order structure of pre-Dynkinsystems can be mapped to the space of finitely additive probabilities.This provides a (lattice) duality for coherent imprecise probabilities with precise probabilities on pre-Dynkin-systems.More concretely, the duality allows for the interpolation from imprecise probabilities which are precise on "all" events to imprecise probabilities which are precise only on the empty set and the entire set.
In the following discussion, we assume, in addition to the technicalities presented in Section 1.1, that a fixed finitely additive probability on 2 Ω , which we call ψ, is given.The finitely additive probability ψ with the algebra 2 Ω and the base set Ω constitute our "base measure space" analogous to the choice of a base measure space in the theory of coherent risk measures [Delbaen, 2002].In comparison to the previous sections, we use ψ instead of µ as "reference measure" to emphasize the difference that µ was defined on a relatively arbitrary pre-Dynkin-systems D on Ω, while ψ is defined and fixed on the algebra 2 Ω on Ω.

Credal Set Function Maps From Pre-Dynkin-Systems to Coherent Probabilities
Equipped with a reference measure ψ we define the credal set function.The name arises due to its close link to the credal set as defined in Corollary 4.9.We stress that although not notated explicitly, the credal set function depends upon the choice of ψ.For a fixed ψ on 2 Ω , the credal set function m maps a subset of the algebra 2 Ω to the set of all finitely additive probabilities which coincide with ψ on this subset.It should be noticed that by definition of ψ, m(A ) ̸ = for every non-empty This defined mapping now simplifies our discussion about how pre-Dynkin-systems and imprecise probabilities correspond.For instance, one can easily see that the extreme case A = 2 Ω corresponds to m(A ) = {ψ} and A = to m(A ) = ∆.More generally, we observe the following two properties of the credal set function.Proof.We need to show that The set inclusion of the right hand side in the left hand side is trivial.For the reverse direction, consider an element ν ∈ ∆ such that ν(A) = ψ(A) for A ∈ A .Let By Theorem 3.1 H is a pre-Dynkin-system.Since A ⊆ H we know D(A ) ⊆ H . Hence, ν(A) = µ(A) for A ∈ D(A ).This gives the desired inclusion.We remark that for A = the equality still holds, since D( ) = { , Ω}.
Proposition 5.5 (Credal Set Function Maps to Weak ⋆ -Closed Convex Sets).Let m be the credal set function.For every non-empty Proof.The reference probability ψ is by definition coherent.Hence, for all non-empty A ⊆ 2 Ω , the set m(A ) is the set of all ν ∈ ∆ which dominate ψ.This set is, by Theorem 3.6.1 in [Walley, 1991], weak ⋆ -closed and convex.
In words, the credal set on some set system coincides with the credal set on its generated pre-Dynkin-system.And the credal set of probabilities is always weak ⋆ -closed and convex.Proposition 5.3 allows us to work with credal sets of arbitrary set systems instead of the entire pre-Dynkin-system.Thus, it resembles the well-known π-λ-Theorem, which is fundamental to classical probability theory [Williams, 1991, Lemma A.1.3].On the other hand, this result justifies our focus on pre-Dynkin-systems instead of arbitrary set systems.We do not lose generality when considering pre-Dynkin-systems instead of non-structured sets of sets.
Proposition 5.5 guarantees that the images of the credal set function behave "nicely".Specifically, these weak ⋆ -closed convex sets correspond to coherent previsions, i.e. generalizations of coherent probabilities as already stated by Walley [1991, Theorem 3.6.1].We elaborate this observation in Section 5.4.In conclusion, credal set functions map pre-Dynkin-systems to coherent probabilities.What about the reverse mapping?

The Dual Credal Set Function
The following is a natural definition of a dual credal set function.We justify this name by Proposition 5.7 below.
Definition 5.6 (Dual Credal Set Function).Let ∆ be the set of all finitely additive probabilities on 2 Ω .Fix a finitely additive probability ψ on 2 Ω .We call The dual credal set function also depends upon ψ, but we do not notate this explicitly.The dual credal set function maps an arbitrary set of finitely additive probability measures on 2 Ω to the (largest) set of events on which all contained probabilities coincide.We remark that each set of finitely additive probability measures can be linked to an imprecise probability.
We suggestively called the antagonist to the credal set function the "dual credal set function".The duality appearing here is a well-known fundamental relationship between partially ordered sets: a Galois connection.A Galois connection is a pair of mappings f : X → Y and f • : Y → X on partially ordered sets (Y , ≤) and (X , ≤) which preserves order structure (cf.Corollary 5.8).More formally, f and f • are a Galois connection if and only if for all x ∈ X , y ∈ Y , x ≤ f (y) ⇔ y ≤ f • (x) [Birkhoff, 1940, §V.8].Galois connections, even though they do not form an order isomorphism, induce a lattice duality.We exploit this lattice duality to provide an order-theoretic interpolation from no compatibility at all to full compatability.For this purpose, we first establish the Galois connection.

Proposition 5.7 (Galois Connection by (Dual) Credal Set Function). The credal set function m and the dual credal set function m
• form a Galois connection.[Birkhoff, 1940, §V.8].First, we show the left to right implication.We assume A ⊆ m • (Q), i.e. every ν ∈ Q coincides with ψ on A .Hence,

Proof. m and m • form a Galois connection if and only if
In case of the right to left implication we suppose Q ⊆ m(A ).Thus, The mappings involved in the Galois connection are antitone, i.e. they reverse the order structure from domain to codomain.Their pairwise application is extensive, i.e. the image of an object contains the object.In summary, the following rules of calculation hold:

Corollary 5.8 (Rules for (Dual) Credal Set Function). Let m be the credal set function and m • be the dual credal set function. For arbitrary A
Proof.[Birkhoff, 1940, §V.7 and V.8] Proposition 5.7 provides a tool to further investigate the dual credal set function.The reader might have noticed the similarity of the dual credal set function and the main question of Section 3: given a lower and upper probability, on which set systems do both coincide?In fact, we obtain an analogous result to Theorem 3.1, again an imprecise probability is mapped to the set of events on which it is precise.Proposition 5.9 (Dual Credal Set Function Maps to Pre-Dynkin-Systems). Let m • be the dual credal set function.For all non-empty Q ⊆ ∆, m • (Q) is a pre-Dynkin-system.

Bipolar-closed sets are sets
Most importantly, the bipolar-closed sets form two antitone isomorphic lattices ordered by set inclusion [Birkhoff, 1940, Theorem V.8.20].This relationship gives us a lattice duality between set systems and credal sets of probabilities.See Figure 2 for an illustration of bipolar-closed sets and the Galois connection.
Example 5.12.The pre-Dynkin-system D({12, 3}) already discussed in Example 5.4 and Example 5.11 is a bipolar-closed set.In contrast, the set {12, 3} cannot be a bipolar-closed set, as it is not a pre-Dynkin-system.
More precisely, bipolar-closed sets in the set of finitely additive probability distributions are weak ⋆ -closed convex (Proposition 5.5).These map to bipolar-closed subsets of 2 Ω , which are pre-Dynkin-systems (Proposition 5.9).All of the stated properties of bipolar-closed sets are necessary.But are they sufficient?
Figure 2: Galois connection between the lattice of pre-Dynkin-systems and the set of credal sets.In the illustrated case, we have m ).The set containment on both sides follows from Proposition 5.3, Corollary 5.8 and Proposition 5.10.

Sufficient Conditions for Bipolar-Closed Sets
In the search for sufficient conditions for bipolar-closed sets we focus on bipolar-closed subsets of 2 Ω .Bipolar-closed subsets of ∆ require further investigation.It is already difficult to characterize sufficient conditions for bipolar-closed subsets of 2 Ω .Corollary 5.13.Let m be the credal set function and m • be the dual credal set function.For an arbitrary subset A ⊆ 2 Ω we have . By Corollary 5.8, the statement follows.
This corollary gives rise to the follow-up question: under which circumstances does D(A ) = m • (m(A ))?As the following theorem demonstrates, this question is closely connected to the sets of measure zero of the base probability ψ and its problems (cf.[Rota, 2001]).

Proposition 5.14 ("Closedness" under Measure Zero Sets). Let m be the credal set function and m • be the dual credal set function. Let
A pre-Dynkin-system D(A ) can only coincide with m • (m(A )) if subsets of measure zero sets are included.Thus, Proposition 5.14 provides a further necessary condition for bipolar-closed subsets of 2 Ω .Yet, it turns out that the sets of measure zero as well can give a sufficient condition for bipolar-closed sets at least in a finite setting.
Let us consider an arbitrary B ∈ 2 [n] \ D. Without loss of generality (Lemma A.7) we can decompose where B D ∈ D and A ∉ D is a weak atom with respect to D (Definition A.6). Thus, we can leverage Lemma A.8: there exists ν ∈ m(D) such that ν(A) ̸ = ψ(A).Thus, Whether this theorem can be extended to more general sets Ω is an open question.Seemingly, proofs along the line of Theorem 5.15 are doomed to fail, since one cannot argue via probabilities on atoms of Ω.

Interpolation From Algebra to Trivial Pre-Dynkin-System
In probability theory there is a choice to be made regarding which events should get assigned probabilities [Kolmogorov, 1927[Kolmogorov, /1929, page 52], page 52].This significant choice has (mathematically) been standardized to form a (σ-)algebra (cf.standard probability space).But, already Kolmogorov, the "father" of modern probability theory, emphasized that this choice is not universal, but should depend on the problem at hand.More recently, Khrennikov [2016] argued that a more appropriate probabilistic modeling should appeal to weaker domains for probabilities to, for instance, represent physical observations such as quantum phenomena.
In particular, it cannot always be taken for granted that all events are compatible with all others, as implied by a (σ-)algebra (cf.Section 2.1).For instance, von Mises' axiomatization of probability inherently reflects potential incompatible events in terms of a pre-Dynkin-system [von Mises andGeiringer, 1964, Schurz andLeitgeb, 2008].In other words, there is a choice to be made about the system of precision.Which sets should be compatible to each other, which should not?How do the choices of the systems of precision relate to each other?
We neglect, without loss of generality, arbitrary systems of precision and focus on pre-Dynkinsystems (cf.Proposition 5.3).The range of choices is captured by the system of pre-Dynkin-systems.
Proof.On the one hand, it is easy to show that the intersection of pre-Dynkin-systems forms a pre-Dynkin-system again.On other hand, the smallest pre-Dynkin-system which contains a finite set of pre-Dynkin-systems is by definition the pre-Dynkin-system generated by the union over all elements in this finite set of pre-Dynkin-systems.
Example 5.17.Let Ω 4 be as defined in Example 5.2.The minimal element in D then is { , Ω 4 }.The maximal element is 2 Ω 4 .For the sake of brevity, we omit all further elements in D and remain with the observation that D 4 of Example 2.3 and D({12, 3}) are elements of D.
The lattice D spans a range of choices from D = 2 Ω , i.e. complete compatibility and only a single probability distribution in its credal set, namely m(2 Ω ) = {ψ}, to D = { , Ω}, i.e. no compatibility and the entire space of probability distributions constitute its credal set m({ , Ω}) = ∆.How "close" D is to the algebra 2 Ω determines how "classical" the credal set behaves.In other words, (D, ⊆) parametrizes a family of credal sets.Thus, it parametrizes coherent probabilities.The knob of compatibility can be turned from trivially nothing ({ , Ω}), to everything (2 Ω ).How does the "amount of compatibility" of the pre-Dynkin-system map to the credal sets?Or, e.g.given two pre-Dynkin-systems on which a probability is defined, what is the credal set of the union of these systems?
Proposition 5.18 (Lattice of Dynkin-Systems and Credal Sets).The credal set function m (Definition 5.1) together with the lattice (D, ⊆) provides a parametrized family of credal sets for which hold Proof.Concerning the first equality, we observe that for arbitrary The second line follows by the definition of infimum on the lattice of pre-Dynkin-systems and simple set containment: Unfortunately, the mentioned interpolation is slightly improper.It turns out that there are pre-Dynkin-systems The reason for this collision of credal sets is that not every pre-Dynkin-system D ⊆ 2 Ω is a bipolar-closed set.
Proposition 5.20 (Credal Set Function is Injective on Bipolar-Closed Sets).Let m be the credal set and m • be the dual credal set function.Let D 1 , D 2 ∈ D be pre-Dynkin-systems, which are bipolar-closed.
Proof.We prove the claim by contraposition: Hence, it is reasonable to focus on the set of bipolar-closed sets contained in 2 Ω .We define the set of interpolating pre-Dynkin-systems C := {C ⊆ 2 Ω : C = m • (m(C ))}.We know that C ⊆ D (Proposition 5.9) and C is even a lattice contained in D.

In particular, this lattice C is antitone isomorphic to the lattice of bipolar-closed sets in
Proof.By Theorem V.8.20 in [Birkhoff, 1940] C is a lattice and m an antitone lattice isomorphism on C. The equations hold by simple manipulations and Note that C is not generally a sublattice of D. It is a lattice contained in the lattice D, but the closure operator for the supremum is distinct.Interestingly, the order structure which both lattices, D and C, induce on the set of credal sets via m is identical.Every set m(A ) for arbitrary A ⊆ 2 Ω is bipolar-closed (Corollary 5.8).Hence, the lattice of bipolar-closed sets in 2 ∆ is the domain of m for elements in D and C. In other words, the lattices D and C provide one and the same parametrized family of credal sets, thus one and the same parametrized family of imprecise probabilities.
In comparison to other parametrized families of imprecise probability, such as distortion risk measures [Wirch and Hardy, 2001], which heavily rely on convex analysis, the duality used here is structurally weaker.Lattice isomorphisms give a glimpse of structure to the involved dual spaces.Convex dualities as exploited in [Fröhlich and Williamson, 2022] are far more informative, but apparently not able to handle the structural knob which we presented in this work: the set of sets which get assigned precise probabilities.Nevertheless, a natural question arises from this lattice duality: how does this lattice duality relate to a convex duality?We leave this question open to further research.A first attempt to an answer is discussed in Appendix C, where we link the parametrized family of distorted probabilities to the pre-Dynkin-system family of imprecise probabilities.

A More General Perspective -The Set of Gambles With Precise Expectation
To this point, we have exclusively focused on probabilities and set systems of events to which we assign probabilities.In fact, there is a more general story to be told.In the literature on imprecise probability focus often lies on expectation-type functionals instead of probabilities and on sets of gambles (bounded functions from the base set Ω to the real numbers) instead of sets of events.One can easily see that the latter is more general and can recover the former.Indicator functions of events are gambles.An expectation-type functional evaluated on an indicator gamble of an event corresponds to a generalized probability of the event.The converse direction, i.e. recovering a unique expectation-type functional from an imprecise probability, however, is not always possible [Walley, 1991, §2.7.3].In the following, we reiterate several questions which we asked in the preceding sections for probabilities and set systems.

Partial Expectations Generalize Finitely Additive Probabilities on (Pre-)Dynkin-Systems
We propose the following definition of partial expectation and show afterwards that it is a natural generalization of finitely additive probabilities defined on (pre-)Dynkin-systems.
Definition 6.1 (Partial Expectation).Let {L i } i ∈I be a non-empty family of linear subspaces of B(Ω).
We call E : i ∈I L i → R a partial expectation if and only if all of the following conditions are fulfilled: (a) for any i ∈ I and for all f , g ∈ L i , then E ( f + g ) = E ( f ) + E (g ), (Partial Linearity), (b) for any i ∈ I and any f ∈ L i , then E ( f ) ≥ inf f , (Coherence).
We remark that for this definition we leveraged the requirements for a linear prevision (Definition 6.5) on a linear space given in [Walley, 1991, Theorem 2.8.4].In other words, a partial expectation is a functional which is defined on a union of linear subspaces and behaves like a "classical" (finitely additive) expectation on each of the subspaces, but not necessarily on all simultaneously.It is a linear prevision when restricted to one of the subspaces L i (cf.Definition 6.5).
There is a one-to-one correspondence of linear previsions and coherent additive probabilities.For every coherent additive probability ν defined on an algebra A , there is a unique linear prevision, which we equivalently denote ν, the set of all A -measurable gambles, which agrees with the probability ν on the indicator gambles of the sets in A [Walley, 1991, Theorem 3.2.2].The following Proposition exploits this correspondence.A finitely additive probability defined on a pre-Dynkinsystem relates one-to-one to a partial expectation which is defined on the set of linear spaces induced by the simple gambles on the blocks of the pre-Dynkin-system.To this end, we introduce the following two notations: let F ⊆ 2 Ω be an algebra.Then S(Ω, F ) ⊆ B(Ω) denotes the linear subspace of simple gambles on F , i.e. scaled and added indicator gambles of a finite number of disjoint sets (cf. [Rao and Rao, 1983, Definition 4.2.12]).Let F σ ⊆ 2 Ω be a σ-algebra.Then B(Ω, F σ ) ⊆ B(Ω) denotes the linear subspace of all bounded, real-valued, F σ -measurable gambles.Proposition 6.2 (Finitely Additive Probability on Pre-Dynkin-System and its Partial Expectation).Let D ⊆ 2 Ω be a pre-Dynkin-system.Let µ : D → [0, 1] be a finitely additive probability defined on the pre-Dynkin-system with block structure {A i } i ∈I .Then µ is in one-to-one correspondence to a partial expectation E : i ∈I S(Ω, A i ) → R defined on the union of linear spaces of simple gambles induced by all blocks A i of D.
Proof.By Theorem 2.7 we can decompose the pre-Dynkin-system D into a set of blocks {A i } i ∈I .Since, A i ⊆ 2 Ω for all i ∈ I , each of the blocks induces a the linear subspace of simple gambles S(Ω, A i ) ⊆ B(Ω).Given a finitely additive measure µ : D → [0, 1], we now define For every i ∈ I , E | S(Ω,A i ) is a linear prevision in one-to-one correspondence to the finitely additive probability µ| A i [Walley, 1991, Theorem 3.2.2].Hence, conditions (a) and (b) in Definition 6.1 are met [Walley, 1991, Theorem 2.8.4 Thus, E is well-defined and there is no other partial expectation which agrees with µ on the indicator gambles of the sets in D.
The attentive reader might have noticed that we defined the partial expectation in Proposition 6.2 on very specific linear subspaces of B(Ω), namely the linear subspaces of simple gambles.In fact, the statement would still hold when enlarging the linear subspaces of simple gambles S(Ω, A i ) for every i ∈ I to linear subspaces of functions which are "convergence in measure"-approximated by gambles in S(Ω, A i ).For more details we refer the reader to [Rao and Rao, 1983, Definition 4.4.5 and Corollary 4.4.9].
But, it is not the case that we can extend the definition to all sets of bounded, A i -measurable functions, i.e. bounded functions whose pre-images of sets in the smallest algebra which contains all open sets of the real numbers are contained in A i .For an algebra A ⊆ 2 Ω the set of A -measurable gambles is not necessarily a linear subspace of B(Ω) [Walley, 1991, page 129].This is different for a σ-algebra A σ ⊆ 2 Ω .The set of bounded, A σ -measurable functions B(Ω, A σ ) forms a linear subspace of B(Ω) [Walley, 1991, page 129].Here, measurability is defined as the pre-image of every Borel-measurable set in R is in A .In this case, the set of linear spaces on which the partial expectation is defined is given by all bounded, measurable functions on the σ-blocks.Proposition 6.3 (Finitely Additive Probability on Dynkin-Systems and its Partial Expectation).Let D σ ⊆ 2 Ω be a Dynkin-system on the base set Ω. Let µ : D σ → [0, 1] be a finitely additive probability defined on the Dynkin-system.Then µ is in one-to-one correspondence to a partial expectation E : i ∈I B(Ω, A i ) → R defined on the union of linear spaces of measurable gambles induced by all σ-blocks A i of D σ .
Proof.By Theorem D.1 we can decompose the Dynkin-system D σ into a set of σ-blocks {A i } i ∈I .Since, A i ⊆ 2 Ω for all i ∈ I , each of the σ-blocks induce a linear subspace of B(Ω), which we denote as B(Ω, A i ).Given a finitely additive measure µ : For every i ∈ I , E | B(Ω,A i ) is a linear prevision in one-to-one correspondence to the finitely additive probability µ| A i [Walley, 1991, Theorem 3.2.2].Hence, conditions (a) and (b) in Definition 6.1 are met [Walley, 1991, Theorem 2.8.4] Thus, E is well-defined and there is no other partial expectation which agrees with µ on the indicator gambles of the sets in D.
It remains to emphasize that there are partial expectations defined on families of linear subspaces which are not induced by finitely additive probabilities on Dynkin-systems.A simple example is given by a linear space which does not contain the constant gamble corresponding to the indicator gamble of the set Ω. Hence, the definition of a partial expectation is indeed a generalization of the definition of a finitely additive probability on a pre-Dynkin-system.
Under the name "partially specified probabilities" Lehrer [2012] introduced a closely related notion to our partial expectation.Lehrer, however, assumed that there is by definition an underlying probability distribution over the entire base set (or better said, a σ-algebra on the base set).Hence, his partially specified probabilities are by definition extendable (see Definition 6.6), a fact, which he implicitly exploited by re-defining the natural extension following [Walley, 1991, Lemma 3.1.3 (e)] of partially specified probabilities [Lehrer, 2007, §3.2].Lehrer did not ask for the structure of the set of gambles with precise expectations, nor did he draw any connection to Walley's work, nor did he link his "partially specified probabilities" to finitely additive probabilities on pre-Dynkin-systems.

System of Precision -The Space of Gambles With Precise Expectations
In Section 3 we have shown that imprecise probabilities are precise on (pre-)Dynkin-systems.The natural analogue of this set structure of precision is the space of gambles with precise expectation, which actually forms a linear subspace.Theorem 6.4.(Imprecise Expectations Are Precise on a Linear Subspace of Precise Gambles) Let B(Ω) be the linear space of bounded, real-valued functions on Ω.Let L : B(Ω) → R and U : B(Ω) → R be two functionals, for which all the following properties hold: Then L and U coincide on a linear space S ⊆ B(Ω), the space of gambles with precise expectation, which contains all constant gambles.Proof.We define and show that S forms a linear subspace of B(Ω).First, let f , g ∈ S , then we have, Third, S contains all constant gambles by (a), (b) and (e).Hence, we have shown that S forms a linear subspace of B(Ω) which contains all constant gambles.
The choice of properties for the lower and upper expectation functional is not arbitrary.We tried to resemble the properties involved in the analogous statement for lower and upper probabilities (Theorem 3.1).One can easily check that a lower and upper expectation with the given properties (a) -(d) forms a lower and upper probability as required in Theorem 3.1 if the expectation is restricted to indicator gambles.However, we added property (e), positive homogeneity.
Without the property of positive homogeneity, the resulting set of gambles with precise expectations would not form a proper linear subspace, as then one can only guarantee closedness of S under rational multiplication.The condition of positive homogeneity "fills up" the gaps with all real-scaled functions.Instead of positive homogeneity one can as well demand a continuity assumption of the lower and upper functional L and U , e.g.[Walley, 1991, Property (l) Theorem 2.6.1].We emphasize that coherent previsions (see Definition 6.5) fulfill all of the demanded properties [Walley, 1991, Theorem 2.6.1].
Interestingly, the restriction L| S = U | S is not necessarily a partial expectation.Otherwise it would form a coherent linear prevision (see Definition 6.5).This is different compared to Theorem 3.1, where the lower and upper probability actually define a finitely additive probability on the set structure of precision, which is not necessarily coherent.However, those two statements are not in contradiction.Any pair of lower and upper expectations, as we defined them here, induce a unique lower and upper probability.The resulting finitely additive probability on the set structure of precision gives rise to a partial expectation (Proposition 6.2) on a set of linear subspaces contained in the space of gambles with precise expecation S of L and U .
The converse direction, however, is not true.There is no unique lower and upper expectation functional with the given properties associated to a lower and upper probability fulfilling the axioms of Theorem 3.1 [Walley, 1991, §2.7.3].Concluding, lower and upper expectation as defined here are not the "perfect" analogues of lower and upper probabilities.
This as well explains the mismatch between systems of precision for probabilities and expectations.A lower and upper expectation fulfills the properties of a lower and upper probability but not vice-versa.Hence, only weaker statements about the system of precision are possible for probabilities.As a result, the analogue of the set structure of precision, a pre-Dynkin-system, is the space of gambles with precise expectations, a single linear subspace.In Definition 6.1, however, we equated pre-Dynkin-systems with sets of linear subspaces.In this case, a one-to-one correspondence between a finitely additive probability on a pre-Dynkin-system, and a generalized expectation, concretely a partial expectation, can be established.Hence, the analogy of pre-Dynkin-systems and linear subspaces of gambles depends on the correspondence of probability and expectation.

Generalized Extendability is Equivalent to Coherence
Partial expectations are, as we have shown, a natural generalization of finitely additive probabilities on pre-Dynkin-systems.Hence, it is not far-fetched to ask for definitions of coherence and extendability again, now in the more general context.It turns out that the same story can be re-told on a more general scale: The definition of coherent probabilities (Definition 4.7) is in fact just the reduction of the following definition of a coherent prevision to indicator gambles.Definition 6.5 (Coherent Prevision).[Walley, 1991, Definition 2.5.1]Let L ⊆ B(Ω) be an arbitrary subset of the linear space of bounded functions.A functional E : L → R is a coherent lower prevision if and only if This definition of coherent previsions is substantiated by consistency of gamblers regarding their betting behavior on gambles with uncertain outcome, e.g.[Walley, 1991, §2.3.1].Importantly, the rather opaque, but general definition of coherence can be simplified greatly for coherent previsions defined on linear subspaces of B(Ω).Theorem 2.5.5 in [Walley, 1991] shows that coherence for lower previsions on linear subspaces can be expressed as superadditivity, positive homogeneity, and accepting sure gains (see [Walley, 1991, Definition 2.3.3]).
Having introduced the notion of a coherent prevision, we now envisage the link between partial expectations and coherent previsions.We introduced extendability for finitely additive probabilities on pre-Dynkin-systems as a useful property.It guarantees that the probability can "nicely" be embedded into "larger" finitely additive probability which is defined on an encompassing algebra.Hence, the analogue for partial expectations is straightforward.

Definition 6.6 (General Extendability). A partial expectation E
Interestingly, the extendability condition provided in Theorem 4.5 has a (more general) cousin adapted to the setting of gambles instead of events.Proposition 6.7 (Extendability Condition for Previsions).(cf.[Maharam, 1972, Theorem 6.1]) Let {L i } i ∈I be a non-empty family of linear subspaces of B(Ω).A partial expectation E : i ∈I L i → R is extendable if and only if for every finite collection of functions f 1 , . . ., Proof.It seems that Theorem 6.1 [Maharam, 1972] is equivalent to our statement.However, there is a subtlety which we want to argue here is indeed irrelevant.Extendability of a partial expectation requires the existence of a positive, normed, linear functional on B(Ω), whose restriction on the according linear subspaces coincides with the partial expectation.Theorem 6.1 in [Maharam, 1972] only guaratees that a positive, linear functionals exists.But, normedness of such functional is automatically given if χ Ω ∈ L i for some i ∈ I .Otherwise, we extend the partial expectation E to Then again, Theorem 6.1 [Maharam, 1972] applies.
Against the background that extendability and coherence define the same concept for finitely additive probabilities on pre-Dynkin-systems, the resulting equivalence of extendability and coherence for partial expectations is of little surprise.Proposition 6.8 (Extendability is Equivalent to Coherence).Let {L i } i ∈I be a non-empty family of linear subspaces of B(Ω).The partial expectation E : i ∈I L i → R is extendable if and only if E is a linear prevision, i.e. is coherent.
Proof.If E is a linear prevision on i ∈I L i , then there is a linear prevision [Walley, 1991, Theorem 3.4.2].Conversely, if E is an extendable partial expectation, then its extension is obviously a linear prevision, hence it is coherent.The restriction of a coherent linear prevision to any subset of gambles is coherent (and linear).

A Duality Theory for Previsions and Families of Linear Subspaces
In Section 5 we step by step spelled out an order relationship between the set structure of precision and credal sets, a model for (coherent) imprecise probabilities.Naturally the presented generalization begs the question whether a related relationship between credal sets and the spaces of gambles with precise expectations exists.We answer affirmatively.We redefine the credal and dual credal set function and shortly discuss its analogous properties.Again, we require a "reference measure".In this case, it is a fixed linear prevision ψ on the space of all gambles B(Ω), which is indeed in one-to-one correspondence to a finitely additive probability measure on 2 Ω .Definition 6.9 (Generalized Credal Set Function).Let ∆ be the set of linear previsions on the Banach space B(Ω).For a fixed linear prevision ψ ∈ ∆ we call Why can we call those functions "generalized"?Simply, because any system of sets is equivalently represented as its set of indicator gambles which span their own linear space of simple gambles, i.e. linear combinations of indicator gambles.
The generalized credal set function maps, as the credal set function in Definition 5.1, to weak ⋆closed, convex subsets of ∆.The generalized dual credal set function, however, reveals a first subtlety.It maps to linear subspaces of B(Ω).The dual credal set function following Definition 5.6 mapped to pre-Dynkin-systems.In Proposition 6.2 and Proposition 6.3 families of linear subspaces were the analogues of (pre-)Dynkin-systems. Here, a single linear subspace is the analogue of a pre-Dynkin-system.For a first step towards an explanation of this asymmetry see Section 6.2.Finally, the pair of functions constitute a Galois connection.Proof.(a) We have fixed ψ to a linear prevision.Hence, it is coherent.For any G ⊆ B(Ω), m(G ) is the set of all linear previsions which dominate ψ on G .Theorem 3.6.1 in [Walley, 1991] then states that this set is weak ⋆ -closed and convex.
Then, for all ν ∈ ∆, For homogeneity we need the easy fact that a linear prevision is not only positive homogeneous, but generally homogeneous.For this consider a linear prevision ν and any gamble f ∈ B(Ω) with α < 0, then (c) The two functions constitute a Galois connection (cf.Proposition 5 To this end, we show the left to right implication, and the right to left implication, This concludes the proof.
Again it seems to be more intricate than expected to characterize bipolar-closed sets.For pre-Dynkin-systems and finitely additive measures we already collected some first hints that sets of measure zero play an important role in the characterization of bipolar-closed sets.In the case of linear subspaces and linear previsions we observe a similar "combinatorial restriction".In order to improve understanding, let us replace 2 ∆ by 2 ba(Ω) in Definition 6.9, Definition 6.10 and Proposition 6.1123 , which is equivalent to stating that linear previsions are not necessarily normalized, nor positive.Then, by leveraging the Hahn-Banach-type Theorem 1.5.14 in [Rao and Rao, 1983], one can easily see that linearity of a subset G ⊆ B(Ω) is not only a necessary, but as well a sufficient condition for bipolar-closedness for those modified "credal set functions".Thus, the restriction to actual linear previsions makes the characterization of bipolar-closed sets more complex.A compelling, more exhaustive answer still waits to be found.
Analogous to the discussion in Section 5.4, it is possible to provide a lattice duality and interpolation scheme via the generalized (dual) credal set functions.Instead of the lattice of pre-Dynkin-systems (D, ⊆) the interpolation is directed by the lattice of linear subspaces (L , ⊆) of B(Ω).As commonly known, the lattice of linear subspaces has the two operations L 1 ∧ L 2 := L 1 ∩ L 2 and L 1 ∨L 2 := lin(L 1 ∪L 2 ) 24 .Its minimal element is the trivial zero vector linear subspace {0}.Its maximal element is the entire space of all gambles B(Ω).Due to higher generality of the here presented (dual) credal set function, the interpolation provided is more fine-grained than for the previously given interpolation by pre-Dynkin-systems.The following set containment (trivially) holds: where m(D) = m({χ D : D ∈ D}).In other words, the lattice of pre-Dynkin-systems is "contained" in the lattice of of linear subspaces.However, as for pre-Dynkin-systems the interpolation via linear subspaces is improper.The reason for this is again that not every linear subspace is bipolar-closed.By restriction to linear, bipolar-closed subspaces one can clean up the setup.For details we refer to Section 5.4.We do not make explicit the detailed reiteration of the same argument here.
In summary, we confirmed our findings of Section 5 extended to previsions.The generalized dual lattice setup underlines the structural consistency between the system of precision and its corresponding imprecise probability.

Conclusion and Open Questions
In this paper, we have explicated relations between the systems of precision and imprecise probabilities (respectively expectations).First, we have shown that the system of precision forms a pre-Dynkin-system (respectively a linear subspace).This structural insight raises a series of followup questions: How does the system of precision of a coherent prevision relate to the set of desirable gambles of this prevision?How does the preference ordering change the set structure of precision for the corresponding beliefs?What is the role of coherence with respect to the system of precision?
Second, we defined finitely additive probabilities on pre-Dynkin-systems.The equivalence of extendability and coherence of such probabilities strengthens the link between quantum probability and imprecise probability.We speculate that further insights can be obtained by exploiting this relationship.In addition, the generalization of finitely additive probabilities on pre-Dynkin-systems to partial expectations directly opens the door to machine learning applications.In robust machine learning the expected risk minimization framework is extended to more general expectation functionals.Partial expectation can, possibly after more computational investigations, deliver the desired robustness against dependencies in specific domains, such as privacy preservation, "not-missing-at-random" features, restricted data base access or multi-measurement data.
Finally, we developed a duality theory of systems of precision and imprecise probabilities (respectively expectations).A Galois connection defines a parametrized family of imprecise probabilities which follow an order structure provided by the lattice of pre-Dynkin-systems (respectively the lattice of linear subspaces).
In modern statistics, especially in machine learning, probabilistic statements are increasingly tailored to individuals.Individual probabilistic statements, however, require justification.One can interpret probabilities on pre-Dynkin-systems as probabilities which do not allow for such statements in the first place.One could perceive this fact as a weakness.We, in contrast, embrace its strength, when for ethical, legislative or other reasons individualistic ascriptions are harmful, unjustifiable, forbidden or not desirable.We provide a first, rough interpolation scheme via the lattice duality.It demonstrates the space of adjustability of probabilistic assumptions in real-world scenarios.The involved pre-Dynkin-systems are mathematical definitions of levels of group resolution.The question of how to choose such set-systems is related to the questions of intersectionality.
Several fundamental, technical questions remain open: how does the lattice duality imposed by pre-Dynkin-systems or linear spaces relate to other dualities, such as convex duality, exploited in the field of imprecise probability.Can one easily characterize the bipolar-closed sets?Why is there no clear analogy between pre-Dynkin-systems and linear subspaces?

C Credal Sets of Pre-Dynkin-System Probabilities -Credal Sets of Distorted Probabilities
To the best of the authors knowledge, there has not been any attempt to parametrize a family of imprecise probabilities via the set of induced precise probabilities.In fact, a much better known class of imprecise probabilities is parametrized via distortion functions [Wirch and Hardy, 2001].
We use distorted probability functions as they regularly occur as examples of imprecise probabilities [Walley, 1991, Wirch and Hardy, 2001, Fröhlich and Williamson, 2022].In particular, there is a one-to-one correspondence of distorted probabilities as defined in the following and so-called spectral risk measures, an important class of coherent upper previsions often used in economics and finance [Fröhlich and Williamson, 2022].
In the following Proposition we show that the set of events on which all measures of a credal set M (ψ, γ) coincide forms a pre-Dynkin-system.Actually, it is the system of certainty, i.e. the set of all events which get assigned either the value 0 or the value 1.We reuse the notation of the dual credal set function m • which, as we noted earlier, maps a set of probabilities to the set structure on which those probabilities coincide.Proof.We show the equality of all sets via a circular set containment.(c) m • (M (ψ, γ)) ⊆ F 01 We show this set inclusion via contraposition.If F ∈ 2 Ω has measure ψ(F ) ∈ (0, 1), then F ∉ m • (M (ψ, γ)).For this we have to argue that there is a measure ν F ∈ M (ψ, γ) for every F ∈ 2 Ω with ψ(F ) ∈ (0, 1) such that ν F (F ) ̸ = ψ(F ).
We call the set F 01 a system of certainty for ψ.Hence, the system of precision of a distorted probability is the system of certainty.Since the proposition clarifies the relation from distorted imprecise probability to probability on Dynkin-system, the reverse question immediately follows: when is M (ψ| D , D) ⊆ M (ψ, γ)?In other words, given an extendable probability defined on a pre-Dynkin-system, what is a distortion function of an extension of this probability such that the credal set of the former is contained in the credal set of the latter.The simple example below gives an instantiation of this problem for which it is easy to find a solution.However, the problem is harder for more general cases.

D Dynkin-Systems and Countably Additive Probability
In the main text, we presented the majority of the results for the general case: pre-Dynkin-systems and finitely additive probabilities.We did so, as we emphasize clearly here, not for the sake of mathematical generality.There are probabilistic problems which demand for the use of finitely additive probabilities, e.g.von Mises frequentistic notion of probability [von Mises andGeiringer, 1964, Schurz andLeitgeb, 2008].As pre-Dynkin-systems and finitely additive probabilities walk hand in hand, so too do Dynkin-systems and countably additive probabilities.It is possible to strengthen some results when we assume Dynkin-systems and countably additive probabilities.Furthermore, countably additive probabilities are more familiar to students of probability theory.Finitely additive probabilities still eke out an exotic living [Rao and Rao, 1983, Kadane et al., 1986, Chichilnisky, 2010].Nevertheless, they are central in large parts of literature on imprecise probability [Walley, 1991, Augustin et al., 2014].

D.1 Dynkin-Systems
Many of the following results are analogous to the statements for pre-Dynkin-systems.The subsequent theorem is analogous to Theorem 2.7.We remark, however, that we cannot use the same proof-technique as in Theorem 2.7 because the generalization of Lemma A.5 to σ-algebras does not hold.Hence, we use a different technique to prove Theorem D.1 for Dynkin-systems and σ-algebras.Theorem D.1 (Dynkin-Systems are made out of σ-Algebras).Let D σ be a Dynkin-system on an arbitrary set Ω. Then there is a unique family of maximal σ-algebras {A i } i ∈I such that D σ = i ∈I A i .We call these σ-algebras the σ-blocks of D σ .
Proof.Since Dynkin-systems are pre-Dynkin-systems Theorem 2.7 guarantees that D σ is constituted of a set of algebras {A i } i ∈I .Each algebra A i is closed under finite intersection.Thus, it is a "compatible collection" following the terms of Gudder [1973].It follows that A i is contained in a sub-σ-algebra of D σ by Theorem 2.1 in [Gudder, 1973].As any sub-σ-algebra is an algebra and A i is maximal, i.e. there is no algebra contained in D such that A i is a strict sub-algebra of this algebra, A i itself is a σ-algebra.Hence, A i are the σ-blocks of D σ .
Proof.Proposition D.4 separates the Dynkin probability space into a set of Kolmogorov probability spaces.Then Theorem 8.1 in [Maharam, 1972] applies.As technical requirements, we emphasize that Ω is a Hausdorff space and all restrictions µ i are inner regular.Thus, there exists a probability measure on the algebra generated by D σ .Since the probability measure is countably additive, Caratheodory's Extension Theorem [Williams, 1991, Theorem 1.7] states that it can be uniquely extended to the Borel-σ-algebra F σ = σ(D σ ).
Similar and related results can be found in [Vorob'ev, 1962, Kellerer, 1964, Horn and Tarski, 1948].In particular, every so-called marginal scenario can be represented as a countably additive probability on a Dynkin-system (cf. [Vorob'ev, 1962] [Gudder, 1984, Example 4.2]).Marginal scenarios often arise in practical setups.They are defined as settings in which for a collection of random variables only specific partial joint distributions of the random variables are given.For those scenarios there are purely combinatorial conditions on the Dynkin-system sufficient for the extendability [Vorob'ev, 1962, Kellerer, 1964].

D.5 Credal Set of Countably Additive Probabilities on Dynkin-Systems
The credal set of probabilities which we defined in Corollary 4.9 contains finitely additive probabilities, some of which might not be countably additive.For this reason we redefine the credal set for countably additive probabilities.The credal set is the set of all countably additive probabilities which coincide with the reference probability µ σ on D σ .Proposition D.7 (σ-Credal Set for Probabilities on Dynkin-Systems).Let F σ be the Borel-σ-algebra on a Polish space Ω and D σ a Dynkin-system contained in this σ-algebra.Let µ σ be a countably additive probability on D σ and ∆ σ the set of all countably additive probabilities on F σ .We call Closedness We assumed Ω to be a Polish space.It follows that ∆ σ itself is a Polish space [Huber, 1981, page 29].Every Polish space is metrizable, consequently Lemma 21.2 in [Munkres, 2014] applies.Closedness of a set Q ⊆ ∆ σ can be identified via the convergence of sequences in Q.
Rather obviously the following corollary holds.

Figure 1 :
Figure1: Illustration of the running example.The dark elements are contained in the pre-Dynkinsystem D on Ω = {1, 2, 3, 4}.The lower and upper coherent extension, respectively the inner and outer extension are denoted at the sides of the elements in the set system as shown in the example in the left upper corner.Elements in D possess a precise probability.
Proposition 5.3 (Credal Set Function is Invariant to Pre-Dynkin-Hull).Let m be the credal set function.For any A ⊆ 2 Ω m(A ) = m(D(A )).

Proposition 6 .
11 (Properties of Generalized (Dual) Credal Set Function).Let m be a generalized credal set function and m • be a generalized dual credal set function.All the following properties hold: (a) The generalized credal set function m maps to weak ⋆ -closed, convex sets.(b) The generalized dual credal set function m • maps to a linear subspace.(c) The generalized credal set function m and generalized dual credal set function m • form a Galois connection.

Table 1 :
Summary of important, used notations.