#
Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

When posing problems in probability calculus, it should be required to indicate for which events the probabilities are assumed to exist.—Andreĭ Kolmogorov [1] (p. 52)

## 1. Introduction

- (a)
- A machine learning algorithm has access to a restricted subset of attributes. It cannot jointly query all attributes simultaneously. This is called “learning on partial, aggregated information” [3]. The reasons might be manifold: for privacy preservation, “not-missing-at-random” features, restricted data base access for acceleration or multi-measurement data sets.
- (b)
- Quantum physical quantities, e.g., location and impulse, are (statistically) incompatible [4].
- (c)

- 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.
- Second, we define probabilities on pre-Dynkin systems in accordance with the literature on quantum probability, in particular [19]. We argue that probabilities on pre-Dynkin systems, 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 [6,19,20] to imprecise probabilities [2,21]. 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.
- 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.

- First, we propose a generalization of a finitely additive probability defined on a pre-Dynkin system. 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.
- Second, we show that under some properties, imprecise expectations are precise on a linear subspace of the linear space of gambles. (cf. Section 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.
- Last, analogous to the lattice duality (A lattice is a poset with pairwise existing minimum and maximum. The duality is expressed via an antitone lattice isomorphism.) 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.

#### Notation and Technical Details

## 2. What Is a (Pre-)Dynkin System?

**Definition 1**((Pre-)Dynkin system)

**.**

- (a)
- $\varnothing \in \mathcal{D}$,
- (b)
- $D\in \mathcal{D}$ implies ${D}^{c}:=\mathsf{\Omega}\setminus D\in \mathcal{D}$
- (c)
- $C,D\in \mathcal{D}$ with $C\cap D=\varnothing $ implies $C\cup D\in \mathcal{D}$.

- (c’)
- let ${\left\{{D}_{i}\right\}}_{i\in \mathbb{N}}\subseteq {\mathcal{D}}_{\sigma}$, if for all $i,j\in \mathbb{N}$ with $i\ne j$ it holds ${D}_{i}\cap {D}_{j}=\varnothing $ then ${\bigcup}_{i\in \mathbb{N}}{D}_{i}\in {\mathcal{D}}_{\sigma}$,

**Lemma 1**(Closedness under Set Difference)

**.**

**Proof.**

**Example**

**1.**

#### 2.1. Compatibility

**Definition 2**(Compatibility)

**.**

**Lemma 2**(Cup gives Cap gives Cup)

**.**

**Proof.**

- For the “⇒”-direction: $A\cup B=(A\setminus (A\cap B\left)\right)\cup B$. The fact $A,A\cap B,B\in \mathcal{D}$ implies $(A\setminus (A\cap B\left)\right)\cup B\in \mathcal{D}$.
- For the “⇐”-direction: $A\cap B=A\setminus \left(\right(A\cup B)\setminus B)$. The fact $A,A\cup B,B\in \mathcal{D}$ implies $A\setminus \left(\right(A\cup B)\setminus B)\in \mathcal{D}$. (A related result for Dynkin systems is given in [19] (5.1).) □

**Example**

**2.**

**Theorem**

**1**

**Proof.**

**Example**

**3.**

#### 2.2. Probabilities on Pre-Dynkin Systems

**Definition 3**(Probability Measure on a Pre-Dynkin System)

**.**

- (a)
- Normalization: $\mu (\varnothing )=0$ and $\mu \left(\mathsf{\Omega}\right)=1$.
- (b)
- Additivity: let $A,B\in \mathcal{D}$ and $A\cap B=\varnothing $. Then, $\mu (A\cup B)=\mu \left(A\right)+\mu \left(B\right)$.

## 3. Imprecise Probabilities Are Precise on a Pre-Dynkin System

**Theorem 2**(Imprecise Probability Induces a (Pre-)Dynkin System)

**.**

- (a)
- Normalization: $u(\varnothing )=\ell (\varnothing )=0$.
- (b)
- Conjugacy: $u\left(A\right)=1-\ell \left({A}^{c}\right)$ for $A,{A}^{c}\in {2}^{\mathsf{\Omega}}$.
- (c)
- Subadditivity of u: for $A,B\in {2}^{\mathsf{\Omega}}$ such that $A\cap B=\varnothing $, then $u(A\cup B)\le u\left(A\right)+u\left(B\right)$.
- (d)
- Superadditivity of ℓ: for $A,B\in {2}^{\mathsf{\Omega}}$ such that $A\cap B=\varnothing $, then $\ell (A\cup B)\ge \ell \left(A\right)+\ell \left(B\right)$.

- (e)
- Continuity from below: for ${A}_{n}\in {2}^{\mathsf{\Omega}}$ with ${A}_{n}\subseteq {A}_{n+1}$ such that ${\bigcup}_{n=1}^{\infty}{A}_{n}=A\in {2}^{\mathsf{\Omega}}$, then${lim}_{n\to \infty}u\left({A}_{n}\right)=u\left(A\right)$,

- (e’)
- Continuity from above: for ${A}_{n}\in {2}^{\mathsf{\Omega}}$ with ${A}_{n+1}\subseteq {A}_{n}$ such that ${\bigcap}_{n=1}^{\infty}{A}_{n}=A\in {2}^{\mathsf{\Omega}}$, then${lim}_{n\to \infty}\ell \left({A}_{n}\right)=\ell \left(A\right)$,

**Proof.**

**Example**

**4.**

## 4. Extending Probabilities on Pre-Dynkin Systems

#### 4.1. Inner and Outer Extension

**Proposition**

**1**

**.**Let $\mathcal{D}$ be a pre-Dynkin system on Ω and μ a finitely additive probability measure on $\mathcal{D}$. The inner probability measure

- (a)
- Normalization: ${\mu}^{*}(\varnothing )=0$, ${\mu}_{*}\left(\mathsf{\Omega}\right)=1$.
- (b)
- Conjugacy: ${\mu}^{*}\left(A\right)=1-{\mu}_{*}\left({A}^{c}\right),\phantom{\rule{4pt}{0ex}}\forall A\in {2}^{\mathsf{\Omega}}$.
- (c)
- Monotonicity: for $A,B\in {2}^{\mathsf{\Omega}}$, if $A\subseteq B$, then ${\mu}^{*}\left(A\right)\le {\mu}^{*}\left(B\right)$.

**Example**

**5.**

**Example**

**6.**

#### 4.2. Extendability and Its Equivalence to Coherence

**Definition 4**(Extendability)

**.**

**Theorem**

**3**

**Example**

**7.**

**Definition 5**(Coherent Probability)

**.**

**Theorem**

**4**

**Proof.**

#### 4.2.1. Extendability, Compatibility and Contextuality

#### 4.2.2. Extendability and Marginal Problem

#### 4.3. Coherent Extension

**Corollary 1**(Coherent Extension of Probability)

**.**

**Example**

**8.**

#### 4.4. Inner and Outer Extension Is More Pessimistic Than Coherent Extension

**Theorem**

**5**

**Proof.**

## 5. The Credal Set and Its Relation to Pre-Dynkin System Structure

#### 5.1. Credal Set Function Maps from Pre-Dynkin Systems to Coherent Probabilities

**Definition 6**(Credal Set Function)

**.**

**Example**

**9.**

**Proposition 2**(Credal Set Function is Invariant to Pre-Dynkin Hull)

**.**

**Proof.**

**Example**

**10.**

**Proposition 3**(Credal Set Function Maps to Weak${}^{\u2605}$-Closed Convex Sets)

**.**

**Proof.**

#### 5.2. The Dual Credal Set Function

**Definition 7**(Dual Credal Set Function)

**.**

**Proposition 4**(Galois Connection by (Dual) Credal Set Function)

**.**

**Proof.**

**Corollary 2**(Rules for (Dual) Credal Set Function)

**.**

**Proof.**

**Proposition 5**(Dual Credal Set Function Maps to Pre-Dynkin Systems)

**.**

**Proof.**

**Proposition 6**(Dual Credal Set Function is Invariant to Weak${}^{\u2605}$-Closed Convex Hull)

**.**

**Proof.**

**Example**

**11.**

#### 5.3. Bipolar-Closed Sets

**Example**

**12.**

#### Sufficient Conditions for Bipolar-Closed Sets

**Corollary**

**3.**

**Proof.**

**Proposition 7**(“Closedness” under Measure Zero Sets)

**.**

**Proof.**

**Theorem**

**6**

**Proof.**

**Example 13**(Bipolar-Closure is Strictly Greater Than Pre-Dynkin Hull)

**.**

#### 5.4. Interpolation from Algebra to Trivial Pre-Dynkin System

**Proposition 8**(Set of Pre-Dynkin Systems is a Lattice)

**.**

**Proof.**

**Example**

**14.**

**Proposition 9**(Lattice of Dynkin Systems and Credal Sets)

**.**

**Proof.**

**Example 15**(Non-Injectivity of Credal Set Function)

**.**

**Proposition 10**(Credal Set Function is Injective on Bipolar-Closed Sets)

**.**

**Proof.**

**Proposition 11**(Lattice of Bipolar-Closed Sets and Credal Sets)

**.**

**Proof.**

## 6. A More General Perspective—The Set of Gambles with Precise Expectation

#### 6.1. Partial Expectations Generalize Finitely Additive Probabilities on (Pre-)Dynkin Systems

**Definition 8**(Partial Expectation)

**.**

- (a)
- for any $i\in I$ and for all $f,g\in {L}_{i}$, then $E(f+g)=E\left(f\right)+E\left(g\right)$, (Partial Linearity),
- (b)
- for any $i\in I$ and any $f\in {L}_{i}$, then $E\left(f\right)\ge inff$, (Coherence).

**Proposition 12**(Finitely Additive Probability on Pre-Dynkin System and its Partial Expectation)

**.**

**Proof.**

**Proposition 13**(Finitely Additive Probability on Dynkin Systems and its Partial Expectation)

**.**

**Proof.**

#### 6.2. System of Precision—The Space of Gambles with Precise Expectations

**Theorem**

**7.**

- (a)
- Normalization: $L\left({\chi}_{\mathsf{\Omega}}\right)=U\left({\chi}_{\mathsf{\Omega}}\right)=1$.
- (b)
- Conjugacy: $U\left(f\right)=-L(-f)$ for $f\in B\left(\mathsf{\Omega}\right)$.
- (c)
- Subadditivity of U: for $f,g\in B\left(\mathsf{\Omega}\right)$, we have $U(f+g)\le U\left(f\right)+U\left(g\right)$.
- (d)
- Superadditivity of L: for $f,g\in B\left(\mathsf{\Omega}\right)$, we have $L(f+g)\ge L\left(f\right)+L\left(g\right)$.
- (e)
- Positive Homogeneity: for $\alpha \in [0,\infty )$ and $f\in B\left(\mathsf{\Omega}\right)$, we have $L\left(\alpha f\right)=\alpha L\left(f\right)$ and $U\left(\alpha f\right)=\alpha U\left(f\right)$.

**Proof.**

#### 6.3. Generalized Extendability Is Equivalent to Coherence

**Definition 9**

**.**Let $L\subseteq B\left(\mathsf{\Omega}\right)$ be an arbitrary subset of the linear space of bounded functions. A functional $\underline{\mathbb{E}}:L\to \mathbb{R}$ is a coherent lower prevision if and only if

**Definition 10**(General Extendability)

**.**

**Proposition 14**(Extendability Condition for Previsions)

**.**

**Proof.**

**Proposition 15**(Extendability is Equivalent to Coherence)

**.**

**Proof.**

#### 6.4. A Duality Theory for Previsions and Families of Linear Subspaces

**Definition 11**(Generalized Credal Set Function)

**.**

**Definition 12**(Generalized Dual Credal Set Function)

**.**

**Proposition 16**(Properties of Generalized (Dual) Credal Set Function)

**.**

- (a)
- The generalized credal set function ${m}_{\psi}$ maps to weak${}^{\u2605}$-closed, convex sets.
- (b)
- The generalized dual credal set function ${m}_{\psi}^{\circ}$ maps to a linear subspace.
- (c)
- The generalized credal set function ${m}_{\psi}$ and generalized dual credal set function ${m}_{\psi}^{\circ}$ form a Galois connection.

**Proof.**

- (b)
- Let $\mathcal{Q}\subseteq \Delta $.
**Additivity**- Let $f,g\in {m}_{\psi}^{\circ}\left(\mathcal{Q}\right)$. Then, for all $\nu \in \Delta $,$$\begin{array}{c}\hfill \nu (f+g)=\nu \left(f\right)+\nu \left(g\right)=\psi \left(f\right)+\psi \left(g\right)=\psi (f+g),\end{array}$$
**Homogeneity**- Let $f\in {m}_{\psi}^{\circ}\left(\mathcal{Q}\right)$ and $\alpha \in \mathbb{R}$. Then, for all $\nu \in \Delta $,$$\begin{array}{c}\hfill \nu \left(\alpha f\right)=\alpha \nu \left(f\right)=\alpha \psi \left(f\right)=\psi \left(\alpha f\right),\end{array}$$

- (c)
- The two functions constitute a Galois connection (cf. Proposition 4), $\mathcal{G}\subseteq {m}_{\psi}^{\circ}\left(\mathcal{Q}\right)\iff \mathcal{Q}\subseteq {m}_{\psi}\left(\mathcal{G}\right)$. To this end, we show the left to right implication,$$\begin{array}{c}\hfill \nu \in \mathcal{Q}\Rightarrow \nu \left(g\right)=\psi \left(g\right),\forall g\in \mathcal{G}\Rightarrow \nu \in {m}_{\psi}\left(\mathcal{G}\right),\end{array}$$$$\begin{array}{c}\hfill g\in \mathcal{G}\Rightarrow \nu \left(g\right)=\psi \left(g\right),\forall \nu \in \mathcal{Q}\Rightarrow g\in {m}_{\psi}^{\circ}\left(\mathcal{Q}\right).\end{array}$$

**Example**

**16.**

## 7. Conclusions and Open Questions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Lemmas and Proofs

#### Appendix A.1. Compatibility Structure

**Example A1**(Union of Algebras is not Necessarily Pre-Dynkin System)

**.**

**Definition A1**($\pi $-System)

**.**

**Definition A2**(Compatibility Structure)

**.**

- (a)
- ${\bigcup}_{i\in I}{\mathcal{I}}_{i}=\mathcal{A}$
- (b)
- For every π-system $\mathcal{I}\subseteq \mathcal{A}$ there exist $i\in I$ such that $\mathcal{I}\subseteq {\mathcal{I}}_{i}$.

**Theorem A1**(Union of Algebras is Pre-Dynkin System if Compatibility Structure)

**.**

**Proof.**

#### Appendix A.2. Supremum of a Chain of Algebras Is an Algebra

**Lemma A1**(Supremum of a Chain of Algebras is an Algebra)

**.**

**Proof.**

#### Appendix A.3. Sufficient Conditions for Bipolar-Closed Sets

**Definition A3**(Weak Atom with Respect to Pre-Dynkin System)

**.**

**Lemma A2**(Decomposition into Pre-Dynkin Set and Atom)

**.**

**Proof.**

**Lemma A3**(Weak Atoms Are Not in the Bipolar-Closed Sets – Finite, Discrete Setting)

**.**

**Proof.**

#### Appendix A.4. Intersection of Linear Subspaces

**Lemma A4**(Intersection of Simple Gamble Spaces)

**.**

**Proof.**

**Lemma A5**(Intersection of Measurable Gamble Spaces)

**.**

**Proof.**

## Appendix B. Names of Pre-Dynkin Systems and Dynkin Systems

## Appendix C. Credal Sets of Pre-Dynkin System Probabilities—Credal Sets of Distorted Probabilities

**Definition A4**(Credal Set of Distorted Probability)

**.**

**Lemma A6**(Distortion Lemma)

**.**

**Proof.**

**Proposition A1**(Events of Precise Probability for Distorted Probabilities)

**.**

**Proof.**

- (a)
- ${\mathcal{F}}_{01}\subseteq D\left({\mathcal{F}}_{0}\right)$If $F\in {2}^{\mathsf{\Omega}}$ such that $\psi \left(F\right)=0$, then clearly $F\in {\mathcal{F}}_{0}\subseteq D\left({\mathcal{F}}_{0}\right)$. If $F\in {2}^{\mathsf{\Omega}}$ such that $\psi \left(F\right)=1$, then $\psi \left({F}^{c}\right)=0$. Thus, ${F}^{c}\in {\mathcal{F}}_{0}$ because pre-Dynkin systems are closed under complement $F\in D\left({\mathcal{F}}_{0}\right)$.
- (b)
- $D\left({\mathcal{F}}_{0}\right)\subseteq {m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)$First, we write out$$\begin{array}{c}\hfill {m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)=\{F\in {2}^{\mathsf{\Omega}}:\nu \left(F\right)=\psi \left(F\right),\phantom{\rule{4pt}{0ex}}\forall \nu \in \{{\nu}^{\prime}\in \Delta :{\nu}^{\prime}\left(F\right)\le \gamma \left(\psi \left(F\right)\right),\phantom{\rule{4pt}{0ex}}\forall F\in {2}^{\mathsf{\Omega}}\}\}\end{array}$$Since $\gamma \left(0\right)=0$, it is easy to see that ${\mathcal{F}}_{0}\subseteq {m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)$. By Proposition 5, we know that ${m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)$ is a pre-Dynkin system. Thus, $D\left({\mathcal{F}}_{0}\right)\subseteq {m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)$.
- (c)
- ${m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)\subseteq {\mathcal{F}}_{01}$We show this set inclusion via contraposition. If $F\in {2}^{\mathsf{\Omega}}$ has measure $\psi \left(F\right)\in (0,1)$, then $F\notin {m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)$. For this we have to argue that there is a measure ${\nu}_{F}\in M(\psi ,\gamma )$ for every $F\in {2}^{\mathsf{\Omega}}$ with $\psi \left(F\right)\in (0,1)$ such that ${\nu}_{F}\left(F\right)\ne \psi \left(F\right)$.Observe that $\gamma \circ \psi $ defines a normalized, monotone, submodular set function on ${2}^{\mathsf{\Omega}}$ [43] (p. 17). Furthermore, any normalized, monotone, submodular set function induces a translation equivariant, monotone, positively homogeneous and subadditive functional ${L}_{\gamma \circ \psi}$ on all $f\in B\left(\mathsf{\Omega}\right)$ such that ${L}_{\gamma \circ \psi}\left({\chi}_{F}\right)=(\gamma \circ \psi )\left(F\right)$ for all $F\in {2}^{\mathsf{\Omega}}$ [80] (p. 260), [2] (p. 130), [43] (Proposition 5.1, Theorem 6.3). Hence, ${L}_{\gamma \circ \psi}$ is a coherent upper prevision [2] (p. 65). Thus, Walley’s extreme point theorem applies [2] (Theorem 3.6.2 (c)). (Even though this theorem is stated in terms of coherent lower previsions, it applies to coherent upper previsions, too. The weak${}^{\u2605}$-compactness of the credal set $M(\psi ,\gamma )$, which is given by the coherence of ${L}_{\gamma \circ \psi}$ and [2] (Theorem 3.6.1), is crucial.) For any function $f\in B\left(\mathsf{\Omega}\right)$, in particular any ${\chi}_{F}$ with $F\in {2}^{\mathsf{\Omega}}$, there is a linear prevision $f\mapsto \langle f,\nu \rangle $ with $\nu \in \Delta $ on $B\left(\mathsf{\Omega}\right)$ dominated by ${L}_{\gamma \circ \psi}$ such that $\langle {\chi}_{F},\nu \rangle ={L}_{\gamma \circ \psi}\left({\chi}_{F}\right)$. More concretely, for any $F\in {2}^{\mathsf{\Omega}}$ there is a ${\nu}_{F}\in M(\psi ,\gamma )$ such that ${\nu}_{F}\left(F\right)=(\gamma \circ \psi )\left(F\right)$. If $\psi \left(F\right)\in (0,1)$, then Lemma A6 applies and ${\nu}_{F}\left(F\right)=(\gamma \circ \psi )\left(F\right)=\gamma \left(\psi \left(F\right)\right)>\psi \left(F\right)$ gives the desired inequality. In conclusion, there is no $F\in {2}^{\mathsf{\Omega}}$ with measure $\psi \left(F\right)\in (0,1)$ such that $F\in {m}_{\psi}^{\circ}\left(M(\psi ,\gamma )\right)$. This implication finalizes the proof.

**Example**

**A2.**

## Appendix D. Dynkin Systems and Countably Additive Probability

#### Appendix D.1. Dynkin Systems

**Theorem A2**(Dynkin Systems are made out of $\sigma $-Algebras)

**.**

**Proof.**

**Theorem A3**(Union of $\sigma $-Algebras is Dynkin system if Compatibility Structure)

**.**

**Proof.**

#### Appendix D.2. Technical Setup

**Table A3.**Summary of used notations in Appendix D.

$\mathsf{\Omega}$ | Polish Space |

${\mathcal{D}}_{\sigma}$ | Dynkin system on $\mathsf{\Omega}$ (Definition 1) |

${\mu}_{\sigma}$ | Countably additive probability defined on ${\mathcal{D}}_{\sigma}$ (Definition 3) |

$\sigma \left(\mathcal{A}\right)$ | $\sigma $-algebra hull of set system $\mathcal{A}\subseteq {2}^{\mathsf{\Omega}}$ |

${\mathcal{F}}_{\sigma}$ | Borel-$\sigma $-algebra on $\mathsf{\Omega}$ |

$ca(\mathsf{\Omega},{\mathcal{F}}_{\sigma})$ | Set of bounded, signed, countably additive measures on ${\mathcal{F}}_{\sigma}$ |

${\Delta}_{\sigma}(\mathsf{\Omega},{\mathcal{F}}_{\sigma})$ respectively ${\Delta}_{\sigma}$ | Set of countably additive probability measures on ${\mathcal{F}}_{\sigma}$ |

${M}_{\sigma}({\mu}_{\sigma},{\mathcal{D}}_{\sigma})$ | $\sigma $-Credal set of ${\mu}_{\sigma}$ on ${\mathcal{D}}_{\sigma}$ (Proposition A3) |

${\underline{\mu}}_{{\mathcal{D}}_{\sigma}}$, ${\overline{\mu}}_{{\mathcal{D}}_{\sigma}}$ | Lower respectively upper coherent $\sigma $-extension (Proposition A4) |

#### Appendix D.3. Dynkin Probability Spaces

**Definition A5**(Dynkin Probability Space

**.**The triple $(\mathsf{\Omega},{\mathcal{D}}_{\sigma},{\mu}_{\sigma})$ is called a Dynkin probability space if and only if (a) Ω is a non-empty base space, (b) ${\mathcal{D}}_{\sigma}$ is a Dynkin system on Ω and (c) ${\mu}_{\sigma}$ is a countably additive probability measure on ${\mathcal{D}}_{\sigma}$ following Definition 3.

**Proposition**

**A2.**

#### Appendix D.4. Conditions for Extendability for Countably Additive Probabilities

**Definition A6**($\sigma $-Extendability)

**.**

**Theorem A4**($\sigma $-Extendability Condition for Countably Additive Probability)

**.**

**Proof.**

#### Appendix D.5. Credal Set of Countably Additive Probabilities on Dynkin Systems

**Proposition A3**($\sigma $-Credal Set for Probabilities on Dynkin Systems)

**.**

**Proof.**

**Convexity**Let ${\left\{{\nu}_{i}\right\}}_{i\in \left[n\right]}\subseteq {M}_{\sigma}({\mu}_{\sigma},{\mathcal{D}}_{\sigma})$ and ${\alpha}_{i}\in [0,1]$ for all $i\in \left[n\right]$ with ${\sum}_{i=1}^{n}{\alpha}_{i}$, then ${\sum}_{i=1}^{n}{\alpha}_{i}{\nu}_{i}\in {M}_{\sigma}({\mu}_{\sigma},{\mathcal{D}}_{\sigma})$, because

**Closedness**- We assumed $ca(\mathsf{\Omega},{\mathcal{F}}_{\sigma})$ to be equipped with the total variation distance. Hence, of a set $Q\subseteq ca(\mathsf{\Omega},{\mathcal{F}}_{\sigma})$ can be identified via the convergence of sequences in Q [84] (Lemma 21.2).Suppose ${\nu}_{1},{\nu}_{2}\dots \in {M}_{\sigma}({\mu}_{\sigma},{\mathcal{D}}_{\sigma})$ is a sequence of probability measures such that ${lim}_{n\to \infty}{\nu}_{n}\stackrel{TV}{=}\nu $. This directly implies set-wise convergence,$$\begin{array}{c}\hfill \nu \left(A\right)=\underset{n\to \infty}{lim}{\nu}_{n}\left(A\right)=\underset{n\to \infty}{lim}{\mu}_{\sigma}\left(A\right)={\mu}_{\sigma}\left(A\right),\phantom{\rule{2.em}{0ex}}\forall A\in {\mathcal{D}}_{\sigma}.\end{array}$$It follows that $\nu \in {M}_{\sigma}({\mu}_{\sigma},{\mathcal{D}}_{\sigma})$.

**Corollary A1**($\sigma $-Extendability and $\sigma $-Credal Set)

**.**

**Proposition A4**(Coherent $\sigma $-Extension of Probability)

**.**

**Proof.**

**Corollary A2**(Extension Theorem—Countably Additive Case)

**.**

**Proof.**

## Appendix E. From Set Systems to Logical Structures and Back

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**Figure 2.**Illustration of the running example. The dark elements are contained in the pre-Dynkin system $\mathcal{D}$ on $\mathsf{\Omega}=\{1,2,3,4\}$. 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 $\mathcal{D}$ possess a precise probability.

**Figure 3.**Galois connection between the lattice of pre-Dynkin systems and the set of credal sets. In the illustrated case, we have ${m}_{\psi}^{\circ}\left(Q\right)={m}_{\psi}^{\circ}\left({m}_{\psi}\left({m}_{\psi}^{\circ}\left(\mathcal{A}\right)\right)\right)$, respectively, ${m}_{\psi}\left(\mathcal{A}\right)={m}_{\psi}\left({m}_{\psi}^{\circ}\left({m}_{\psi}\left(Q\right)\right)\right)$. The set containment on both sides follows from Proposition 2, Corollary 2 and Proposition 6.

$\mathsf{\Omega},{2}^{\mathsf{\Omega}}$ | Base set and its power set |

$\left[n\right]$ | Set $\{1,\dots ,n\}$ |

$\mathcal{D}$ | Pre-Dynkin system on $\mathsf{\Omega}$ (Definition 1) |

${\mathcal{D}}_{\sigma}$ | Dynkin system on $\mathsf{\Omega}$ (Definition 1) |

$D\left(\mathcal{A}\right)$ | Pre-Dynkin hull of a set system $\mathcal{A}\subseteq {2}^{\mathsf{\Omega}}$ (Definition 1) |

$\mu $ | Finitely additive probability defined on $\mathcal{D}$ (Definition 3) |

${\mu}_{*}$, ${\mu}^{*}$ | Inner respectively outer extension (Proposition 1) |

${\underline{\mu}}_{\mathcal{D}}$, ${\overline{\mu}}_{\mathcal{D}}$ | Lower respectively upper coherent extension (Corollary 1) |

$M(\mu ,\mathcal{D})$ | Credal set of $\mu $ on $\mathcal{D}$ (Corollary 1) |

$\nu $ | Finitely additive probability defined on ${2}^{\mathsf{\Omega}}$ |

$\psi $ | Fixed, finitely additive probability defined on ${2}^{\mathsf{\Omega}}$ |

${\chi}_{A}$ | Indicator function of the set $A\subset \mathsf{\Omega}$ |

$\Delta $ | Set of finitely additive probability measures on ${2}^{\mathsf{\Omega}}$, set of linear previsions |

${m}_{\psi}:{2}^{{2}^{\mathsf{\Omega}}}\to {2}^{\Delta}$ | Credal set function (Definition 6) |

${m}_{\psi}^{\circ}:{2}^{\Delta}\to {2}^{{2}^{\mathsf{\Omega}}}$ | Dual credal set function (Definition 7) |

$\overline{\mathrm{co}}\phantom{\rule{4pt}{0ex}}$ | Convex, Weak${}^{\u2605}$ Closure |

$B\left(\mathsf{\Omega}\right)$ | Set of real-valued, bounded functions on $\mathsf{\Omega}$ |

$ba\left(\mathsf{\Omega}\right)$ | Set of bounded, signed, finitely additive measures on ${2}^{\mathsf{\Omega}}$ |

E | Partial Expectation (Definition 8) |

$S(\mathsf{\Omega},\mathcal{A})$ | Linear space of simple gambles on the set system $\mathcal{A}$ |

$B(\mathsf{\Omega},{\mathcal{F}}_{\sigma})$ | Linear space of bounded, ${\mathcal{F}}_{\sigma}$-measurable functions |

$\underline{\mathbb{E}}$ | Coherent lower prevision (Definition 9) |

$\overline{\mathbb{E}}$ | Coherent upper prevision (Definition 9) |

$\nu $ | Linear prevision defined on $B\left(\mathsf{\Omega}\right)$ (equivalent to $\nu $ above) |

$\psi $ | Fixed, linear prevision defined on $B\left(\mathsf{\Omega}\right)$ (equivalent to $\psi $ above) |

${m}_{\psi}:{2}^{B\left(\mathsf{\Omega}\right)}\to {2}^{\Delta}$ | Generalized credal set function (Definition 11) |

${m}_{\psi}^{\circ}:{2}^{\Delta}\to {2}^{B\left(\mathsf{\Omega}\right)}$ | Generalized dual credal set function (Definition 12) |

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**MDPI and ACS Style**

Derr, R.; Williamson, R.C.
Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces. *Entropy* **2023**, *25*, 1283.
https://doi.org/10.3390/e25091283

**AMA Style**

Derr R, Williamson RC.
Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces. *Entropy*. 2023; 25(9):1283.
https://doi.org/10.3390/e25091283

**Chicago/Turabian Style**

Derr, Rabanus, and Robert C. Williamson.
2023. "Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces" *Entropy* 25, no. 9: 1283.
https://doi.org/10.3390/e25091283