## Abstract

On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary relation (‘qualitative probability’) that is tied to *measure*. It expresses that one region is *smaller than or equal in size* to another. Algebraic models of our theory are *separation* *σ*-*algebras with qualitative probability*: \((\mathbb {B}, \ll , \preceq )\), where \(\mathbb {B}\) is a Boolean *σ*-algebra, ≪ is a separation relation on \(\mathbb {B}\), and ≼ is a qualitative probability on \(\mathbb {B}\). We show that from algebraic models of this kind we can, in an interesting class of cases, recover a compact Hausdorff topology *X*, together with a countably additive measure *μ* on a *σ*-field of Borel subsets of that topology, and that \((\mathbb {B}, \ll , \preceq )\) is isomorphic to a ‘standard model’ arising out of the pair (*X*, *μ*). It follows from one of our main results that any closed ball in Euclidean space, \(\mathbb {R}^{n}\), together with Lebesgue measure arises in this way from a separation *σ*-algebra with qualitative probability.

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## Acknowledgments

I thank Tinko Tinchev for a careful reading of the manuscript and helpful comments. I thank the TEAM conference at Princeton University and the philosophy departments at Duke University and Stanford University for the opportunity to present this paper.

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I owe this beautiful phrase to Halmos, who uses it in his ‘automathography’ [13].

## Appendix A: Proof of Theorem 18

### Appendix A: Proof of Theorem 18

In this appendix, we prove Theorem 18: the extension of Villegas’s representation theorem (Theorem 17) to abstract Boolean *σ*-algebras. We begin by recalling the Loomis-Sikorski representation theorem [17] and [23]:

###
**Theorem 121**

If \(\mathbb {B}\) is a Boolean *σ*-algebra, then there is a *σ*-field of sets, \(\mathbb {F}\), and a *σ*-ideal *I* of \(\mathbb {F}\) such that \(\mathbb {B}\) is isomorphic to the quotient \(\mathbb {F} \slash I\).

Throughout this appendix, let \(\mathbb {B}\) be a Boolean *σ*-algebra, and let ≼ be a monotonely continuous, bottomless qualitative probability on \(\mathbb {B}\). By Theorem 120, we know that there is a *σ*-field \(\mathbb {F}\) of subsets of a set Ω, a *σ*-ideal *I* of \(\mathbb {F}\) and an isomorphism:

We denote by upper-case letters *A*, *B*, *C*, … elements of the *σ*-field \(\mathbb {F}\), and by |*A*| the equivalence class in \(\mathbb {F} \slash I\) containing the set *A*. Note that since \(\mathbb {F}\) is a *σ*-field and *I* is a *σ*-ideal, the map \(|\cdot |: \mathbb {F} \to \mathbb {F} \slash I\) defined by *A*↦|*A*| is a Boolean *σ*-homomorphism.

Define the relation \(\preceq ^{\prime }\) on \(\mathbb {F} \slash I\) by putting: \(|A| \preceq ^{\prime } |B|\) iff *φ*^{− 1}(|*A*|) ≼ *φ*^{− 1}(|*B*|). Then clearly \(\preceq ^{\prime }\) is a monotonely continuous, bottomless qualitative probability on \(\mathbb {F} \slash I\). We now define the relation \(\preceq _{\mathbb {F}}\) on \(\mathbb {F}\) by putting:

Note that for any \(A, B \in \mathbb {F}\), \(A \prec _{\mathbb {F}} B\) iff \(|A| \prec ^{\prime } |B|\).

###
**Lemma 122**

\(\preceq _{\mathbb F}\) is a monotonely continuous, bottomless qualitative probability on \(\mathbb {F}\).

###
*Proof*

The reader can verify that \(\preceq _{\mathbb F}\) is a qualitative probability on \(\mathbb {F}\). To see that \(\preceq _{\mathbb F}\) is monotonely continuous, suppose that \(A_{1} \subseteq A_{2} \subseteq A_{3} \dots \) and \(A_{n} \preceq _{\mathbb F} B\) for all \(n \in \mathbb N\). Since \(|\cdot |: \mathbb {F} \to \mathbb {F} \slash I\) is a Boolean homomorphism, \(|A_{1}| \leq |A_{2}| \leq |A_{3}| \dots \), and by definition of \(\preceq _{\mathbb {F}}\), \(|A_{n}| \preceq ^{\prime } |B|\) for all \(n \in \mathbb N\). So by monotone continuity of \(\preceq ^{\prime }\), \(\bigvee _{n \in \mathbb {B}} |A_{n}| \preceq ^{\prime } |B|\). But since |⋅| is a Boolean *σ*-homomorphism, \(\bigvee _{n \in \mathbb {B}} |A_{n}|= |\bigcup _{n \in \mathbb N} A_{n} |\). Therefore \(\bigcup _{n \in \mathbb N} A_{n} \preceq _{\mathbb {F}} B\).

To see that \(\preceq _{\mathbb {F}}\) is bottomless, suppose that \(A \in \mathbb {F}\) and \(\emptyset \prec _{\mathbb {F}} A\). Then \(|\emptyset | \prec ^{\prime } |A|\). Since \(\preceq ^{\prime }\) is bottomless, there exists \(b \in \mathbb {F} \slash I\) such that *b* ≤|*A*| and \(|\emptyset | \prec ^{\prime } b \prec ^{\prime } |A|\). Let *b* = |*B*|. We can assume WLOG that \(B \subseteq A\). (If not, let \(B^{\prime }=B \cap A\). Then clearly \(B^{\prime } \subseteq A\). Since |*B*|≤|*A*|, we have: \(b=|B| = |B| \wedge |A| = |B \cap A| = |B^{\prime }|\).) Since \(|\emptyset | \prec ^{\prime } |B| \prec ^{\prime } |A|\), we have \(\emptyset \prec _{\mathbb {F}} B \prec _{\mathbb {F}} A\). □

###
**Lemma 123**

There is a unique probability measure on \(\mathbb {F}\) that represents \(\preceq _{\mathbb {F}}\), and it is countably additive.

###
*Proof*

Immediate from Lemma 121 and Theorem 17. □

Let \(m_{\mathbb {F}}\) be the unique probability measure on \(\mathbb {F}\) that represents \(\preceq _{\mathbb {F}}\). Note that for any *A* ∈ *I*, \(m_{\mathbb {F}}(A)=0\). Indeed, if *A* ∈ *I*, then |*A*| = |*∅*|, so \(|A| \preceq ^{\prime } |\emptyset |\), and therefore \(A \preceq _{\mathbb {F}} \emptyset \). Since \(m_{\mathbb {F}}\) represents \(\preceq _{\mathbb {F}}\), \(m_{\mathbb {F}}(A) \leq m_{\mathbb {F}}(\emptyset ) =0\).

Define the function \(m^{\prime }\) on \(\mathbb {F} \slash I\) by putting:

Note that \(m^{\prime }\) is well-defined, because if \(|A|=|A^{\prime }|\), then \(A \bigtriangleup A^{\prime } \in I\), so \(m_{\mathbb {F}}(A \bigtriangleup A^{\prime }) =0\), and \(m_{\mathbb {F}}(A) = m_{\mathbb {F}}(A^{\prime })\).

###
**Proposition 124**

\(m^{\prime }\) is the unique probability measure on \(\mathbb {F} \slash I \) that represents \(\preceq ^{\prime }\), and it is countably additive.

###
*Proof*

We first show that \(m^{\prime }\) is a countably additive probability measure that represents \(\preceq ^{\prime }\), and then prove the uniqueness claim. Note that \(m^{\prime }(1) = m^{\prime }(|{\Omega }|) = m_{\mathbb {F}}({\Omega })=1\), since \(m_{\mathbb {F}}\) is a probability measure on \(\mathbb {F}\). Clearly also \(m^{\prime }\) is real-valued and non-negative, since \(m_{\mathbb {F}}\) is. Suppose that \(\{a_{n} | n \in \mathbb N\}\) is a pairwise disjoint set of elements in \(\mathbb {F} \slash I\). Then *a*_{n} = |*A*_{n}| for some \(A_{n} \in \mathbb {F}\). WLOG we can assume that the *A*_{n}’s are pairwise disjoint. (If not, let *B*_{0} = *A*_{0} and let \(B_{n+1}=A_{n+1} \setminus (A_{0} \cup {\dots } \cup A_{n})\), for \(n \in \mathbb N\). Then the *B*_{n}’s are pairwise disjoint, and |*A*_{n}| = |*B*_{n}| for all \(n \in \mathbb N\).) So we have:

Thus \(m^{\prime }\) is a countably additive probability measure on \(\mathbb {F} \slash I\). To see that \(m^{\prime }\) represents \(\preceq ^{\prime }\), note that:

For the uniqueness claim, suppose that *μ* is a probability measure on \(\mathbb {F} \slash I\) that represents \(\preceq ^{\prime }\). We must show that \(\mu =m^{\prime }\). Define the function *ν* on \(\mathbb {F}\) by putting: *ν*(*A*) = *μ*(|*A*|). Then *ν* is a probability measure on \(\mathbb {F}\). Indeed, *ν* is real-valued and non-negative, since *μ* is; *ν*(Ω) = *μ*(|Ω|) = *μ*(1) = 1; and finally, if *A* and *B* are disjoint elements of \(\mathbb {F}\), then |*A*| and |*B*| are disjoint elements of \(\mathbb {F} \slash I\), so *ν*(*A* ∪ *B*) = *μ*(|*A*|∨|*B*|) = *μ*(|*A*|) + *μ*(|*B*|) = *ν*(*A*) + *ν*(*B*). Moreover, *ν* represents \(\preceq _{\mathbb {F}}\), since:

But \(m_{\mathbb {F}}\) is the unique probability measure on \(\mathbb {F}\) that represents \(\preceq _{\mathbb {F}}\). Therefore, \(\nu =m_{\mathbb {F}}\). But then \(\mu =m^{\prime }\). We have shown that \(m^{\prime }\) is the unique probability measure on \(\mathbb {F} \slash I \) that represents \(\preceq ^{\prime }\). □

Recall that *φ* is a Boolean isomorphism from \(\mathbb {B}\) to \(\mathbb {F} \slash I\). Define the function *m* on \(\mathbb {B}\) by putting:

Since *φ* is an isomorphism, we have the following.

###
**Proposition 125**

*m* is the unique probability measure on \(\mathbb {B}\) that represents ≼, and it is countably additive.

We have shown that if \(\mathbb {B}\) is a Boolean *σ*-algebra and ≼ is a monotonely continuous, bottomless qualitative probability on \(\mathbb {B}\), then there is a unique probability measure on \(\mathbb {B}\)—namely *m* defined above—that represents ≼, and it is countably additive. Thus Theorem 18 is proved.

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Lando, T. Valueless Measures on Pointless Spaces.
*J Philos Logic* **52**, 1–52 (2023). https://doi.org/10.1007/s10992-022-09652-w

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DOI: https://doi.org/10.1007/s10992-022-09652-w

### Keywords

- Region-based theories of space
- Boolean contact algebras
- Topology
- Measure algebras