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.

在我们对空间的普通表示中，空间是由不可分割的、无量纲的点组成的；扩展区域被理解为无限的点集。基于区域的空间理论颠倒了这种原子图景，将扩展区域上的几个关系视为原始关系，并将点恢复为区域的高阶抽象。多年来，此类理论几乎只关注空间的拓扑结构和几何结构。我们向基于区域的空间理论介绍了一种新的原始二元关系（“定性概率”），它与测量有关。它表示一个区域的大小小于或等于另一个区域。我们理论的代数模型是分离 σ -具有定性概率的代数：\((\mathbb {B}, \ll , \preceq )\)，其中\(\mathbb {B}\)是布尔σ -代数，≪ 是\(\mathbb上的分离关系{B}\) ，并且 ≼ 是\(\mathbb {B}\)上的定性概率。我们表明，在一类有趣的情况下，我们可以从此类代数模型中恢复紧凑的 Hausdorff 拓扑X以及该拓扑的 Borel 子集的σ域上的可数加性测度μ ，并且\(( \mathbb {B}, \ll , \preceq )\)同构于由对 ( X , μ)产生的“标准模型”). 从我们的一个主要结果可以看出，欧几里德空间中的任何闭球\(\mathbb {R}^{n}\)连同勒贝格测度都是以这种方式从具有定性概率的分离σ -代数中产生的。