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- Title
Valueless Measures on Pointless Spaces.
- Authors
Lando, Tamar
- 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 separationσ-algebras with qualitative probability: (B , ≪ , ≼) , where B is a Boolean σ-algebra, ≪ is a separation relation on B , and ≼ is a qualitative probability on 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 (B , ≪ , ≼) 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, ℝ n , together with Lebesgue measure arises in this way from a separation σ-algebra with qualitative probability.
- Subjects
BOREL subsets; LEBESGUE measure; MODEL theory; POINT set theory; TOPOLOGY
- Publication
Journal of Philosophical Logic, 2023, Vol 52, Issue 1, p1
- ISSN
0022-3611
- Publication type
Article
- DOI
10.1007/s10992-022-09652-w