Valueless Measures on Pointless Spaces

Journal of Philosophical Logic 52 (1):1-52 (2022)
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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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Tamar Lando
Columbia University

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The Foundations of Statistics.Leonard J. Savage - 1954 - Synthese 11 (1):86-89.
La Prévision: Ses Lois Logiques, Ses Sources Subjectives.Bruno de Finetti - 1937 - Annales de l'Institut Henri Poincaré 7 (1):1-68.

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