Valueless Measures on Pointless Spaces

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 : ( 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.