Hajime Ishihara: A constructive theory of uniform spaces and its application to integration theory

Abstract: We introduce the constructive notion of a uniform space with the spirit of Sambin's notion of a basic pair, and construct a completion of a uniform space and a product of uniform spaces. We show some natural properties of the completion and the product. Then we define topological linear spaces and topological vector lattices as linear spaces and vector lattices equipped with uniform structures, and show that these algebraic and topological structures are preserved under the completion. We introduce the notion of an abstract integration space consisting of a vector lattice and a positive linear functional. By defining two uniform structures on an abstract integration space, we define corresponding topological vector lattices, and spaces of integration and measurable functions as the completion of these topological vector lattices. Finally, we show some convergence theorems on these spaces such as Lebesgue's monotone and dominated convergence theorems and Fatou's lemma.