Rings of maps: sequential convergence and completion

Abstract

The ring B(R) of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring C(R) of all continuous functions and, similarly, the ring $$\mathbb{B}$$ of all Borel measurable subsets of R is a sequential ring completion of the subring $$\mathbb{B}_0 $$ of all finite unions of half-open intervals; the two completions are not categorical. We study $$L_0^* $$ -rings of maps and develop a completion theory covering the two examples. In particular, the σ-fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets $$\mathbb{A}$$ , the generated σ-field $$\sigma (\mathbb{A})$$ yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative $$L_0^* $$ -groups.