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
of all Borel measurable subsets of R is a sequential ring completion of the subring
of all finite unions of half-open intervals; the two completions are not categorical. We study
-rings of maps and develop a completion theory covering the two examples. In particular, the -fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets
, the generated -field
yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative
-groups. |