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Rings of maps: sequential convergence and completion. (English) Zbl 0949.54003
Summary: The ring $$B(\mathbb R)$$ of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring $$C(\mathbb R)$$ of all continuous functions and, similarly, the ring $$\mathbb B$$ of all Borel measurable subsets of $$\mathbb R$$ 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 $$\mathcal L_0^*$$-rings of maps and develop a completion theory covering the two examples. In particular, the $$\sigma$$-fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets $$\mathbb A$$, the generated $$\sigma$$-field $$\sigma (\mathbb A)$$ yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative $$\mathcal L_0^*$$-groups.

##### MSC:
 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 54B30 Categorical methods in general topology 54H13 Topological fields, rings, etc. (topological aspects) 60A99 Foundations of probability theory
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