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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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