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Title: Rings of maps: sequential convergence and completion (English)
Author: Frič, Roman
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 49
Issue: 1
Year: 1999
Pages: 111-118
Summary lang: English
Category: math
Summary: 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 $\mathcal L_0^*$-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. (English)
Keyword: Rings of sets
Keyword: completion of sequential convergence rings
Keyword: $Z(2)$-generation
Keyword: $Z(2)$-completion
Keyword: $\sigma $-rings of maps
Keyword: epireflection
Keyword: fields of events
Keyword: foundation of probability
MSC: 54A20
MSC: 54B30
MSC: 54H13
MSC: 60A99
idZBL: Zbl 0949.54003
idMR: MR1676833
Date available: 2009-09-24T10:20:25Z
Last updated: 2020-07-03
Stable URL:
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