Rings of maps: sequential convergence and completion

Roman Frič

Czechoslovak Mathematical Journal (1999)

  • Volume: 49, Issue: 1, page 111-118
  • ISSN: 0011-4642

Abstract

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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 𝔹 of all Borel measurable subsets of R is a sequential ring completion of the subring 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 𝔸 , the generated σ -field σ ( 𝔸 ) yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative ℒ 0 * -groups.

How to cite

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Frič, Roman. "Rings of maps: sequential convergence and completion." Czechoslovak Mathematical Journal 49.1 (1999): 111-118. <http://eudml.org/doc/30469>.

@article{Frič1999,
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 $\mathcal \{L\}_0^*$-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.},
author = {Frič, Roman},
journal = {Czechoslovak Mathematical Journal},
keywords = {Rings of sets; completion of sequential convergence rings; $Z(2)$-generation; $Z(2)$-completion; $\sigma $-rings of maps; epireflection; fields of events; foundation of probability; rings of sets; completion of sequential convergence rings; -generation; -completion; -rings of maps; epireflection; fields of events; foundation of probability},
language = {eng},
number = {1},
pages = {111-118},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Rings of maps: sequential convergence and completion},
url = {http://eudml.org/doc/30469},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Frič, Roman
TI - Rings of maps: sequential convergence and completion
JO - Czechoslovak Mathematical Journal
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 1
SP - 111
EP - 118
AB - 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.
LA - eng
KW - Rings of sets; completion of sequential convergence rings; $Z(2)$-generation; $Z(2)$-completion; $\sigma $-rings of maps; epireflection; fields of events; foundation of probability; rings of sets; completion of sequential convergence rings; -generation; -completion; -rings of maps; epireflection; fields of events; foundation of probability
UR - http://eudml.org/doc/30469
ER -

References

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