Rings of maps: sequential convergence and completion

Frič, Roman

Czechoslovak Mathematical Journal, Tome 49 (1999) no. 1, pp. 111-118.

Résumé

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.

Détail

Classification : 54A20, 54B30, 54H13, 60A99
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