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Keywords:
$MV$-algebra; bold algebra; field of sets; Łukasiewicz tribe; sequential convergence; sequential continuity; measure; extension of measures; sequential envelope; absolute sequentially closed bold algebra; epireflective subcategory
$MV$-algebra; bold algebra; field of sets; Łukasiewicz tribe; sequential convergence; sequential continuity; measure; extension of measures; sequential envelope; absolute sequentially closed bold algebra; epireflective subcategory
Summary:
We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean $MV$-algebra) can be uniquely extended to a sequentially continuous measure on the generated Łukasiewicz tribe and, in a natural way, the extension is maximal. We prove that for normed measures on Łukasiewicz tribes monotone (sequential) continuity implies sequential continuity, hence the assumption of sequential continuity is not restrictive. This yields a characterization of the Łukasiewicz tribes as bold algebras absolutely sequentially closed with respect to the extension of probabilities. The result generalizes the relationship between fields of sets and the generated $\sigma $-fields discovered by J. Novák. We introduce the category of bold algebras and sequentially continuous homomorphisms and prove that Łukasiewicz tribes form an epireflective subcategory. The restriction to fields of sets yields the epireflective subcategory of $\sigma $-fields of sets.
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