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 \L ukasiewicz tribe and, in a natural way, the
extension is maximal. We prove that for normed measures on
\L ukasiewics tribes monotone (sequential) continuity
implies sequential continuity, hence the assumption of
sequential continuity is not restrictive. This yields a
characterization of the \L ukasiewicz 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. Nov\'ak. We introduce the
category of bold algebras and sequentially continuous
homomorphisms and prove that \L ukasiewicz tribes form an
epireflective subcategory. The restriction to fields of sets
yields the epireflective subcategory of $\sigma$-fields of
sets.
