We show that  every $\sigma$-complete MV-algebra is an
MV-$\sigma$-homo\-morphic image of some $\sigma$-complete
MV-algebra of fuzzy sets, called a tribe, which is a system
of fuzzy sets of a crisp set $\Omega$ containing $1_\Omega$
and closed under fuzzy complement and formation of
$\min\{\sum_n f_n, 1\}.$ Since a tribe is a direct
generalization of a $\sigma$-algebra of crisp subsets, the
representation theorem is an analogue of the
Loomis--Sikorski theorem for MV-algebras. In addition, this
result will be extended also for Dedekind $\sigma$-complete
$\ell$-groups.
