The aim of the present paper is to study the extensions of the $\bot$-decomposable measure $m$, with respect to a continuous $f$-conorm $\bot$, from a ring $\mathcal R$ to the $\sigma$-ring $\Sigma$ generated by $\mathcal R$ and to the class $\mathcal R_\sigma$ of sets which are limits of increasing sequences from $\mathcal R$.