Such a construction can be made rigorous by considering restrictions of partitions to the first $n$ integers and by a consistency argument. Then $\Pi ^{0}$ has the law of the fragmentation with parameters $(0,0,\nu )$.

Out of this “homogeneous” fragmentation, we construct the $(\alpha ,0,\nu
) $-fragmentation by introducing a time-change. Call $\lambda _{i}(t)$ the asymptotic frequency of the block of $\Pi ^{0}(t)$ that contains $i$, and write $$T_{i}(t)=\inf \left\{ u\geq 0:\int_{0}^{u}\lambda _{i}(r)^{-\alpha }\mathrm{d} r>t\right\} .  \label{1}$$ Last, for every $t\geq 0$ we let $\Pi (t)$ be the random partition such that $i,j$ are in the same block of $\Pi (t)$ if and only if they are in the same block of $\Pi ^{0}(T_{i}(t))$, or equivalently of $\Pi ^{0}(T_{j}(t))$. Then $(\Pi (t),t\geq 0)$ is the wanted version. Let $\left( \mathcal{G}(t),t\geq
0\right) $ be the natural filtration generated by $\Pi $ completed up to $P$-null sets. According to [@bertsfrag02], the fragmentation property holds actually for $\mathcal{G}$-stopping times and we shall refer to it as the *strong fragmentation property.* In the homogeneous case, we will rather call $\mathcal{G}^{0}$ the natural filtration.

When $\alpha<0$, the loss of mass in the ranked fragmentations shows up at the level of partitions by the fact that 