n}\prod_{i\geq 1}\mathbf{1}_{\left\{ m\in \pi
_{k(i,j)}^{(i)}\right\} }\text{ a.s}.$$ On the other hand, the random variables $\prod_{i\geq 1}\mathbf{1}_{\{ m\in
\pi _{k(i,j)}^{(i)}\} }=\mathbf{1}_{\left\{ m\in \pi _{j}\right\} }, $ $m\geq j+1,$ are i.i.d. conditionally on $\left| \pi _{j}\right| $ with mean $\left| \pi _{j}\right| $ and then the limit above converges a.s. to $\left|
\pi _{j}\right| ,$ again by the law of large numbers. $\hfill \square$

We now turn our attention to partitions-valued fragmentations.

Let $(\Pi (t),t\geq 0)$ be a Markovian $\mathcal{P}_{\infty }$-valued process with $\Pi (0)=\{\mathbb{N},\varnothing ,\varnothing ,\ldots \}$ that is continuous in probability and exchangeable as a process (meaning that the law of $\Pi $ is invariant by the action of permutations). Call it a partition-valued self-similar fragmentation with index $\alpha \in \mathbb{R}
$ if moreover $\Pi (t)$ admits asymptotic frequencies for all $t$, a.s., if the process $(|\Pi (t)|,t\geq 0)$ is continuous in probability, and if the following fragmentation property is satisfied. For $t,t^{\prime }\geq 0$, given $\Pi (t)=(\pi _{1},\pi _{2},\ldots )$, the sequence $\Pi (t+t^{\prime
})$ has the same law as the partition with blocks $\pi _{1}\cap \Pi
^{(1)}(|\pi _{1}|^{\alpha }t^{\prime }),\pi _{2}\cap \Pi ^{(2)}(|\pi
_{2}|^{\alpha }t^{\prime }),\ldots $, where $(\Pi ^{(i)},i\geq 1)$ are independent copies of $\Pi $.

Bertoin 