  For a separable rearrangement invariant space $X$ on $[0,1]$ of fundamental
type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is
finitely represented in $X$ in such a way that the unit basis vectors of
$\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and
equimeasurable functions. This can be treated as a follow up of a paper by the
first-named author related to separable rearrangement invariant spaces on
$(0,\infty)$.
