  Let $\cal M$ be a semi-finite von Neumann algebra equipped with a
distinguished faithful, normal, semi-finite trace $\tau$. We introduce the
notion of equi-integrability in non-commutative spaces and show that if a
rearrangement invariant quasi-Banach function space $E$ on the positive
semi-axis is $\alpha$-convex with constant 1 and satisfies a non-trivial lower
$q$-estimate with constant 1, then the corresponding non-commutative space of
measurable operators $E({\cal M}, \tau)$ has the following property: every
bounded sequence in $E({\cal M}, \tau)$ has a subsequence that splits into a
$E$-equi-integrable sequence and a sequence with pairwise disjoint projection
supports. This result extends the well known Kadec-Pe\l czy\'nski subsequence
decomposition for Banach lattices to non-commutative spaces. As applications,
we prove that for $1\leq p <\infty$, every subspace of $L^p(\cal M, \tau)$
either contains almost isometric copies of $\ell^p$ or is strongly embedded in
$L^p(\cal M, \tau)$.
