  We discuss the Mayer-Vietoris spectral sequence as an invariant in the
context of persistent homology. In particular, we introduce the notion of
$\varepsilon$-acyclic carriers and $\varepsilon$-acyclic equivalences between
filtered regular CW-complexes and study stability conditions for the associated
spectral sequences. We also look at the Mayer-Vietoris blowup complex and the
geometric realization, finding stability properties under compatible noise; as
a result we prove a version of an approximate nerve theorem. Adapting work by
Serre we find conditions under which $\varepsilon$-interleavings exist between
the spectral sequences associated to two different covers.
