  We introduce a natural stratification of the space of projective classes of
measured laminations on a complete hyperbolic surface of finite area. We prove
a rigidity result, namely, the group of self-homeomorphisms of the space of
projective measured laminations that preserve such a stratification is in
general identified with the extended mapping class group of the corresponding
surface. We use this approach to fill a gap in the proof of the rigidity of the
action of the extended mapping class group on the unmeasured laminations space.
