  In this paper, we study the existence of infinite dimensional closed linear
subspaces of a rearrangement invariant space on [0,1] every nonzero element of
which does not belong to any included rearrangement invariant space of the same
class such that the inclusion operator is disjointly strictly singular. We
consider Lorentz, Marcinkiewicz and Orlicz spaces. The answer is affirmative
for Marcinkiewicz spaces and negative for Lorentz and Orlicz spaces. Also, the
same problem is studied for Nakano spaces assuming different hypothesis.
