  We give two new versions of the LS category for the set-up of measurable
laminations defined by Berm\'udez. Both of these versions must be considered as
"tangential categories". The first one, simply called (LS) category, is the
direct analogue for measurable laminations of the tangential category of
(topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For
the measurable lamination that underlies any lamination, our measurable
tangential category is a lower bound of the tangential category. The second
version, called the measured category, depends on the choice of a transverse
invariant measure. We show that both of these "tangential categories" satisfy
appropriate versions of some well known properties of the classical category:
the homotopy invariance, a dimensional upper bound, a cohomological lower bound
(cup length), and an upper bound given by the critical points of a smooth
function.
