Ls category, foliated spaces and transverse invariant measure

The LS category is a homotopy invariant given by the minimum of open subsets, contractible within a topological space, needed to cover it. It was introduced by Lusternik and Schnirelmann in 1934 in the setting of variational calculus. Many variants of this notion has been given. In particular, E. Macías and H. Colman introduced a tangential version for foliations, where they used leafwise contractions to transversals. In this work, we introduce and study several new versions of the tangential LS category. The first two of them, called measurable category and measurable _-category, are defined for measurable laminations, which are laminations where the ambient topology is removed and only the leaf topology and ambient measurable structure remain; thus we use tangential deformations of open sets to transversals that are leafwise continuous and measurable on the ambient space.