  In this paper we study the liftability property for piecewise continuous maps
of compact metric spaces, which admit inducing schemes in the sense of Pesin
and Senti [PS05, PS06]. We show that under some natural assumptions on the
inducing schemes - which hold for many known examples - any invariant ergodic
Borel probability measure of sufficiently large entropy can be lifted to the
tower associated with the inducing scheme. The argument uses the construction
of connected Markov extensions due to Buzzi [Buz99], his results on the
liftability of measures of large entropy, and a generalization of some results
by Bruin [Bru95] on relations between inducing schemes and Markov extensions.
We apply our results to study the liftability problem for one-dimensional cusp
maps (in particular, unimodal and multimodal maps) and for some
multidimensional maps.
