  A blender for a surface endomorphism is a hyperbolic basic set for which the
union of the local unstable manifolds contains robustly an open set. Introduced
by Bonatti and D{\'i}az in the 90s, blenders turned out to have many powerful
applications to differentiable dynamics. In particular, a generalization in
terms of jets, called parablenders, allowed Berger to prove the existence of
generic families displaying robustly infinitely many sinks. In this paper, we
introduce analogous notions in a measurable point of view. We define an almost
blender as a hyperbolic basic set for which a prevalent perturbation has a
local unstable set having positive Lebesgue measure. Almost parablenders are
defined similarly in terms of jets. We study families of endomorphisms of R2
leaving invariant the continuation of a hyperbolic basic set. When some
inequality involving the entropy and the maximal contraction along stable
manifolds is satisfied, we obtain an almost blender or parablender. This
answers partially a conjecture of Berger. The proof is based on thermodynamic
formalism: following works of Mihailescu, Simon, Solomyak and Urba{\'n}ski, we
study families of fiberwise unipotent skew-products and we give conditions
under which these maps have limit sets of positive measure inside their fibers.
