  Given a monotone convex function on the space of essentially bounded random
variables with the Lebesgue property (order continuity), we consider its
extension preserving the Lebesgue property to as big solid vector space of
random variables as possible. We show that there exists a maximum such
extension, with explicit construction, where the maximum domain of extension is
obtained as a (possibly proper) subspace of a natural Orlicz-type space,
characterized by a certain uniform integrability property. As an application,
we provide a characterization of the Lebesgue property of monotone convex
function on arbitrary solid spaces of random variables in terms of uniform
integrability and a "nice" dual representation of the function.
