  The Lebesgue property (order-continuity) of a monotone convex function on a
solid vector space of measurable functions is characterized in terms of (1) the
weak inf-compactness of the conjugate function on the order-continuous dual
space, (2) the attainment of the supremum in the dual representation by
order-continuous linear functionals. This generalizes and unifies several
recent results obtained in the context of convex risk measures.
