Before revision:  We also apply a variant of the C-property to establish a w^*-representation theorem for proper convex increasing functionals on dual Banach lattices that satisfy a suitable version of Delbaen's Fatou property  ]  .
After revision:  We also apply a variant of the C-property to establish a w^*-representation theorem for proper convex increasing functionals on dual Banach lattices that satisfy a suitable version of Delbaen's Fatou property ] which is not L_1[0,1] .