Schur convex functionals: Fatou property and representation

The Fatou property for every Schur convex lower semicontinuous (l.s.c.) functional on a general probability space is established. As a result, the existing quantile representations for Schur convex l.s.c. positively homogeneous convex functionals, established onfor eitherp= 1orp=∞and with the requirement of the Fatou property, are generalized for, with no requirement of the Fatou property. In particular, the existing quantile representations for law invariant coherent risk measures and law invariant deviation measures on an atomless probability space are extended for a general probability space.