Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : A representation result is provided for concave Schur concave functions on L∞( ).
In particular, it is proven that any monotone concave Schur concaveweakly upper semicontinuous
function is the infinimum of a family of nonnegative affine combinations
of Choquet integrals with respect to a convex continuous distortion of the underlying
probability. The method of proof is based on the concave Fenchel transform and on
Hardy and Littlewoodâ€™s inequality. Under the assumption that the probability space is
nonatomic, concave, weakly upper semicontinuous, lawinvariant functions are shown
to coincide with weakly upper semicontinuous concave Schur concave functions. A
representation result is, thus, obtained for weakly upper semicontinuous concave lawinvariant
functions. 





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