Representation of bounded c-quasilinear functional defined on the set of H-Continuous interval valued functions

Abstract

Every bounded linear functional from the set C[0,1] of continuous functions on [0,1], endowed with the supremum norm, can be written as a Riemann-Stieltjes integral on [0,1], according to a valuable statement proposed and proven by F. Riesz in 1909. This result is now known as Riesz’s representation theorem. The above result has been generalized from various class C(X) of continuous functions on various topological spaces X. Moreover,ithasbeengeneralizedinthesetofp-integrablerealvaluedfunctionsdefinedon X. ThegoalofthisworkistoextendtheRiesz’srepresentationtheoremtakingintoaccount that the set of continuous real valued functions C(X) is replaced by the set of interval valued functions and the notion of linear functional is replaced by the notion of quasilinear functional. It has been shown that the bounded convex quasilinear functional defined on thesetofHausdorffcontinuousfunctionson [0,1] canberepresentedasHenstock-Stieltjes integral of interval valued function on [0,1].