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Abstract

In this thesis we are concerned with obtaining an integral representation of a class of nonlinear additive and biadditive functionals on function spaces of measurable functions and on L[superscript] p-spaces, p > 0 . The associated measure space is essentially atom-free finite and o-finite. Also we are concerned to the extend the presence of atoms in a measure space complicates the representation theory for functionals of the type under consideration here. A class of nonlinear transformations on L[superscript] p-spaces, 1 ≤ p ≤ ∞, called Urysohn operators. [11] taking measurable functions to measurable functions is studied and we describe an integral representation for this class when the associated measure space is an arbitrary 0-finite measure space and this characterization extends our previous results where the measure space considered was atom-free.