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Abstract. Let (X,%, ß) be a u-finite measure space and 3 £ be a linear subspace of S?x{ft) with supp ^ = X. The following inverse problem is treated: Which sets A £ 21 are "^-determined " within the class of all functions g e S?<x{ff) satisfying 0 < g < 1, i.e. when is g = \A the unique solution of f fgdß = J flAdfi, f £ J? ? Recent results of Fishburn et al. and Kemperman show that the condition A = { /> 0} for some f £ 5? is sufficient but not necessary for uniqueness. To obtain a complete characterization of all ^"-determined sets, 3 £ has to be enlarged to some hull X* by extending the usual weak convergence to limits not in.25 (ß). Then one of the main results states that A is ^-determined if and only if there is a representation A = {/ *> 0} and X\A = {/ * < 0} for some / * £ 3£ *

Year: 2013

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