On Sets That Are Uniquely Determined by a Restricted Set of Integrals

In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset 5 of a measure space (X, X) such as R" , or an unknown function 0 < (j> < 1 on X , having known moments (integrals) relative to a specified class F of functions f:X—*R. Usually, these F-moments do not fully determine the object S or function . We will say that 5 is a set of uniqueness if no other function 0 < y < 1 has the same F -moments as 5 in so far as the latter moments exist. Here, S is identified with its indicator function. Within this general setting and with no further assumptions, we develop a powerful sufficient condition for uniqueness, called generalized additivity. When F is a finite class, this condition of generalized additivity is shown to be also necessary for uniqueness. For each 0 , which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same Fmoments as 4>, provided (X, X, F) is nonatomic or regular and, moreover, 'strongly rich', a condition which is satisfied in many applications. Using such general results, we also study the uniqueness problem for measures with given marginals relative to a given system of projections n : X —► Y■ (j € J). Here, one likes to know, for instance, what subsets S of X are uniquely determined by the corresponding set of projections (J-ray pictures). It is allowed that X(S) = oo . Our results are also relevant to a wide class of optimization problems.