### On Sets that are Uniquely Determined by a Restricted Set of Integrals

J. H. B. Kemperman
1990 Transactions of the American Mathematical Society
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
more » ... 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. . Primary 44A60, 68P20, 28A35, 49D35, 49A55, 49B36. Key words and phrases. Sets with given moments, generalized additive sets, sets of uniqueness, reconstructing a set from its projections, strongly rich systems, optimal bang-bang control, tomography. 7 7 J J J J J J J