On sets that are uniquely determined by a restricted set of integrals

Abstract:

In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset $S$ of a measure space $(X,\lambda )$ such as ${{\mathbf {R}}^n}$, or an unknown function $0 \leqslant \phi \leqslant 1$ on $X$, having known moments (integrals) relative to a specified class $F$ of functions $f:X \to {\mathbf {R}}$. Usually, these $F$-moments do not fully determine the object $S$ or function $\phi$. We will say that $S$ is a set of uniqueness if no other function $0 \leqslant \psi \leqslant 1$ has the same $F$-moments as $S$ 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 $\phi$, which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same $F$-moments as $\phi$, provided $(X,\lambda ,F)$ 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 ${\pi _j}:X \to {Y_j}(j \in J)$. Here, one likes to know, for instance, what subsets $S$ of $X$ are uniquely determined by the corresponding set of projections (X-ray pictures). It is allowed that $\lambda (S) = \infty$. Our results are also relevant to a wide class of optimization problems.