Uniqueness in Bounded Moment Problems

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? 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£* . Introduction It is an immediate consequence of the Fourier inversion formula that a finite mass distribution p0 on R" is uniquely determined by all its one-dimensional projections, i.e. by the image measures tp{po) with respect to all linear maps cp : W —► R. In applications as in tomography, however, only a finite number of these projections can be observed. For simplicity let po be absolutely continuous with respect to Lebesgue measure; then it turns out that—except for the trivial case po = 0—a reconstruction is never possible (see §6). In most applications, however, some upper bound p for po is available. Thus, given projections <px, ... , cpk , the following "bounded moment problem" arises: Which measures po are uniquely determined by their images <p¡{po) under the side condition p0 < p ? Since the set of measures meeting these constraints is convex, it is no surprise that a necessary condition requires po to be a restriction of p to some subset A (see §6). Translated via the Radon-Nikodym theorem from measures to functions the problem takes the following form: The indicator functions of which sets A cl" are—up to null sets—uniquely determined, within the class of all functions g on R" satisfying 0 < g < 1, by the integrals fxfotpjlAdp, 1 < i < k , for all bounded functions / on R? Kuba and Volcic [12] and Fishburn et al. [3, 4] seem to be the first who studied this problem, specializing it to the classical "marginal" situation, i.e. the case where p is (a restriction of) Lebesgue measure and the canonical projections nx, ... , n„ play the role of the maps cpx, ... , <pk . In [12] the authors restrict the study to dimension 2 and, making essential use of a result due to Lorentz Received by the editors December 21, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 44A60; Secondary 28A35, 46G10, 52A07. ©1993 American Mathematical Society 0002-9947/93 $1.00+ $.25 per page