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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Uniqueness in bounded moment problems


Author: Hans G. Kellerer
Journal: Trans. Amer. Math. Soc. 336 (1993), 727-757
MSC: Primary 44A60; Secondary 44A12, 46G99
MathSciNet review: 1145730
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Abstract: Let $ (X,\mathfrak{A},\mu )$ be a $ \sigma $-finite measure space and $ \mathcal{K}$ be a linear subspace of $ {\mathcal{L}_1}(\mu )$ with $ \mathcal{K} = X$. The following inverse problem is treated: Which sets $ A \in \mathfrak{A}$ are " $ \mathcal{K}$-determined" within the class of all functions $ g \in {\mathcal{L}_\infty }(\mu )$ satisfying $ 0 \leq g \leq 1$ , i.e. when is $ g = {1_A}$ the unique solution of $ \smallint fg\;d\mu = \smallint f{1_A}\;d\mu $, $ f \in \mathcal{K}?$ Recent results of Fishburn et al. and Kemperman show that the condition $ A = \{ f \geq 0\} $ for some $ f \in \mathcal{K}$ is sufficient but not necessary for uniqueness. To obtain a complete characterization of all $ \mathcal{K}$-determined sets, $ \mathcal{K}$ has to be enlarged to some hull $ {\mathcal{K}^{\ast} }$ by extending the usual weak convergence to limits not in $ {\mathcal{L}_1}(\mu )$. Then one of the main results states that $ A$ is $ \mathcal{K}$-determined if and only if there is a representation $ A = \{ {f^{\ast} } > 0\} $ and $ X\backslash A = \{ {f^{\ast} } < 0\} $ for some $ {f^{\ast}} \in {\mathcal{K}^{\ast} }$ .


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1145730-2
PII: S 0002-9947(1993)1145730-2
Article copyright: © Copyright 1993 American Mathematical Society