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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Uniqueness in bounded moment problems
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by Hans G. Kellerer PDF
Trans. Amer. Math. Soc. 336 (1993), 727-757 Request permission

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 ${f^{\ast }} \in {\mathcal {K}^{\ast } }$ .
References
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 336 (1993), 727-757
  • MSC: Primary 44A60; Secondary 44A12, 46G99
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1145730-2
  • MathSciNet review: 1145730