Uniqueness in bounded moment problems
HTML articles powered by AMS MathViewer
- 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
- R. Anantharaman and K. M. Garg, Some topological properties of vector measures and their integral maps, J. Austral. Math. Soc. Ser. A 23 (1977), no. 4, 453–466. MR 486398, DOI 10.1017/s1446788700019601
- Robert Chen and Larry A. Shepp, On the sum of symmetric random variables, Amer. Statist. 37 (1983), no. 3, 237. MR 713833, DOI 10.2307/2683381
- P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, Sets uniquely determined by projections on axes. I. Continuous case, SIAM J. Appl. Math. 50 (1990), no. 1, 288–306. MR 1036243, DOI 10.1137/0150017
- P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, Sets uniquely determined by projections on axes. II. Discrete case, Discrete Math. 91 (1991), no. 2, 149–159. MR 1124762, DOI 10.1016/0012-365X(91)90106-C
- R. J. Gardner, Measure theory and some problems in geometry, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 1, 51–72. MR 1111758
- Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR 0410335
- Hans G. Kellerer, Schnittmass-Funktionen in mehrfachen Produkträumen, Math. Ann. 155 (1964), 369–391 (German). MR 167587, DOI 10.1007/BF01350747
- J. H. B. Kemperman, On sets that are uniquely determined by a restricted set of integrals, Trans. Amer. Math. Soc. 322 (1990), no. 2, 417–458. MR 1076178, DOI 10.1090/S0002-9947-1990-1076178-4
- J. H. B. Kemperman, Sets of uniqueness and systems of inequalities having a unique solution, Pacific J. Math. 148 (1991), no. 2, 275–301. MR 1094491
- J. F. C. Kingman and A. P. Robertson, On a theorem of Lyapunov, J. London Math. Soc. 43 (1968), 347–351. MR 224768, DOI 10.1112/jlms/s1-43.1.347 I. Kluvánek and G. Knowles, Vector measures and control systems, North-Holland, Amsterdam and Oxford, 1975.
- A. Kuba and A. Volčič, Characterisation of measurable plane sets which are reconstructable from their two projections, Inverse Problems 4 (1988), no. 2, 513–527. MR 954907
- A. Liapounoff, Sur les fonctions-vecteurs complètement additives, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 465–478 (Russian, with French summary). MR 0004080
- Joram Lindenstrauss, A remark on extreme doubly stochastic measures, Amer. Math. Monthly 72 (1965), 379–382. MR 181728, DOI 10.2307/2313497
- G. G. Lorentz, A problem of plane measure, Amer. J. Math. 71 (1949), 417–426. MR 28925, DOI 10.2307/2372255 S. Marx, Schnittmaßprobleme im ${R^d}$ bezüglich endlich vieler Teilräume, Diploma thesis, Universität Erlangen, 1988.
- I. V. Romanovskiĭ and V. N. Sudakov, On the existence of independent partitions, Trudy Mat. Inst. Steklov 79 (1965), 5–10 (Russian). MR 0220321
- Gordon Simons, An unexpected expectation, Ann. Probability 5 (1977), no. 1, 157–158. MR 436268, DOI 10.1214/aop/1176995902
- Andrzej Spakowski, Openness of vector measures and their integral maps, J. Austral. Math. Soc. Ser. A 45 (1988), no. 3, 351–359. MR 957200
- V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math. 2 (1979), i–v, 1–178. Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976). MR 530375
- Rudolf Wegmann, Der Wertebereich von Vektorintegralen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 14 (1969/70), 203–238 (German, with English summary). MR 262457, DOI 10.1007/BF01111418
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