Abstract

Let μ\bm{\mu} be a non-zero, non-atomic vector measure on the measurable space (X,Ω)(\bm{X}, \bm{\Omega}) taking values in Rn\bm{R}^n. Liapounov's convexity theorem gives that the range R(μ)R(\mu) is convex, an immediate consequence of this is that there exist uncountably many smaller collections S⊂ΩS\subset \Omega with preservation of the range, that is R(μ)=R(μ/S)\bm{R}(\bm{\mu}) = \bm{R}(\bm{\mu} / S). In case X\bm{X} is a topological space and Ω\bm{\Omega} the class of Borel sets, such reductions consisting of open sets or other sets related to continuous functions on X\bm{X} have been obtained by H. Render and H. Stroetmann [Arch. Math. 67 (1996) 331–336], D. Wulbert [Proc. Amer. Math. Soc. 108 (1990) 955–960; Proc. Amer. Math. Soc. 128 (2000) 2431–2438; Israel J. Math. 126 (2001) 363-380; J. Functional Analysis 182 (2001) 1-15] and H. G. Kellerer [Arch. Math. 72 (1999) 206–213; Proc. Amer. Math. Soc. 130 (2002) 2305–2309]. J. M. Gouweleeuw [Indag. Math. N.S. 4 (1993) 141–161; Proc. London Math. Soc., III. Ser. 70 (1995) 336–362] gave a decomposition of an Rn\bm{R}^n-valued vector measure μ=(μ1,μ2,…,μn)\bm{\mu} = (\mu_1, \mu_2, \ldots, \mu_n) on the measurable space (X,Ω)(\bm{X}, \bm{\Omega}), where each μi\mu_i is a non-negative real measure on (X,Ω)(\bm{X}, \bm{\Omega}), based on the atoms of the measure μ\bm{\mu}. She also characterized those μ\bm{\mu} which have a convex range. J. van Mill and A. Ran [Indag. Mathem., N.S. 7 (1996) 227–242] gave various interesting variants and generalizations of the Gouweleeuw decomposition and convexity results. It is the purpose of this paper to apply these decompositions and make attempts to shorten the domain Ω\bm{\Omega} to various minimal subsets of Ω\bm{\Omega} called shortenings, which preserve the range of μ\bm{\mu}. We also make use of the Rényi criteria and the work on interval filling sequences of Z. Daróczy, A. Járai, I. Katái and T. Szabó.

Author information

Contact details are reproduced from the original publication and may be historical.

Laltanpuia

Dept. of Mathematics, University of Delhi, Delhi 110007, India
and: Dept. of Mathematics, Mizoram University, Aizawl 796001, Mizoram, India

puia1000@yahoo.com

Ajit Iqbal Singh

Dept. of Mathematics, University of Delhi, South Campus, Benito Juarez Road, New Delhi 110021, India

aisingh@sify.com

Keywords

  • Vector measure
  • preservation of the range
  • shortening of the domain
  • Gouweleeuw decomposition
  • van Mill and Ran decomposition
  • interval filling sequence

Mathematics Subject Classification

28B05; 26A51, 46G10, 52A05

References

  1. Z. Daróczy, A. Járai, I. Katái: Intervallfüllende Folgen und volladditive Funktionen, Acta Sci. Math. 50 (1986) 337–350.Google Scholar
  2. Z. Daróczy, I. Katái: Interval filling and additive functions, Acta Sci. Math. 52 (1988) 337–347.https://doi.org/10.1007/BF01951575Google Scholar
  3. Z. Daróczy, I. Katái, T. Szabó: On completely additive functions related to interval-filling sequences, Arch. Math. 54 (1990) 173–179.https://doi.org/10.1007/BF01198114Google Scholar
  4. J. Diestel: Sequences and Series in Banach Spaces, Springer, New York (1984).https://doi.org/10.1007/978-1-4612-5200-9Google Scholar
  5. J. Diestel, J. J. Uhl, Jr.: Vector Measures, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  6. J. M. Gouweleeuw: A generalization of Lyapounov’s theorem, Indag. Math., New Ser. 4(2) (1993) 141–161.https://doi.org/10.1016/0019-3577(93)90036-XGoogle Scholar
  7. J. M. Gouweleeuw: A characterization of vector measures with convex range, Proc. Lond. Math. Soc., III. Ser. 70(2) (1995) 336–362.https://doi.org/10.1112/plms/s3-70.2.336Google Scholar
  8. Hans G. Kellerer: A topological version of Liapounov’s theorem, Arch. Math. 72 (1999) 206–213.https://doi.org/10.1007/s000130050323Google Scholar
  9. Hans G. Kellerer: Continuous transformation of Baire measure into Lebesgue measure, Proc. Amer. Math. Soc. 130 (8) (2002) 2305–2309.https://doi.org/10.1090/S0002-9939-02-06505-XGoogle Scholar
  10. Laltanpuia: M. Phil. dissertation, Mathematics Department, University of Delhi (2003).Google Scholar
  11. Laltanpuia, Ajit Iqbal Singh: Geometry of the range of a vector measure, East-West J. Math. 7(2) (2005) 117–140.Google Scholar
  12. A. Liapounov: On completely additive vector functions, Izv. Akad. Nauk. SSSR 4 (1940) 465–478.Google Scholar
  13. A. Liapounov: On completely additive vector functions, Izv. Akad. Nauk. SSSR 10 (1940) 277–279.Google Scholar
  14. J. van Mill, A. Ran: On a generalization of Lyapounov’s theorem, Indag. Math., New Ser. 7(2) (1996) 227–242.https://doi.org/10.1016/0019-3577(96)85092-0Google Scholar
  15. H. Render, H. Stroetmann: Lyapounov’s convexity theorem for topological measures, Arch. Math. 67 (1996) 331–336.https://doi.org/10.1007/BF01197598Google Scholar
  16. A. Rényi: Probability Theory, North-Holland, Amsterdam (1970).Google Scholar
  17. P. Ribenboim: Representation of real numbers by means of Fibonacci numbers, Enseign. Math., II. Sér. 31 (1985) 249–255.Google Scholar
  18. I. Singer: Bases in Banach Spaces, Springer, Berlin (1970).https://doi.org/10.1007/978-3-642-51633-7Google Scholar
  19. J. J. Uhl, Jr.: The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969) 158–163.https://doi.org/10.1090/S0002-9939-1969-0264029-1Google Scholar
  20. D. Wulbert: Liapunov’s and related theorems, Proc. Amer. Math. Soc. 108(8) (1990) 955–960.https://doi.org/10.1090/S0002-9939-1990-1005716-8Google Scholar
  21. D. Wulbert: Annihilating a subspace of Ł1 with signs of Continuous Functions, Proc. Amer. Math. Soc. 128(8) (2000) 2431–2438.https://doi.org/10.1090/S0002-9939-99-05317-4Google Scholar
  22. D. Wulbert: A general Hobby-Rice theorem and cake cutting, Israel J. Math. 126 (2001) 363–380.https://doi.org/10.1007/BF02784162Google Scholar
  23. D. Wulbert: The range of vector measures over topological sets and a unified Liapunov theorem, J. Funct. Anal. 182 (2001) 1–15.https://doi.org/10.1006/jfan.2000.3717Google Scholar

Suggested citation

Laltanpuia, A. I. Singh. “Preservation of the Range of a Vector Measure under Shortenings of the Domain.” Journal of Convex Analysis 15 (2008), No. 2, 313–324. https://doi.org/10.68381/jca15022

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.