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$ L\sb{p}(\mu ,X)$ $ (1<p<\infty )$ has the Radon-Nikodým property if $ X$ does by martingales


Authors: Barry Turett and J. J. Uhl
Journal: Proc. Amer. Math. Soc. 61 (1976), 347-350
MSC: Primary 46E40; Secondary 28A45
DOI: https://doi.org/10.1090/S0002-9939-1976-0423069-3
MathSciNet review: 0423069
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Abstract: Using the fact that $ {L_p}[0,\;1]\;(1 < p < \infty )$ has an unconditional basis, Sundaresan has shown that $ {L_p}(\mu ,\;X)$ has the Radon-Nikodým property if $ 1 < p < \infty $ and $ X$ has the Radon-Nikodým property. In this note, Sundaresan's theorem is proved by direct martingale methods. Then it is shown how to adapt this argument to the context of Orlicz spaces in which Sundaresan's argument is not applicable.


References [Enhancements On Off] (What's this?)

  • [1] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504. MR 34 #8456. MR 0208647 (34:8456)
  • [2] S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces, Math. Scand. 22 (1968), 21-41. MR 39 #7645. MR 0246341 (39:7645)
  • [3] L. Dor and E. Odell, Monotone bases in $ {L_p}$, Pacific J. Math (to appear). MR 0399832 (53:3674)
  • [4] J. Diestel and J. J. Uhl, Jr., The theory of vector measures, Math. Surveys, Amer. Math. Soc. Providence, R.I. (to appear). MR 0453964 (56:12216)
  • [5] N. Dunford and J. T. Schwartz, Linear operators. Part I, Interscience, New York, 1958. MR 22 #8302.
  • [6] V. F. Gapoškin, Existence of absolute bases in Orlicz spaces, Funkctional.Anal. i Priložen. 1 (1967), 26-32 = Functional Anal. Appl. 1 (1967), 278-284. MR 36 #5678. MR 0222628 (36:5678)
  • [7] K. Sundaresan, The Radon-Nikodym theorem for Lebesgue-Bochner function spaces (preprint). MR 0450956 (56:9246)
  • [8] Barry Turett, Fenchel-Orlicz spaces, Thesis, Univ. of Illinois at Urbana-Champaign, 1976.
  • [9] J. J. Uhl, Jr., Applications of Radon-Nikodým theorems to martingale convergence, Trans. Amer. Math. Soc. 145 (1969), 271-285. MR 40 #4983. MR 0251756 (40:4983)

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DOI: https://doi.org/10.1090/S0002-9939-1976-0423069-3
Article copyright: © Copyright 1976 American Mathematical Society

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