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Proceedings of the American Mathematical Society
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On the Radon-Nikodým theorem and locally convex spaces with the Radon-Nikodým property


Author: G. Y. H. Chi
Journal: Proc. Amer. Math. Soc. 62 (1977), 245-253
MSC: Primary 28A45; Secondary 60G99, 46G10
MathSciNet review: 0435338
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Abstract: Let F be a quasi-complete locally convex space, $ (\Omega ,\Sigma ,\mu )$ a complete probability space, and $ {L^1}(\mu ;F)$ the space of all strongly integrable functions $ f:\Omega \to F$ with the Egoroff property. If F is a Banach space, then the Radon-Nikodým theorem was proved by Rieffel. This result extends to Fréchet spaces. If F is dual nuclear, then the Lebesgue-Nikodým theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodým theorem is not available in general. It is shown in this article that the Radon-Nikodým theorem for the strong integral can be established for quasi-complete locally convex spaces F having the following property:

(CM) For every bounded subset $ B \subset l_N^1\{ F\} $, the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset $ M \subset F$ such that $ \Sigma _{i = 1}^\infty {p_M}({x_i}) < 1,\forall ({x_i}) \in B$.

In fact, these spaces have the Radon-Nikodým property, and they include the Montel (DF)-spaces, the strong duals of metrizable Montel spaces, the strong duals of metrizable Schwartz spaces, and the precompact duals of separable metrizable spaces. When F is dual nuclear, the Radon-Nikodým theorem reduces to the Lebesgue-Nikodým theorem. An application to probability theory is considered.


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  • [1] K. Brauner, Duals of Fréchet spaces and a generalization of the Banach-Dieudonné theorem, Duke J. Math. 40 (1973), 845-855. MR 48 #9323. MR 0330988 (48:9323)
  • [2] G. Y. H. Chi, The Radon-Nikodým theorem for vector measures with values in the duals of some nuclear spaces, Vector and Operator Valued Measures and Applications (Proc. Sympos., Snowbird Resort, Alta, Utah, 1972), Academic Press, New York, 1973, pp. 85-95. MR 49 #5833. MR 0341083 (49:5833)
  • [3] -, The Radon-Nikodým theorem for Fréchet spaces, 1973 (preprint).
  • [4] -, A geometric characterization of Fréchet spaces with the Radon-Nikodým property, Proc. Amer. Math. Soc. 48 (1975), 371-380. MR 0357730 (50:10198)
  • [5] S. D. Chatterji, Sur l'intégrabilite de Pettis 1973 (preprint).
  • [6] N. Dinculeanu, Vector measures, Internat. Ser. of Monographs in Pure and Appl. Math., vol. 95, Pergamon Press, New York; VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 34 #6011b. MR 0206190 (34:6011b)
  • [7] J. Horvath, Topological vector spaces and distributions. Vol. 1, Addison-Wesley, Reading, Mass., 1966. MR 34 #4863. MR 0205028 (34:4863)
  • [8] R. E. Huff, Dentability and the Radon-Nikodým property, Duke J. Math. 41 (1974), 111-114. MR 49 #5783. MR 0341033 (49:5783)
  • [9] G. Köthe, Topologische linear Räume. I, Die Grundlehren der math. Wissenchaften, Band 107, Springer-Verlag, Berlin; English transl., Die Grundlehren der math. Wissenschaften, Band 159, Springer-Verlag, New York, 1969. MR 24 #A411; 40 #1750.
  • [10] J. Kupka, Radon-Nikodým theorems for vector valued measures, Trans. Amer. Math. Soc. 169 (1972), 197-217. MR 47 #433. MR 0311871 (47:433)
  • [11] D. G. Larmen and C. A. Rogers, The normability of metrizable sets, Bull. London Math. Soc. 5 (1973), 39-48. MR 0320681 (47:9217)
  • [12] D. R. Lewis, On the Radon-Nikodým theorem, 1971 (preprint).
  • [13] H. B. Maynard, A geometric characterization of Banach spaces having the Radon-Nikodým property, Trans. Amer. Math. Soc. 185 (1972), 493-500. MR 0385521 (52:6382)
  • [14] B. MacGibbon, A criterion for the metrizability of a compact convex set in terms of the set of extreme points, J. Functional Analysis 11 (1972), 385-392. MR 49 #7723. MR 0342979 (49:7723)
  • [15] M. Métvier, Martingales à valeurs vectorielles. Applications à la dérivation des mesures vectorielles, Ann. Inst. Fourier (Grenoble) 17 (1967), fasc. 2, 175-208 (1968). MR 40 #926. MR 0247663 (40:926)
  • [16] A. Pietsch, Nuclear locally convex spaces, Springer-Verlag, New York, 1972. MR 0350360 (50:2853)
  • [17] D. Randtke, Characterization of precompact maps, Schwartz spaces and nuclear spaces, Trans. Amer. Math. Soc. 165 (1972), 87-101. MR 46 #4139. MR 0305009 (46:4139)
  • [18] M. A. Rieffel, The Radon-Nikodým theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466-487. MR 36 #5297. MR 0222245 (36:5297)
  • [19] -, Dentable subsets of Banach spaces, with applications to a Radon-Nikodým theorem, Functional Analysis (Proc. Conf., Irvine, Calif., 1966), Thompson Book, Washington, D. C.; Academic Press, London, 1967, pp. 71-77. MR 36 #5668. MR 0222618 (36:5668)
  • [20] H. H. Schaeffer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [21] M. Sion, Theory of semi-group valued measures, Lecture Notes in Math., vol. 355, Springer-Verlag, New York, 1973. MR 0450503 (56:8797)
  • [22] C. Swartz, Vector measures and nuclear spaces, Rev. Roumaine Math. Pures Appl. 18 (1973), 1261-1268. MR 49 #7772. MR 0343028 (49:7772)
  • [23] T. Terzioğelu, On Schwartz spaces, Math. Ann. 182 (1969), 236-242. MR 40 #683. MR 0247417 (40:683)
  • [24] G. E. F. Thomas, The Lebesgue-Nikodým theorem for vector valued Radon measures, Mem. Amer. Math. Soc. No. 139 (1974).
  • [25] I. Tweddle, Vector valued measures, Proc. London Math. Soc. (3) 20 (1970), 469-489. MR 41 #3707. MR 0259065 (41:3707)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0435338-2
PII: S 0002-9939(1977)0435338-2
Keywords: Radon-Nikodým theorem, vector measures, nuclear spaces, Montel (DF)-spaces, Schwartz spaces, metrizable spaces
Article copyright: © Copyright 1977 American Mathematical Society