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A Radon-Nikodým theorem and \(L_ p\) completeness for finitely additive vector measures. (English) Zbl 0601.28005

Let \(\mu\) be a bounded finitely additive real-valued measure and \(\nu\) be a finitely additive measure with values in a Banach space, defined on a common field of sets. Necessary and sufficient conditions are obtained for the existence of a \(\mu\)-integrable function f such that \(\nu (E)=\int_{E}fd\mu\) for all sets E in the field. The conditions involve absolute continuity, properties of the range of \(\nu\) /\(\mu\), and certain exhaustion and boundedness properties investigated by the author. Using the main results, it is also shown that those bounded finitely additive measures for which the corresponding \(L_ p\) spaces of vector-valued functions are complete are precisely the ones for which each pair of disjoint components is separated in the sense of concentration on ideals.

MSC:

28A15 Abstract differentiation theory, differentiation of set functions
28B05 Vector-valued set functions, measures and integrals
28A25 Integration with respect to measures and other set functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

[1] W. C. Bell and J. W. Hagood; W. C. Bell and J. W. Hagood · Zbl 0592.28005
[2] Bochner, S., Additive set functions on groups, Ann. of Math., 40, 769-799 (1939) · JFM 65.1317.01
[3] Darst, R. B., A decomposition of finitely additive set functions, J. Reine Angew. Math., 210, 31-37 (1962) · Zbl 0105.26904
[4] Diestel, J.; Uhl, J. J., Vector Measures, (Math. Surveys No. 15 (1977), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0369.46039
[5] Dunford, N.; Schwartz, J. T., Linear Operators, Part I (1958), Interscience: Interscience New York
[6] Maynard, H. B., A geometrical characterization of Banach spaces with the Radon-Nikodym property, Trans. Amer. Math. Soc., 185, 493-500 (1973) · Zbl 0278.46040
[7] Maynard, H. B., A general Radon-Nikodym theorem, (“Vector and Operator Valued Measures and Applications” Proc. Sympos.. “Vector and Operator Valued Measures and Applications” Proc. Sympos., Snowbird Resort, Alta, Utah, 1972 (1973), Academic Press: Academic Press New York), 233-246 · Zbl 0301.28006
[8] Maynard, H. B., A Radon-Nikodym theorem for finitely additive bounded measures, Pacific J. Math., 83, 401-413 (1979) · Zbl 0453.28004
[9] Metivier, M., Martingales a valeurs vectorielles. Application a la derivation des measures vectorielles, Ann. Inst. Fourier (Grenoble), 17, 175-208 (1967) · Zbl 0162.48801
[10] Phillips, R. S., On weakly compact subsets of a Banach space, Amer. J. Math., 65, 108-136 (1943) · Zbl 0063.06212
[11] Rieffel, M. A., The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc., 131, 466-487 (1968) · Zbl 0169.46803
[12] Rieffel, M. A., Dentable subsets of Banach spaces with applications to a Radon-Nikodym theorem, (Functional Analysis Proc. Conf.. Functional Analysis Proc. Conf., Irvine, Calif., 1966 (1967), Thompson: Thompson Washington, D.C), 71-77 · Zbl 0213.13703
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