Boundedness of vector measures with values in the spaces $L_{0}$ of Bochner measurable functions
HTML articles powered by AMS MathViewer
- by Lech Drewnowski PDF
- Proc. Amer. Math. Soc. 91 (1984), 581-588 Request permission
Abstract:
Let ${L_0}(Z)$ be the $F$-space of all Bochner measurable functions from a probability space to a Banach space $Z$. We prove that every countably additive vector measure taking values in ${L_0}(Z)$ has bounded range. This generalizes a recent result due to M. Talagrand and, independently, N. J. Kalton, N. T. Peck and J. W. Roberts, asserting the same for the case when $Z$ is the space of scalars.References
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
- N. J. Kalton, N. T. Peck, and James W. Roberts, $L_{0}$-valued vector measures are bounded, Proc. Amer. Math. Soc. 85 (1982), no. 4, 575–582. MR 660628, DOI 10.1090/S0002-9939-1982-0660628-X
- Stefan Rolewicz, Metric linear spaces, Monografie Matematyczne, Tom 56. [Mathematical Monographs, Vol. 56], PWN—Polish Scientific Publishers, Warsaw, 1972. MR 0438074
- Michel Talagrand, Les mesures vectorielles à valeurs dans $L^{0}$ sont bornées, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 4, 445–452 (1982) (French). MR 654206, DOI 10.24033/asens.1414
- Philippe Turpin, Une mesure vectorielle non bornée, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), A509–A511 (French). MR 385556
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 581-588
- MSC: Primary 46G10; Secondary 28B05, 46E40
- DOI: https://doi.org/10.1090/S0002-9939-1984-0746094-6
- MathSciNet review: 746094