A note on the range of vector measures
-
Niccolò Urbinati
and
Hans Weber
Abstract
We give another proof for Kluvanek and Knowles’ characterization of Liapounoff measures [KLUVANEK, I.—KNOWLES, G.: Vector Measures and Control Systems. North-Holland Mathematics Studies 20, Amsterdam, 1976] and of the fact that the range of an exhaustive measure with values in a complete locally convex space is relatively weakly compact.
References
[1] Diestel, J.—Uhl, J. J.: Vector Measures, Amer. Math. Soc., Rode Island, Providence, 1977.10.1090/surv/015Search in Google Scholar
[2] Fremlin, D.: Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.10.1017/CBO9780511897207Search in Google Scholar
[3] Jarchow, H.: Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.10.1007/978-3-322-90559-8Search in Google Scholar
[4] Kluvanek, I.—Knowles, G.: Vector Measures and Control Systems. North-Holland Mathematics Studies 20, Amsterdam, 1976.Search in Google Scholar
[5] Lindenstrauss, J.: A short proof of Liapounoff’s convexity theorem, J. Math. Mech. 15 (1966), 971–972.10.1512/iumj.1966.15.15064Search in Google Scholar
[6] Sikorski, R.: Boolean Algebras, Springer-Verlag, Heidelberg, 1960.10.1007/978-3-662-01507-0Search in Google Scholar
[7] Tweddle, I.: Weak compactness in locally convex spaces, Glasg. Math. J. 9 (1968), 123–127.10.1017/S0017089500000409Search in Google Scholar
[8] Uhl Jr., J. J.: The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969), 158–163.10.1090/S0002-9939-1969-0264029-1Search in Google Scholar
[9] Weber, H.: FN-topologies and group-valued measures. In: Handbook of Measure Theory, vol. I (E. Pap ed.), North-Holland, Amsterdam, 2002, pp. 703–743.10.1016/B978-044450263-6/50017-8Search in Google Scholar
[10] Wnuk, W.: The converse of Lyapunov convexity theorem, Comment. Math. 21 (1979), 389–390.Search in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
Abstract
We give another proof for Kluvanek and Knowles’ characterization of Liapounoff measures [KLUVANEK, I.—KNOWLES, G.: Vector Measures and Control Systems. North-Holland Mathematics Studies 20, Amsterdam, 1976] and of the fact that the range of an exhaustive measure with values in a complete locally convex space is relatively weakly compact.
References
[1] Diestel, J.—Uhl, J. J.: Vector Measures, Amer. Math. Soc., Rode Island, Providence, 1977.10.1090/surv/015Search in Google Scholar
[2] Fremlin, D.: Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.10.1017/CBO9780511897207Search in Google Scholar
[3] Jarchow, H.: Locally Convex Spaces, B. G. Teubner, Stuttgart, 1981.10.1007/978-3-322-90559-8Search in Google Scholar
[4] Kluvanek, I.—Knowles, G.: Vector Measures and Control Systems. North-Holland Mathematics Studies 20, Amsterdam, 1976.Search in Google Scholar
[5] Lindenstrauss, J.: A short proof of Liapounoff’s convexity theorem, J. Math. Mech. 15 (1966), 971–972.10.1512/iumj.1966.15.15064Search in Google Scholar
[6] Sikorski, R.: Boolean Algebras, Springer-Verlag, Heidelberg, 1960.10.1007/978-3-662-01507-0Search in Google Scholar
[7] Tweddle, I.: Weak compactness in locally convex spaces, Glasg. Math. J. 9 (1968), 123–127.10.1017/S0017089500000409Search in Google Scholar
[8] Uhl Jr., J. J.: The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969), 158–163.10.1090/S0002-9939-1969-0264029-1Search in Google Scholar
[9] Weber, H.: FN-topologies and group-valued measures. In: Handbook of Measure Theory, vol. I (E. Pap ed.), North-Holland, Amsterdam, 2002, pp. 703–743.10.1016/B978-044450263-6/50017-8Search in Google Scholar
[10] Wnuk, W.: The converse of Lyapunov convexity theorem, Comment. Math. 21 (1979), 389–390.Search in Google Scholar
© 2017 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Paolo de Lucia
- Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
- A note on field-valued measures
- Topologies and uniformities on d0-algebras
- On some properties of 𝓙-approximately continuous functions
- Porous subsets in the space of functions having the Baire property
- Rademacher’s theorem in Banach spaces without RNP
- Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
- Measure games on pseudo-D-lattices
- On some properties of k-subadditive lattice group-valued capacities
- Lp Spaces in vector lattices and applications
- The Choquet integral with respect to fuzzy measures and applications
- A note on the range of vector measures
- Ideal convergent subsequences and rearrangements for divergent sequences of functions
- Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
- A generalization of the exponential sampling series and its approximation properties
- Monotonicity and total boundedness in spaces of “measurable” functions
- Vector lattices in synaptic algebras
- Density, ψ-density and continuity
- Feather topologies
- On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
- Rate of convergence of empirical measures for exchangeable sequences
- On non-additive probability measures
- Equi-topological entropy curves for skew tent maps in the square
- On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
