Compactness and Weak Compactness in Spaces of Compact-Range Vector Measures

Graves, William H. ; Ruess, Wolfgang

Canadian journal of mathematics, Tome 36 (1984) no. 6, pp. 1000-1020

Résumé

This paper features strong and weak compactness in spaces of vector measures with relatively compact ranges in Banach spaces. Its tools are the measure-operator identification of [16] and [24] and the description of strong and weak compactness in spaces of compact operators in [10], [11], and [29].Given a Banach space X and an algebra of sets, it is shown in [16] that under the usual identification via integration of X-valued bounded additive measures on with X-valued sup norm continuous linear operators on the space of -simple scalar functions, the strongly bounded, countably additive measures correspond exactly to those operators which are continuous for the coarser (locally convex) universal measure topology τ on . It is through the latter identification that the results on strong and weak compactness in [10], [11], and [29] can be applied to X-valued continuous linear operators on the generalized DF space to yield results on strong and weak compactness in spaces of vector measures.

Détail

DOI : 10.4153/CJM-1984-057-9

Bibliographie

[1] 1. Batt, J., On weak compactness in spaces of vector-valued measures and Bochner integrable functions in connection with the Radan-Nikodym property of Banach spaces. Rev. Roumaine Math. Pures Appl. 19 (1974), 285–304. Google Scholar

[2] 2. Batt, J. and Hiermeyer, W., On compactness in L(μ, X), in the weak topology and in the topology σ(L(μ, X), L(n, X′)), Math. Z. 182 (1983), 409–423. Google Scholar

[3] 3. Bourgain, F. and Talagrand, M., Compacité extremale, Proc. Amer. Math. Soc. 80 (1980), 68–70. Google Scholar

[4] 4. Brook, C. H., Projections and measures, Dissertation, Univ. of North Carolina, Chapel Hill, NC (1978). Google Scholar

[5] 5. Brook, C. H., On the universal measure space, J. Math. Anal, and Appl. 85 (1982), 584–598. Google Scholar

[6] 6. Brook, C. H., Continuity properties of vector measures, J. Math. Anal, and Appl. 86 (1982), 268–280. Google Scholar

[7] 7. Brook, C. H. and Graves, W. H., The range of a vector measure, J. Math. Anal, and Appl. 79 (1980), 219–237. Google Scholar

[8] 8. Brook, C. H., Closed measures, Proc. Conf. of Integration, Topology, and Geometry in Linear Spaces, Amer. Math. Soc. Contemporary Math. 2 (Amer. Math. Soc, Providence, RI, 1980). Google Scholar

[9] 9. Brooks, J. K. and Dinculeanu, N., Weak and strong compactness in the space of Pettis integrable functions, Proc. Conf. of Integration, Topology, and Geometry in Linear Spaces, Amer. Math. Soc. Contemporary Math. 2 (Amer. Math. Soc, Providence, RI, 1980). Google Scholar

[10] 10. Collins, H. S. and Ruess, W., Weak convergence in spaces of compact operators and of vector valued functions, Pacific J. Math. 106 (1983), 45–71. Google Scholar

[11] 11. Collins, H. S. and Ruess, W., Duals of spaces of compact operators, Studia Math. 75 (1982), 213–245. Google Scholar

[12] 12. DeWilde, M., Pointwise compactness in spaces of functions and R. C James' theorem, Math. Ann. 208 (1974), 33–47. Google Scholar

[13] 13. Diestel, J. and Uhl, J. J. Jr., Vector measures, Amer. Math. Soc. Surveys 15 (Amer. Math. Soc, Providence, RI, 1977). Google Scholar | DOI

[14] 14. Dunford, N. and Schwartz, J. T., Linear operators I (Interscience, New York, 1957). Google Scholar

[15] 15. Fremlin, D. H. and Talagrand, M., A decomposition theorem for additive set functions, with applications to Pettis integrals and ergodic means, Math. Z. 168 (1979), 117–142. Google Scholar

[16] 16. Graves, W. H., On the theory of vector measures, Amer. Math. Soc. Memoirs 195 (Amer. Math Soc, Providence, RI, 1977). Google Scholar

[17] 17. Graves, W. H. and Ruess, W., Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded measurable functions, Proc. Conf. on Integration, Topology, and Geometry in Linear Spaces, Amer. Math. Soc. Contemporary Math. 2 (Amer. Math. Soc, Providence, RI, 1980). Google Scholar

[18] 18. Graves, W. H. and Sentilles, F. D., The extension and completion of the universal measure and the dual of the space of measures, J. Math. Anal, and Appl. 68 (1979), 228–264. Google Scholar

[19] 19. Graves, W. H. and Wheeler, R. F., On the Grothendieck and Nikodym properties for algebras of Baire, Borel, and universally measurable sets, Rocky Mountain J. Math. 13 (1983), 333–353. Google Scholar

[20] 20. Grothendieck, A., Sur les espaces (F) et (DF), Summa Brasil. Math. 3 (1954), 57–122. Google Scholar

[21] 21. Grothendieck, A., Produit tensoriels topologiques et espaces nucléaires, Amer. Math. Soc Memoirs 16 (Amer. Math. Soc, Providence, RI, 1955). Google Scholar

[22] 22. Lewis, D. R., Conditional weak compactness in certain inductive tensor products, Math. Ann 201 (1973), 201–209. Google Scholar

[23] 23. Lindenstrauss, J. and Stegall, C., Examples of separable spaces which do not contain l and whose duals are non-separable, Studia Math. 54 (1975), 81–105. Google Scholar

[24] 24. Molnar, S. M., Representing measures with values in locally convex Hausdorff spaces, Thesis, Univ. North Carolina, Chapel Hill (1973). Google Scholar

[25] 25. Rosenthal, H. P., A characterization of Banach spaces containing l1 Proc. Nat. Acad. Sci. USA 71 (1974), 2411–2413. Google Scholar

[26] 26. Ruess, W., On the locally convex structure of strict topologies, Math. Z. 153 (1977), 179–192. Google Scholar

[27] 27. Ruess, W., The strict topology and DF spaces, Proc. Paderborn Conf. on Functional Analysis 1976, North Holland Math. Studies 27, 105–118 (North Holland, New York, 1977). Google Scholar

[28] 28. Ruess, W., [Weakly] compact operators and DF spaces, Pacific J. Math. 98 (1982), 419–441. Google Scholar

[29] 29. Ruess, W., Compactness, and collective compactness in spaces of compact operators, J. Math. Anal, and Appl. 84 (1981), 400–417. Google Scholar

[30] 30. Ruess, W. and Stegall, C., Extreme points in duals of operator spaces, Math. Ann. 261 (1982), 535–546. Google Scholar

[31] 31. Schachermayer, W., On some classical measure-theoretic theorems for non o-comp/ete Boolean algebras, preprint. Google Scholar

[32] 32. Schaefer, H. H., Topological vector spaces (Springer-Verlag, New York-Heidelberg, 1971). Google Scholar | DOI

[33] 33. Schwartz, L., Théorie des distributions a valeurs vectorielles I, Ann. Institut Fourier 7 (1957), 1–140. Google Scholar

[34] 34. Sikorski, R., Boolean algebras (third ed.) (Springer-Verlag, New York-Heidelberg, 1964). Google Scholar