On vector measures

Diestel, J.; Faires, B.

doi:10.1090/S0002-9947-1974-0350420-8

On vector measures
HTML articles powered by AMS MathViewer

by J. Diestel and B. Faires

Trans. Amer. Math. Soc. 198 (1974), 253-271

DOI: https://doi.org/10.1090/S0002-9947-1974-0350420-8

PDF | Request permission

Abstract:

The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space $X$ to have the property that bounded additive $X$-valued maps on $\sigma$-algebras be strongly bounded are presented, namely, $X$ can contain no copy of ${l_\infty }$. The next two sections treat the Jordan decomposition for measures with values in ${L_1}$-spaces on ${c_0}(\Gamma )$ spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of ${c_0}$. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space $X$ has an equivalent very smooth norm (in particular, a Fréchet differentiable norm) then its dual has the Radon-Nikodym property. Consequently, a $C(\Omega )$ space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper.

References

D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35–46. MR 228983, DOI 10.2307/1970554
Tsuyoshi Andô, Convergent sequences of finitely additive measures, Pacific J. Math. 11 (1961), 395–404. MR 137806
John Warren Baker, Ordinal subspaces of topological spaces, General Topology and Appl. 3 (1973), 85–91. MR 324623
R. G. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), 337–352. MR 80721, DOI 10.4064/sm-15-3-337-352
R. G. Bartle, N. Dunford, and J. Schwartz, Weak compactness and vector measures, Canadian J. Math. 7 (1955), 289–305. MR 70050, DOI 10.4153/CJM-1955-032-1
C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164
C. Bessaga and A. Pełczyński, On extreme points in separable conjugate spaces, Israel J. Math. 4 (1966), 262–264. MR 211244, DOI 10.1007/BF02771641
Errett Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27–35. MR 0154092
James K. Brooks, Representations of weak and strong integrals in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 266–270. MR 274697, DOI 10.1073/pnas.63.2.266
James K. Brooks, On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 999–1001. MR 286971, DOI 10.1090/S0002-9904-1971-12834-3
James K. Brooks, Weak compactness in the space of vector measures, Bull. Amer. Math. Soc. 78 (1972), 284–287. MR 324408, DOI 10.1090/S0002-9904-1972-12960-4
James K. Brooks and Robert S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298. MR 269802, DOI 10.1073/pnas.67.3.1294
J. Diestel, Applications of weak compactness and bases to vector measures and vectorial integration, Rev. Roumaine Math. Pures Appl. 18 (1973), 211–224. MR 317042
J. Diestel, On the essential uniqueness of an example of C. E. Rickart, Comment. Math. Prace Mat. 17 (1973), 263–264. MR 322130
J. Diestel, Grothendieck spaces and vector measures, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 97–108. MR 0338774
J. Diestel, The Radon-Nikodym property and the coincidence of integral and nuclear operators, Rev. Roumaine Math. Pures Appl. 17 (1972), 1611–1620. MR 333728
N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
Nelson Dunford and B. J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323–392. MR 2020, DOI 10.1090/S0002-9947-1940-0002020-4
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
T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197–200. MR 341032, DOI 10.1090/S0002-9939-1973-0341032-5
J. R. Giles, On a characterisation of differentiability of the norm of a normed linear space, J. Austral. Math. Soc. 12 (1971), 106–114. MR 0284797
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
N. E. Gretsky and J. J. Uhl Jr., Bounded linear operators on Banach function spaces of vector-valued functions, Trans. Amer. Math. Soc. 167 (1972), 263–277. MR 295110, DOI 10.1090/S0002-9947-1972-0295110-3
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type $C(K)$, Canad. J. Math. 5 (1953), 129–173 (French). MR 58866, DOI 10.4153/cjm-1953-017-4

Vector measures

28

Robert E. Huff, The Yosida-Hewitt decomposition as an ergodic theorem, Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972) Academic Press, New York, 1973, pp. 133–139. MR 0330406
K. John and V. Zizler, A renorming of dual spaces, Israel J. Math. 12 (1972), 331–336. MR 344853, DOI 10.1007/BF02764623
Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
Shizuo Kakutani, Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024. MR 5778, DOI 10.2307/1968778
Gottfried Köthe, Topologische lineare Räume. I, Die Grundlehren der mathematischen Wissenschaften, Band 107, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0130551
D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294–307. MR 291409
Joram Lindenstrauss, Weakly compact sets—their topological properties and the Banach spaces they generate, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N.J., 1972, pp. 235–273. MR 0417761
A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225–238. MR 66558, DOI 10.1090/S0002-9947-1955-0066558-7
Hugh B. Maynard, A geometrical characterization of Banach spaces with the Radon-Nikodym property, Trans. Amer. Math. Soc. 185 (1973), 493–500. MR 385521, DOI 10.1090/S0002-9947-1973-0385521-0
Michel Métivier, Martingales à valeurs vectorielles. Applications à la dérivation des mesures vectorielles, Ann. Inst. Fourier (Grenoble) 17 (1967), no. fasc. 2, 175–208 (1968) (French). MR 247663
S. Moedomo and J. J. Uhl Jr., Radon-Nikodým theorems for the Bochner and Pettis integrals, Pacific J. Math. 38 (1971), 531–536. MR 308359
A. Pełczyński, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641–648. MR 149295
B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), no. 2, 277–304. MR 1501970, DOI 10.1090/S0002-9947-1938-1501970-8
R. S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48 (1940), 516–541. MR 4094, DOI 10.1090/S0002-9947-1940-0004094-3
R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943), 108–136. MR 7938, DOI 10.2307/2371776
M. M. Rao, Smoothness of Orlicz spaces. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 68 = Indag. Math. 27 (1965), 671–680; 681–690. MR 0190704
C. E. Rickart, Decomposition of additive set functions, Duke Math. J. 10 (1943), 653–665. MR 9977
M. A. Rieffel, Dentable subsets of Banach spaces, with application to a Radon-Nikodým theorem, Functional Analysis (Proc. Conf., Irvine, Calif., 1966) Academic Press, London; Thompson Book Co., Washington, D.C., 1967, pp. 71–77. MR 0222618
M. A. Rieffel, The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc. 131 (1968), 466–487. MR 222245, DOI 10.1090/S0002-9947-1968-0222245-2
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
G. L. Seever, Measures on $F$-spaces, Trans. Amer. Math. Soc. 133 (1968), 267–280. MR 226386, DOI 10.1090/S0002-9947-1968-0226386-5
M. H. Stone, Boundedness properties in function-lattices, Canad. J. Math. 1 (1949), 176–186. MR 29091, DOI 10.4153/cjm-1949-016-5

The Lebesgue-Nikodym theorem for vector valued Radon measures

139

Ju. B. Tumarkin, Locally convex spaces with basis, Dokl. Akad. Nauk SSSR 195 (1970), 1278–1281 (Russian). MR 0271694
J. J. Uhl Jr., Applications of Radon-Nikodým theorems to martingale convergence, Trans. Amer. Math. Soc. 145 (1969), 271–285. MR 251756, DOI 10.1090/S0002-9947-1969-0251756-X
J. J. Uhl Jr., Extensions and decompositions of vector measures, J. London Math. Soc. (2) 3 (1971), 672–676. MR 286974, DOI 10.1112/jlms/s2-3.4.672
J. J. Uhl Jr., A note on the Radon-Nikodym property for Banach spaces, Rev. Roumaine Math. Pures Appl. 17 (1972), 113–115. MR 482100
J. J. Uhl Jr., The Radon-Nikodym theorem and the mean convergence of Banach space valued martingales, Proc. Amer. Math. Soc. 21 (1969), 139–144. MR 238364, DOI 10.1090/S0002-9939-1969-0238364-7
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Victor L. Klee Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10–43. MR 54850, DOI 10.1090/S0002-9947-1953-0054850-X

Duality and smoothness in Lebesgue-Bochner function spaces