Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Cover of the book
Tytuł książki

On vector measures which have everywhere infinite variation or noncompact range

Rozprawy Matematyczne tom/nr w serii: 339 wydano: 1995
Warianty tytułu

Let X be an infinite-dimensional Banach space, let Σ be a σ-algebra of subsets of a set S, and denote by ca(Σ,X) the Banach space of X-valued measures on Σ equipped with the uniform norm. We say that a nonzero μ ∈ ca(Σ,X) is everywhere of infinite variation [has everywhere noncompact range] if |μ|(A) = ∞ or 0 [{μ(E): E ∈ Σ, E ⊂ A} is not relatively compact or equals {0}] for every A ∈ Σ. Let λ be a nonatomic probability measure on Σ, and denote by ca(Σ,λ,X) the closed subspace of ca(Σ,X) consisting of λ-continuous measures. Analogously as above, we define measures in ca(Σ,λ,X) that are λ-everywhere of infinite variation or have λ-everywhere noncompact range. Using the Dvoretzky-Rogers theorem, we give two constructions of an absolutely convergent series of λ-simple measures in ca(Σ,λ,X) such that the sum of each of its subseries is λ-everywhere of infinite variation. In particular, the normed space P(λ,X) of Pettis λ-integrable functions with values in X lacks property (K), and so is incomplete. These results refine and improve some earlier results of E. Thomas, and L. Janicka and N. J. Kalton. One of the constructions also yields the existence of an infinite-dimensional closed subspace in ca(Σ,λ,X) all of whose nonzero members are λ-everywhere of infinite variation. Moreover, modifying some ideas of R. Anantharaman and K. M. Garg, we prove that the measures that are λ-everywhere of infinite variation form a dense $G_δ$-set in ca(Σ,λ,X). From this we derive an analogous result on measures that are everywhere of infinite variation and the closed subspace of ca(Σ,X) consisting of nonatomic measures. Similar results concerning measures that have [λ-] everywhere noncompact range are also established. In this case, the existence of X-valued measures with noncompact range must, however, be postulated. We also prove that the measures of σ-finite variation form an $F_{σδ}$-, but not $F_σ$-, subset of ca(Σ,λ,X), and the same is true for P(λ,X) provided that X is separable. Finally, we consider the special case when X is a Banach lattice and, for X nonisomorphic to an AL-space, we note analogues of some of the results above for positive X-valued measures on Σ.
Miejsce publikacji
Rozprawy Matematyczne tom/nr w serii: 339
Liczba stron
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXXIX
  • Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
  • Institute of Mathematics, Polish Academy of Sciences, Wrocław Branch, Kopernika 18, 51-617 Wrocław, Poland
  • [AG] R. Anantharaman and K. M. Garg, The properties of a residual set of vector measures, in: Measure Theory and its Applications, Proc. Conf. Sherbrooke, Québec, 1982, J. M. Belley, J. Dubois and P. Morales (eds.), Lecture Notes in Math. 1033, Springer, Berlin, 1983, 12-35.
  • [Ar] A. P. Artemenko, La forme générale d'une fonctionnelle linéaire dans l'espace des fonctions à variation bornée, Mat. Sb. 6 (48) (1939), 215-220 (in Russian, French summary; MR 1, 239).
  • [BP] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monografie Mat. 58, Polish Scientific Publishers, Warszawa, 1975.
  • [BKL] J. Burzyk, C. Kliś and Z. Lipecki, On metrizable Abelian groups with a completeness-type property, Colloq. Math. 49 (1984), 33-39.
  • [DU] J. Diestel and J. J. Uhl, Jr. Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
  • [D1] L. Drewnowski, Topological rings of sets, continuous set functions, integration. I, II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 269-276 and 277-286.
  • [D2] L. Drewnowski, Quasi-complements in F-spaces, Studia Math. 77 (1984), 373-391.
  • [D3] L. Drewnowski, Another note on copies of $l_∞$ and $c_0$ in ca(Σ,X), and the equality ca(Σ,X) = cca(Σ,X), preprint.
  • [DFP] L. Drewnowski, M. Florencio and P. J. Paúl, The space of Pettis integrable functions is barrelled, Proc. Amer. Math. Soc. 114 (1992), 687-694.
  • [DS] N. Dunford and J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.
  • [Ha] F. Hausdorff, Mengenlehre, Walter de Gruyter, Berlin, 1927.
  • [He] M. Heiliö, Weakly Summable Measures in Banach Spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 1988, no. 66.
  • [JK] L. Janicka and N. J. Kalton, Vector measures of infinite variation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 239-241.
  • [K] R. B. Kirk, Sets which split families of measurable sets, Amer. Math. Monthly 79 (1972), 884-886.
  • [L] D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294-307.
  • [LR] J. Lindenstrauss and H. P. Rosenthal, The $L_p$ spaces, Israel J. Math. 7 (1969), 325-349.
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, Berlin, 1977.
  • [LM] Z. Lipecki and K. Musiał, On the Radon-Nikodym derivative of a measure taking values in a Banach space with basis, Rev. Roumaine Math. Pures Appl. 23 (1978), 911-915.
  • [M-N] P. Meyer-Nieberg, Banach Lattices, Springer, Berlin, 1991.
  • [Ma] R. D. Mauldin, The continuum hypothesis, integration and duals of measure spaces, Illinois J. Math. 19 (1975), 33-40.
  • [Mu] K. Musiał, The weak Radon-Nikodym property, Studia Math. 64 (1979), 151-173.
  • [PB] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, Notas de Matemática 131, North-Holland, Amsterdam, 1987.
  • [Pe] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
  • [Pi] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, Cambridge, 1989.
  • [R-P] L. Rodríguez Piazza, Rango y propriedades de medidas vectoriales. Conjuntos p-Sidon p.s., Doctoral Dissertation, University of Seville, 1991.
  • [R] V. I. Rybakov, On the completeness of the space of Pettis integrable functions, Moskov. Gos. Ped. Inst. Uchen. Zap., 1970, no. 277, 58-64 (in Russian).
  • [S] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974.
  • [T1] E. Thomas, The Lebesgue-Nikodym Theorem for Vector Valued Radon Measures, Mem. Amer. Math. Soc. 139 (1974).
  • [T2] E. Thomas, Totally summable functions with values in locally convex spaces, in: Measure Theory, Proc. Conf. Oberwolfach, 1975, A. Bellow and D. Kölzow (eds.), Lecture Notes in Math. 541, Springer, Berlin, 1976, 117-131.
  • [VTC] N. N. Vakhaniya, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, D. Reidel, Dordrecht, 1987.
  • [W] L. Waelbroeck, Topological Vector Spaces and Algebras, Lecture Notes in Math. 230, Springer, Berlin, 1971.
Języki publikacji
Identyfikator YADDA
Zawartość książki

rozwiń roczniki

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.