Conical measures and properties of a vector measure determined by its range

L. Rodríguez-Piazza; M. Romero-Moreno

Studia Mathematica (1997)

  • Volume: 125, Issue: 3, page 255-270
  • ISSN: 0039-3223

Abstract

top
We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability is not determined by the range and study when every measure having the same range of a given measure has a Pettis derivative.

How to cite

top

Rodríguez-Piazza, L., and Romero-Moreno, M.. "Conical measures and properties of a vector measure determined by its range." Studia Mathematica 125.3 (1997): 255-270. <http://eudml.org/doc/216437>.

@article{Rodríguez1997,
abstract = {We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability is not determined by the range and study when every measure having the same range of a given measure has a Pettis derivative.},
author = {Rodríguez-Piazza, L., Romero-Moreno, M.},
journal = {Studia Mathematica},
keywords = {vector measures; range; conical measures; operator ideal norms; Pettis integral; operator ideal; associated Kluvánek conical measure; range of a vector measure; total variation; -finiteness; Bochner derivability; -summing; -nuclear norm; integration operator; Pettis derivability},
language = {eng},
number = {3},
pages = {255-270},
title = {Conical measures and properties of a vector measure determined by its range},
url = {http://eudml.org/doc/216437},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Rodríguez-Piazza, L.
AU - Romero-Moreno, M.
TI - Conical measures and properties of a vector measure determined by its range
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 3
SP - 255
EP - 270
AB - We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability is not determined by the range and study when every measure having the same range of a given measure has a Pettis derivative.
LA - eng
KW - vector measures; range; conical measures; operator ideal norms; Pettis integral; operator ideal; associated Kluvánek conical measure; range of a vector measure; total variation; -finiteness; Bochner derivability; -summing; -nuclear norm; integration operator; Pettis derivability
UR - http://eudml.org/doc/216437
ER -

References

top
  1. [AD] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221-235. Zbl0749.28006
  2. [C] C. H. Choquet, Lectures on Analysis, Vols. I, II, III, Benjamin, New York, 1969. Zbl0181.39602
  3. [DJT] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995. Zbl0855.47016
  4. [DU] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc. Providence, R.I., 1977. 
  5. [E] G. Edgar, Measurability in a Banach space, Indiana Univ. Math. J. 26 (1977), 663-677. Zbl0361.46017
  6. [FT] D. Fremlin and M. Talagrand, A decomposition theorem for additive set functions and applications to Pettis integral and ergodic means, Math. Z. 168 (1979), 117-142. Zbl0393.28005
  7. [K] I. Kluvánek, Characterization of the closed convex hull of the range of a vector measure, J. Funct. Anal. 21 (1976), 316-329. Zbl0317.46035
  8. [L] D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294-307. Zbl0242.28008
  9. [M] K. Musiał, The weak Radon-Nikodym property in Banach spaces, Studia Math. 64 (1979), 151-174. Zbl0405.46015
  10. [P] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980. 
  11. [R1] L. Rodríguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), 205-214. Zbl0727.28005
  12. [R2] L. Rodríguez-Piazza, Derivability, variation and range of a vector measure, Studia Math. 112 (1995), 165-187. 
  13. [T] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 307 (1984). Zbl0582.46049
  14. [Th] E. Thomas, Integral representations in convex cones, Groningen University Report ZW-7703 (1977). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.