On Denjoy-Dunford and Denjoy-Pettis integrals

José Gámez; José Mendoza

Studia Mathematica (1998)

  • Volume: 130, Issue: 2, page 115-133
  • ISSN: 0039-3223

Abstract

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The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function f : [ a , b ] → c 0 which is not Pettis integrable on any subinterval in [a,b], while ʃ J f belongs to c 0 for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord and Denjoy-Pettis integrals are studied.

How to cite

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Gámez, José, and Mendoza, José. "On Denjoy-Dunford and Denjoy-Pettis integrals." Studia Mathematica 130.2 (1998): 115-133. <http://eudml.org/doc/216547>.

@article{Gámez1998,
abstract = {The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function $f : [a,b] → c_0$ which is not Pettis integrable on any subinterval in [a,b], while $ʃ_J f$ belongs to $c_0$ for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord and Denjoy-Pettis integrals are studied.},
author = {Gámez, José, Mendoza, José},
journal = {Studia Mathematica},
keywords = {Banach-valued functions; Denjoy-Dunford integrals; Denjoy-Pettis integrals},
language = {eng},
number = {2},
pages = {115-133},
title = {On Denjoy-Dunford and Denjoy-Pettis integrals},
url = {http://eudml.org/doc/216547},
volume = {130},
year = {1998},
}

TY - JOUR
AU - Gámez, José
AU - Mendoza, José
TI - On Denjoy-Dunford and Denjoy-Pettis integrals
JO - Studia Mathematica
PY - 1998
VL - 130
IS - 2
SP - 115
EP - 133
AB - The two main results of this paper are the following: (a) If X is a Banach space and f : [a,b] → X is a function such that x*f is Denjoy integrable for all x* ∈ X*, then f is Denjoy-Dunford integrable, and (b) There exists a Dunford integrable function $f : [a,b] → c_0$ which is not Pettis integrable on any subinterval in [a,b], while $ʃ_J f$ belongs to $c_0$ for every subinterval J in [a,b]. These results provide answers to two open problems left by R. A. Gordon in [4]. Some other questions in connection with Denjoy-Dundord and Denjoy-Pettis integrals are studied.
LA - eng
KW - Banach-valued functions; Denjoy-Dunford integrals; Denjoy-Pettis integrals
UR - http://eudml.org/doc/216547
ER -

References

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  1. [1] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, 1984. 
  2. [2] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977. 
  3. [3] N. Dunford and J. T. Schwartz, Linear Operators, part I, Interscience, New York, 1958. 
  4. [4] R. A. Gordon, The Denjoy extension of the Bochner, Pettis, and Dunford integrals, Studia Math. 92 (1989), 73-91. Zbl0681.28006
  5. [5] R. A. Gordon, The integrals of Lebesque, Denjoy, Perron and Henstock, Grad. Stud. Math. 4, Amer. Math. Soc., Providence, 1994. Zbl0807.26004
  6. [6] J. Lindenstrauss and L.Tzafriri, Classical Banach Spaces I, Springer, 1977. Zbl0362.46013
  7. [7] S. Saks, Theory of the Integral, 2nd revised ed., Hafner, New York, 1937. 

Citations in EuDML Documents

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  1. B. Bongiorno, Luisa Di Piazza, Kazimierz Musiał, Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrability of strongly measurable functions
  2. Kirill Naralenkov, On Denjoy type extensions of the Pettis integral
  3. Bianca Satco, Volterra integral inclusions via Henstock-Kurzweil-Pettis integral
  4. Bianca Satco, A Komlós-type theorem for the set-valued Henstock-Kurzweil-Pettis integral and applications
  5. Afif Ben Amar, Some fixed point theorems and existence of weak solutions of Volterra integral equation under Henstock-Kurzweil-Pettis integrability

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