On Denjoy type extensions of the Pettis integral

Kirill Naralenkov

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 3, page 737-750
  • ISSN: 0011-4642

Abstract

top
In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.

How to cite

top

Naralenkov, Kirill. "On Denjoy type extensions of the Pettis integral." Czechoslovak Mathematical Journal 60.3 (2010): 737-750. <http://eudml.org/doc/38039>.

@article{Naralenkov2010,
abstract = {In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.},
author = {Naralenkov, Kirill},
journal = {Czechoslovak Mathematical Journal},
keywords = {scalar derivative; approximate scalar derivative; absolute continuity; bounded variation; $VBG$ function; $ACG$ function; Pettis integral; Denjoy-Pettis integral; scalar derivative; approximate scalar derivative; absolute continuity; bounded variation; function; function; Pettis integral; Denjoy-Pettis integral},
language = {eng},
number = {3},
pages = {737-750},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Denjoy type extensions of the Pettis integral},
url = {http://eudml.org/doc/38039},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Naralenkov, Kirill
TI - On Denjoy type extensions of the Pettis integral
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 737
EP - 750
AB - In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.
LA - eng
KW - scalar derivative; approximate scalar derivative; absolute continuity; bounded variation; $VBG$ function; $ACG$ function; Pettis integral; Denjoy-Pettis integral; scalar derivative; approximate scalar derivative; absolute continuity; bounded variation; function; function; Pettis integral; Denjoy-Pettis integral
UR - http://eudml.org/doc/38039
ER -

References

top
  1. Birkhoff, G., Integration of functions with values in a Banach space, Trans. Am. Math. Soc. 38 (1935), 357-378. (1935) Zbl0013.00803MR1501815
  2. Piazza, L. Di, Musiał, K., 10.4064/sm176-2-4, Stud. Math. 176 (2006), 159-176. (2006) MR2264361DOI10.4064/sm176-2-4
  3. Diestel, J., 10.1007/978-1-4612-5200-9, Springer-Verlag New York-Heidelberg-Berlin (1984). (1984) MR0737004DOI10.1007/978-1-4612-5200-9
  4. Fremlin, D. H., Mendoza, J., 10.1215/ijm/1255986891, Ill. J. Math. 38 (1994), 127-147. (1994) Zbl0790.28004MR1245838DOI10.1215/ijm/1255986891
  5. Gámez, J. L., Mendoza, J., On Denjoy-Dunford and Denjoy-Pettis integrals, Stud. Math. 130 (1998), 115-133. (1998) MR1623348
  6. Gordon, R. A., 10.4064/sm-92-1-73-91, Stud. Math. 92 (1989), 73-91. (1989) Zbl0681.28006MR0984851DOI10.4064/sm-92-1-73-91
  7. Gordon, R. A., 10.1090/gsm/004/09, American Mathematical Society (AMS) Providence (1994). (1994) MR1288751DOI10.1090/gsm/004/09
  8. Hildebrandt, T. H., 10.1090/S0002-9904-1953-09694-X, Bull. Am. Math. Soc. 59 (1953), 111-139. (1953) Zbl0051.04201MR0053191DOI10.1090/S0002-9904-1953-09694-X
  9. Lee, Peng Yee, Výborný, R., The Integral: An Easy Approach after Kurzweil and Henstock. Australian Mathematical Society Lecture Series, Vol. 14, Cambridge University Press Cambridge (2000). (2000) MR1756319
  10. Lindenstrauss, J., Tzafriri, L., Classical Banach Spaces. I. Sequence Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer Berlin-Heidelberg-New York (1977). (1977) MR0500056
  11. Naralenkov, K. M., 10.14321/realanalexch.30.1.0235, Real Anal. Exch. 30 (2004/2005), 235-260. (2004) MR2127529DOI10.14321/realanalexch.30.1.0235
  12. Pettis, B. J., 10.1090/S0002-9947-1938-1501970-8, Trans. Am. Math. Soc. 44 (1938), 277-304. (1938) Zbl0019.41603MR1501970DOI10.1090/S0002-9947-1938-1501970-8
  13. Saks, S., Theory of the Integral, Dover Publications Inc. New York (1964). (1964) MR0167578
  14. Talagrand, M., Pettis Integral and Measure Theory. Mem. Am. Math. Soc. No. 307, (1984). (1984) MR0756174
  15. Wang, Chonghu, Yang, Zhenhua, 10.1216/rmjm/1022008999, Rocky Mt. J. Math. 30 (2000), 393-400. (2000) Zbl0986.46028MR1763820DOI10.1216/rmjm/1022008999

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.