A PU-integral on an abstract metric space

Giuseppa Riccobono

Mathematica Bohemica (1997)

  • Volume: 122, Issue: 1, page 83-95
  • ISSN: 0862-7959

Abstract

top
In this paper, we define a -integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure μ is compatible with its topology in the sense that every open set is μ -measurable. We prove that the -integral is equivalent to μ -integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.

How to cite

top

Riccobono, Giuseppa. "A PU-integral on an abstract metric space." Mathematica Bohemica 122.1 (1997): 83-95. <http://eudml.org/doc/248118>.

@article{Riccobono1997,
abstract = {In this paper, we define a $$-integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure $\mu $ is compatible with its topology in the sense that every open set is $\mu $-measurable. We prove that the $$-integral is equivalent to $\mu $-integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.},
author = {Riccobono, Giuseppa},
journal = {Mathematica Bohemica},
keywords = {PU-integral; partition of unity; PU-integral; partition of unity},
language = {eng},
number = {1},
pages = {83-95},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A PU-integral on an abstract metric space},
url = {http://eudml.org/doc/248118},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Riccobono, Giuseppa
TI - A PU-integral on an abstract metric space
JO - Mathematica Bohemica
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 122
IS - 1
SP - 83
EP - 95
AB - In this paper, we define a $$-integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure $\mu $ is compatible with its topology in the sense that every open set is $\mu $-measurable. We prove that the $$-integral is equivalent to $\mu $-integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.
LA - eng
KW - PU-integral; partition of unity; PU-integral; partition of unity
UR - http://eudml.org/doc/248118
ER -

References

top
  1. A. M. Bruckner, Differentiation of integrals, Amer. Math. Monthly, No. 9 Supplement 78 (1971), 1-51. (1971) Zbl0225.28002MR0293044
  2. D. Caponetti, V. Marraffa, An integral in the real line defined by BV partitions of unity, Atti Semin. Mat. Fis., Univ. Modena, 42 (1994), 69-82. (1994) Zbl0824.26004MR1282323
  3. R. O. Davies, Z. Schuss, 10.1112/jlms/2.Part_3.561, J. London Math. Soc. (2) 2 (1970), 561-562. (1970) Zbl0197.04103MR0265526DOI10.1112/jlms/2.Part_3.561
  4. G. A. Edgar, Measure, topology and fractal geometry, Springer-Verlag, 1990. (1990) Zbl0727.28003MR1065392
  5. H. Hahn, A. Rosenthal, Set functions, Albuquerque, N. Mexico, 1948. (1948) Zbl0033.05301MR0024504
  6. R. Henstock, The general theory of integration, Clarendon Press, Oxford, 1991. (1991) Zbl0745.26006MR1134656
  7. R. Henstock, Measure spaces and division spaces, Real Anal. Exchange 19 (1) (1993/94). 121-128. (1993) MR1268837
  8. J. Jarník, J. Kurzweil, A non absolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (110) (1985), 116-139. (1985) MR0779340
  9. J. Jarník, J. Kurzweil, A new and more powerful concept of the PU-integral, Czechoslovak Math. J. 38 (113) (1988), 8-48. (1988) MR0925939
  10. J. Kurzweil J. Mawhin, W. Pfeffer, An integral defined by approximating BV paгtitions of unity, Czechoslovak Math. J. 41 (116) (1991), 695-712. (1991) MR1134958
  11. W. F. Pfeffer, 10.1090/S0002-9947-1986-0833702-0, Trans. Amer. Math. Soc. 295 (2) (1986), 665-685. (1986) Zbl0596.26007MR0833702DOI10.1090/S0002-9947-1986-0833702-0
  12. W. F. Pfeffer, Wei-Chi Yang, A note on conditionally convergent integrals, Real Anal. Exchange 17 (1991/92), 815-819. (1991) MR1171426
  13. W. F. Pfeffer, The Riemann approach to integration, Cambridge University Pгess, 1993. (1993) Zbl0804.26005MR1268404

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.