The quasi topology associated with a countably subadditive set function

Bent Fuglede

Annales de l'institut Fourier (1971)

  • Volume: 21, Issue: 1, page 123-169
  • ISSN: 0373-0956

Abstract

top
This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space X . The principal aim is the study of the “quasi-topological” properties of subsets of X , or of numerical functions on X , with respect to such a capacity C . Analogues are obtained to various important properties of the fine topology in potential theory, notably the quasi Lindelöf principle (Doob), the existence of a fine support (Getoor), and the theorem on capacity for decreasing families of sets (Brelot). This analogy becomes an actual identity if a certain compatibility is assumed betweeh the capacity C and a new homology (called “fine”) on X . Sufficient conditions are obtained with a convex cone of lower semicontinuous functions on  X .

How to cite

top

Fuglede, Bent. "The quasi topology associated with a countably subadditive set function." Annales de l'institut Fourier 21.1 (1971): 123-169. <http://eudml.org/doc/74024>.

@article{Fuglede1971,
abstract = {This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space $X$. The principal aim is the study of the “quasi-topological” properties of subsets of $X$, or of numerical functions on $X$, with respect to such a capacity $C$. Analogues are obtained to various important properties of the fine topology in potential theory, notably the quasi Lindelöf principle (Doob), the existence of a fine support (Getoor), and the theorem on capacity for decreasing families of sets (Brelot). This analogy becomes an actual identity if a certain compatibility is assumed betweeh the capacity $C$ and a new homology (called “fine”) on $X$. Sufficient conditions are obtained with a convex cone of lower semicontinuous functions on $X$.},
author = {Fuglede, Bent},
journal = {Annales de l'institut Fourier},
keywords = {topology; capacity; fine topology},
language = {eng},
number = {1},
pages = {123-169},
publisher = {Association des Annales de l'Institut Fourier},
title = {The quasi topology associated with a countably subadditive set function},
url = {http://eudml.org/doc/74024},
volume = {21},
year = {1971},
}

TY - JOUR
AU - Fuglede, Bent
TI - The quasi topology associated with a countably subadditive set function
JO - Annales de l'institut Fourier
PY - 1971
PB - Association des Annales de l'Institut Fourier
VL - 21
IS - 1
SP - 123
EP - 169
AB - This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space $X$. The principal aim is the study of the “quasi-topological” properties of subsets of $X$, or of numerical functions on $X$, with respect to such a capacity $C$. Analogues are obtained to various important properties of the fine topology in potential theory, notably the quasi Lindelöf principle (Doob), the existence of a fine support (Getoor), and the theorem on capacity for decreasing families of sets (Brelot). This analogy becomes an actual identity if a certain compatibility is assumed betweeh the capacity $C$ and a new homology (called “fine”) on $X$. Sufficient conditions are obtained with a convex cone of lower semicontinuous functions on $X$.
LA - eng
KW - topology; capacity; fine topology
UR - http://eudml.org/doc/74024
ER -

References

top
  1. [1] M. BRELOT, Eléments de la théorie classique du potentiel. Les cours de Sorbonne. C.D.U., Paris (4e édit. 1969). 
  2. [2] M. BRELOT, Introduction axiomatique de l'effilement, Annali di Mat. 57 (1962), 77-96. Zbl0119.08902MR25 #3187
  3. [3] M. BRELOT, Axiomatique des fonctions harmoniques. Séminaire de mathématiques supérieures, Montréal (1966). Zbl0148.10401
  4. [4] M. BRELOT, Capacity and balayage for decreasing sets. Symposium on Statistics and Probability, Berkeley (1965). Zbl0314.60056
  5. [5] M. BRELOT, Recherches axiomatiques sur un théorème de Choquet concernant l'effilement, Nagoya Math. Journal 30 (1967), 39-46. Zbl0156.12302MR35 #5650
  6. [6] M. BRELOT, On topologies and boundaries in potential theory. (Lectures in Bombay, 1966), Lecture Notes in Math., n° 175. Berlin (1970). Zbl0222.31014
  7. [7] H. CARTAN, Théorie générale du balayage en potentiel newtonien, Ann. Univ. Grenoble 22 (1946), 221-280. Zbl0061.22701MR8,581e
  8. [8] G. CHOQUET, Theory of capacities, Ann. Inst. Fourier 5 (1953), 131-295. Zbl0064.35101MR18,295g
  9. [9] G. CHOQUET, Forme abstraite du théorème de capacitabilité, Ann. Inst. Fourier 9 (1959), 83-89. Zbl0093.29701MR22 #3692b
  10. [10] G. CHOQUET, Sur les points d'effilement d'un ensemble, Ann. Inst. Fourier 9 (1959), 91-101. Zbl0093.29702MR22 #3692c
  11. [11] G. CHOQUET, Démonstration non probabiliste d'un théorème de Getoor, Ann. Inst. Fourier 15, fasc. 2 (1965), 409-414. Zbl0141.30501MR33 #5929
  12. [12] J. L. DOOB, Applications to analysis of a topological definition of smallness of a set, Bull. Amer. Math. Soc. 72 (1966), 579-600. Zbl0142.09001MR34 #3514
  13. [13] B. FUGLEDE, Le théorème du minimax et la théorie fine du potentiel, Ann. Inst. Fourier 15, fasc. 1 (1965), 65-88. Zbl0128.33103MR32 #7781
  14. [14] B. FUGLEDE, Esquisse d'une théorie axiomatique de l'effilement et de la capacité, C.R. Acad. Sci. Paris 261, (1965), 3272-3274. Zbl0192.20401MR32 #7782
  15. [15] B. FUGLEDE, Quasi topology and fine topology. Sém. Brelot-Choquet-Deny : Théorie du potentiel, 10 (1965-1966), n° 12. Zbl0164.14002
  16. [16] B. FUGLEDE, Applications du théorème minimax à l'étude de diverses capacités, C.R. Acad. Sci. Paris 266 (1968), 921-923. Zbl0159.40801MR45 #5395
  17. [17] B. FUGLEDE, Propriétés de connexion en topologie fine. Prépublication, Copenhague (1969). 
  18. [18] R.K. GETOOR, Additive functionals of a Markov process. (Lectures), Hamburg (1964). Zbl0212.20301
  19. [19] C. GOFFMAN, C.J. NEUGEBAUER and T. NISHIURA, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497-505. Zbl0101.15502MR25 #1254
  20. [20] J. RIDDER, Über approximativ stetigen Funktionen, Fund. Math. 13 (1929), 201-209. JFM55.0145.01
  21. [21] S. SAKS, Theory of the Integral, Warsaw (1937). 
  22. [22] A. IONESCU TULCEA and C. IONESCU TULCEA, Topics in the Theory of Lifting, Erg. Math. 48 (1969). Zbl0179.46303MR43 #2185
  23. [23] R.E. ZINK, On semicontinuous functions and Baire functions, Trans. Amer. Math. Soc. 117 (1965), 1-9. Zbl0143.27703MR30 #216

Citations in EuDML Documents

top
  1. Ivan Netuka, Luděk Zajíček, Functions continuous in the fine topology for the heat equation
  2. Gianni Dal Maso, Γ -convergence and μ -capacities
  3. Lars-Inge Hedberg, Thomas H. Wolff, Thin sets in nonlinear potential theory
  4. David R. Adams, John L. Lewis, Fine and quasi connectedness in nonlinear potential theory
  5. Jan Malý, Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points
  6. Juha Heinonen, Terro Kilpeläinen, Olli Martio, Fine topology and quasilinear elliptic equations
  7. Enrico Vitali, Convergence of unilateral convex sets in higher order Sobolev spaces
  8. Jaroslav Lukeš, Luděk Zajíček, When finely continuous functions are of the first class of Baire
  9. Oldřich John, Jan Malý, Jana Stará, Nowhere continuous solutions to elliptic systems

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.