Geodesic
A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem
Canadian journal of mathematics, Tome 43 (1991) no. 3, pp. 526-539

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.
DOI : 10.4153/CJM-1991-032-8
Mots-clés : 26A39, 26B30, 26A36, Perron integral, fundamental theorem, fundamental inequality, absolute continuity, weak variation, null condition 
Jurkat, W. B.; Knizia, R. W. A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem. Canadian journal of mathematics, Tome 43 (1991) no. 3, pp. 526-539. doi: 10.4153/CJM-1991-032-8
@article{10_4153_CJM_1991_032_8,
     author = {Jurkat, W. B. and Knizia, R. W.},
     title = {A {Characterization} of {Multi-Dimensional} {Perron} {Integrals} and the {Fundamental} {Theorem}},
     journal = {Canadian journal of mathematics},
     pages = {526--539},
     year = {1991},
     volume = {43},
     number = {3},
     doi = {10.4153/CJM-1991-032-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-032-8/}
}
TY  - JOUR
AU  - Jurkat, W. B.
AU  - Knizia, R. W.
TI  - A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 526
EP  - 539
VL  - 43
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-032-8/
DO  - 10.4153/CJM-1991-032-8
ID  - 10_4153_CJM_1991_032_8
ER  - 
%0 Journal Article
%A Jurkat, W. B.
%A Knizia, R. W.
%T A Characterization of Multi-Dimensional Perron Integrals and the Fundamental Theorem
%J Canadian journal of mathematics
%D 1991
%P 526-539
%V 43
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-032-8/
%R 10.4153/CJM-1991-032-8
%F 10_4153_CJM_1991_032_8

[1] 1. Bauer, H., Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen, Monatsh. Math. 26 (1915), 153–198. [FM 45, 446] Google Scholar

[2] 2. Besicovitch, A.S., On sufficient conditions for a function to be analytic, and on behaviour of analytic functions in the neighborhood of non-isolated singular point, Proc. London Math. Soc. 32(1931), 1–9. Google Scholar

[3] 3. Henstock, R., The use of convergence factors in Ward integration, Proc. London Math. Soc. 10(1960), 107–121. [MR 22#12197] Google Scholar

[4] 4. Henstock, R., Theory of integration. Butterworths, London, 1963. [MR 28 #1274] Google Scholar

[5] 5. Henstock, R., Linear Analysis. Butterworths, London, 1967. [MR 54 #7725] Google Scholar

[6] 6. Henstock, R., Integration by parts, Aequationes Math. 9(1973), 1–18. [MR 47 #3608] Google Scholar

[7] 7. Henstock, R., Integration, variation and differentiation in division spaces, Proc. Roy. Irish Acad. 78(1978), 69–85. [MR 80d:26011] Google Scholar

[8] 8. Henstock, R., The variation on the real line, Proc. Roy. Irish Acad. (1) 79(1979), 10 pp. [MR 81d:26005] Google Scholar

[9] 9. Henstock, R., Generalized Riemann integration and an intrinsic topology, Can. J. Math. 32(1980), 395–413. [MR 82b: 26010] Google Scholar

[10] 10. Henstock, R., Division spaces, vector-valued functions and backwards martingales, Proc. Roy. Irish Acad. 80(1980), 217–232. [MR 82i:60091] Google Scholar

[11] 11. Kempisty, S., Sur les fonctions absolument continues d'intervalle, Fund. Math. 27(1936), 10–37. [FM 62, 248] Google Scholar

[12] 12. Kempisty, S., Sur les fonctions absolument semi-continues, Fund. Math. 30(1938), 104–127. [FM 64, 200] Google Scholar

[13] 13. Kempisty, S., Fonctions d'intervalle non additives, Actual. Sci. Industr. 824(1939), 62 pp. [FM 65, 1175], [MR 1,207] Google Scholar

[14] 14. Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter,, Czech. Math. J. 7(1957), 418–449. [MR 22 #2735] Google Scholar

[15] 15. Kurzweil, J., Nichtabsolut konvergente Intégrale. Teubner, Leipzig, 1980. [MR 82m:26007] Google Scholar

[16] 16. Mařik, J., Základy théorie integráulu v euklidových prostorech, Casopis Pest. Mat. 77(1952), 1–51. 125— 145, 267–301. [MR 15, 691] Google Scholar

[17] 17. Ridder, J., Über stetige, additive Intervallfunktionen in der Ebene und Hire Derivierten, Nieuw. Arch. Wis. (1) 16(1929), 55–69. [FM 55, 149] Google Scholar

[18] 18. Ridder, J., Ein einheitliches Verfahren zur Definition von Absolut-und Bedingt-Konvergenten Integralen,, Indag. Math. 27(1965), 1–13. 14–30. 31–39. 165–177. 365–375. 705–745. [MR 31 #308; 35 #4354; 32 #187; 32 #5836] Google Scholar

[19] 19. Ridder, J., Ein einheitliches Verfahren zur Definition von Absolut-und Bedingt-Konvergenten Integralen,, Indag. Math. 28(1966), 248–257. [MR 33 #5833] Google Scholar

[20] 20. Ridder, J., Ein einheitliches Verfahren zur Definition von Absolut-und Bedingt-Konvergenten Integralen,, Indag. Math. 29(1967), 1–7. 8–17. 305–316. [MR 35 #4355; 36 #323] Google Scholar

[21] 21. Ridder, J., Ein einheitliches Verfahren zur Definition von Absolut-und Bedingt-Konvergenten Integralen,, Indag. Math. 31(1969), 10–17. [MR 39 #1619] Google Scholar

[22] 22. Ridder, J., Äquivalenzen von Integraldefinitionen im Sinne von Denjoy, von Perron und von Riemann, Indag. Math 31(1969), 201–212. [MR 40 #290] Google Scholar

[23] 23. Romanovski, P., Intégrale de Denjoy dans les espaces abstraits, Mat. Sborr», N.S. 9(1941), 67–120. [FM 68, 170], [MR 2, 354] Google Scholar

[24] 24. Romanovski, P., Intǵrale de Denjoy dans l'espace à n dimensions, Mat. Shorn, N.S. 9(1941), 281–307. [FM 68, 172], [MR 2, 354] Google Scholar

[25] 25. Romanovski, P.,Intégrale relative à un réseau, Mat. Shorn. N.S. 9(1941), 309–316. [FM 68, 173], [MR 2, 354] Google Scholar

[26] 26. Saks, S., Theory of the integral, Dover, New York, 1964. [MR 29 #4850], [FM 63, 183] Google Scholar

[27] 27. Thomson, B.S., Constructive definition for non-absolutely convergent integrals, Proc. London Math. Soc. 20(1970), 699–716. [MR 42 #3248] Google Scholar

[28] 28. Thomson, B.S., Construction of measures and integrals, Trans. Amer. Math. Soc. 160(1971), 287–296. [MR 43 #6385] Google Scholar

[29] 29. Trjitzinsky, W.J., Theory of functions of intervals and applications to junctions of a complex variable, J. Math. Pure Appl. 25(1946), 347-395 (1947). [MR 9, 418] Google Scholar

[30] 30. Trjitzinsky, W.J., Totalisation dans les espaces abstraits. Mem. Sci. Math. CLV, Paris, 131 pp., 1963. [MR 28 #4078] Google Scholar

[31] 31. Trjitzinsky, W.J.,La totale-D de Denjoy et la totale-S symétrique. Mem. Sci. Math. CLV, Paris, 105 pp., (1968). [MR 39 #1604] Google Scholar

[32] 32. Trjitzinsky, W.J., Totalisations abstraites dans espaces vectoriels, Ann. Mat. Pura Appl. 82(1969), 275–379. [MR 55 #8274] Google Scholar

[33] 33. Ward, A.J., On the derivation of additive functions of intervals in m-dimensional space, Fund. Math. 28(1937), 265–279. [FM 63,192] Google Scholar

[34] 34. Wheeden, R.L., Zygmund, A., Measure and integral. Dekker, New York, 1977. [MR 58 #11295] Google Scholar

Cité par Sources :