Geodesic
On Kurzweil-Henstock equiintegrable sequences
Mathematica Bohemica, Tome 121 (1996) no. 2, pp. 189-207

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
For the Kurzweil-Henstock integral the equiintegrability of a pointwise convergent sequence of integrable functions implies the integrability of the limit function and the relation \lim_{m \to\infty}\int_a^bf_m(s)\dd s = \int_a^b\lim_{m \to\infty}f_m(s)\dd s. Conditions for the equiintegrability of a sequence of functions pointwise convergent to an integrable function are presented. These conditions are given in terms of convergence of some sequences of integrals.
For the Kurzweil-Henstock integral the equiintegrability of a pointwise convergent sequence of integrable functions implies the integrability of the limit function and the relation \lim_{m \to\infty}\int_a^bf_m(s)\dd s = \int_a^b\lim_{m \to\infty}f_m(s)\dd s. Conditions for the equiintegrability of a sequence of functions pointwise convergent to an integrable function are presented. These conditions are given in terms of convergence of some sequences of integrals.
DOI : 10.21136/MB.1996.126102
Classification : 26A39
Keywords: equiintegrable sequence; Kurzweil-Henstock integral 
Schwabik, Štefan; Vrkoč, Ivo. On Kurzweil-Henstock equiintegrable sequences. Mathematica Bohemica, Tome 121 (1996) no. 2, pp. 189-207. doi: 10.21136/MB.1996.126102
@article{10_21136_MB_1996_126102,
     author = {Schwabik, \v{S}tefan and Vrko\v{c}, Ivo},
     title = {On {Kurzweil-Henstock} equiintegrable sequences},
     journal = {Mathematica Bohemica},
     pages = {189--207},
     year = {1996},
     volume = {121},
     number = {2},
     doi = {10.21136/MB.1996.126102},
     mrnumber = {1400612},
     zbl = {0863.26009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.126102/}
}
TY  - JOUR
AU  - Schwabik, Štefan
AU  - Vrkoč, Ivo
TI  - On Kurzweil-Henstock equiintegrable sequences
JO  - Mathematica Bohemica
PY  - 1996
SP  - 189
EP  - 207
VL  - 121
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.126102/
DO  - 10.21136/MB.1996.126102
LA  - en
ID  - 10_21136_MB_1996_126102
ER  - 
%0 Journal Article
%A Schwabik, Štefan
%A Vrkoč, Ivo
%T On Kurzweil-Henstock equiintegrable sequences
%J Mathematica Bohemica
%D 1996
%P 189-207
%V 121
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.126102/
%R 10.21136/MB.1996.126102
%G en
%F 10_21136_MB_1996_126102

[1] Gordon R. A.: Another look at a convergence theorem foг the Henstock integгal. Real Analysis Exchange 15 (1989-90), 724-728. | DOI | MR

[2] Gordon R. A.: A general convergence theorem for non-absolute integгals. J. London Math. Soc. 44 (1991), 301-309. | DOI | MR

[3] Gordon R. A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Graduate Studies in Math., Vol. 4, American Mathematical Society, 1994. | DOI | MR | Zbl

[4] Henstock R.: Lectures on the Theory of Integration. Series in Real Analysis, Vol. 1, World Scientific, Singapore, 1988. | MR | Zbl

[5] Kurzweil J.: Nichtabsolut konvergente Integrale. Teubner-Texte zur Mathematik, Band 26, Teubner, Leipzig, 1980. | MR | Zbl

[6] Kurzweil J., Jarník J.: Equiintegrability and controlled convergence of Perron-type integrable functions. Real Anal. Exchange П (1991-92), 110-139.

[7] Lee Peng Yee: Lanzhou Lectures on Henstock Integration. Series in Real Analysis, Vol. 2, World Scientific, Singapore, 1989. | MR | Zbl

[8] McLeod R. M.: The Generalized Riemann Integral. Caгus Mathematical Monographs, No. 20, Mathematical Association of America, 1980. | MR | Zbl

[9] Schwabik Š.: Generalized Ordinary Differential Equations. Series in Real Analysis, Vol. 5, World Scientific, Singapore, 1992. | MR | Zbl

[10] Schwabik Š.: Convergence theorems for the Perron integral and Sklyarenko's condition. Comment. Math. Univ. Carolin. 33,2 (1992), 237-244. | MR | Zbl

Cité par Sources :