Article
Full entry |
PDF
(1.2 MB)
Feedback
PDF
(1.2 MB)
Feedback
Keywords:
equiintegrable sequence; Kurzweil-Henstock integral
equiintegrable sequence; Kurzweil-Henstock integral
Summary:
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.
References:
[1] Gordon R. A.: Another look at a convergence theorem foг the Henstock integгal. Real Analysis Exchange 15 (1989-90), 724-728. DOI 10.2307/44152048 | MR 1059433
[2] Gordon R. A.: A general convergence theorem for non-absolute integгals. J. London Math. Soc. 44 (1991), 301-309. DOI 10.1112/jlms/s2-44.2.301 | MR 1136442
[3] Gordon R. A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Graduate Studies in Math., Vol. 4, American Mathematical Society, 1994. DOI 10.1090/gsm/004/09 | MR 1288751 | Zbl 0807.26004
[4] Henstock R.: Lectures on the Theory of Integration. Series in Real Analysis, Vol. 1, World Scientific, Singapore, 1988. MR 0963249 | Zbl 0668.28001
[5] Kurzweil J.: Nichtabsolut konvergente Integrale. Teubner-Texte zur Mathematik, Band 26, Teubner, Leipzig, 1980. MR 0597703 | Zbl 0441.28001
[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 1050957 | Zbl 0699.26004
[8] McLeod R. M.: The Generalized Riemann Integral. Caгus Mathematical Monographs, No. 20, Mathematical Association of America, 1980. MR 0588510 | Zbl 0486.26005
[9] Schwabik Š.: Generalized Ordinary Differential Equations. Series in Real Analysis, Vol. 5, World Scientific, Singapore, 1992. MR 1200241 | Zbl 0781.34003
[10] Schwabik Š.: Convergence theorems for the Perron integral and Sklyarenko's condition. Comment. Math. Univ. Carolin. 33,2 (1992), 237-244. MR 1189654 | Zbl 0774.26004
Search
Partner of
