Article
| Title: | Convergence of ap-Henstock-Kurzweil integral on locally compact spaces (English) |
| Author: | Kalita, Hemanta |
| Author: | Agarwal, Ravi P. |
| Author: | Hazarika, Bipan |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 103-121 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We introduce an ap-Henstock-Kurzweil type integral with a non-atomic Radon measure and prove the Saks-Henstock type lemma. The monotone convergence theorem, $\mu _{\rm ap}$-Henstock-Kurzweil equi-integrability, and uniformly strong Lusin condition are discussed. (English) |
| Keyword: | ap-Henstock-Kurzweil integral |
| Keyword: | uniformly strong Lusin condition |
| Keyword: | monotone convergence theorem |
| Keyword: | $\mu _{\rm ap}$-Henstock-Kurzweil equi-integrability |
| Keyword: | Henstock's lemma |
| MSC: | 26A39 |
| MSC: | 28A12 |
| DOI: | 10.21136/CMJ.2023.0450-22 |
| . | |
| Date available: | 2025-03-11T15:57:57Z |
| Last updated: | 2025-03-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152899 |
| . | |
| Reference: | [1] Bongiorno, D., Corrao, G.: An integral on a complete metric measure space.Real Anal. Exch. 40 (2015), 157-178. Zbl 1391.26027, MR 3365396, 10.14321/realanalexch.40.1.0157 |
| Reference: | [2] Bullen, P. S.: The Burkill approximately continuous integral.J. Aust. Math. Soc., Ser. A 35 (1983), 236-253. Zbl 0533.26005, MR 0704431, 10.1017/S1446788700025738 |
| Reference: | [3] Burkill, J. C.: The approximately continuous Perron integral.Math. Z. 34 (1932), 270-278. Zbl 0002.38604, MR 1545252, 10.1007/BF01180588 |
| Reference: | [4] Cao, S. S.: The Henstock integral for Banach-valued functions.Southeast Asian Bull. Math. 16 (1992), 35-40. Zbl 0749.28007, MR 1173605 |
| Reference: | [5] Corrao, G.: An Henstock-Kurzweil Type Integral on a Measure Metric Space: Doctoral Thesis.Universita Degli Studi Di Palermo, Palermo (2013). |
| Reference: | [6] Piazza, L. Di, Marraffa, V., Musia{ł}, K.: Variational Henstock integrability of Banach space valued functions.Math. Bohem. 141 (2016), 287-296. Zbl 1413.26019, MR 3499788, 10.21136/MB.2016.19 |
| Reference: | [7] Edwards, R. E.: Functional Analysis: Theory and Applications.Holt Rinehart and Winston, New York (1965). Zbl 0182.16101, MR 0221256 |
| Reference: | [8] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics 4. AMS, Providence (1994). Zbl 0807.26004, MR 1288751, 10.1090/gsm/004 |
| Reference: | [9] Henstock, R.: Linear Analysis.Butterworths, London (1967). Zbl 0172.39001, MR 0419707 |
| Reference: | [10] Henstock, R.: Lectures on the Theory of Integration.Series in Real Analysis 1. World Scientific, Singapore (1988). Zbl 0668.28001, MR 0963249, 10.1142/0510 |
| Reference: | [11] Henstock, R.: The General Theory of Integration.Oxford Mathematical Monographs. Clarendon Press, Oxford (1991). Zbl 0745.26006, MR 1134656 |
| Reference: | [12] Kalita, H., Hazarika, B.: A convergence theorem for $ap$-Henstock-Kurzweil integral and its relation to topology.Filomat 36 (2022), 6831-6839. MR 4563043, 10.2298/FIL2220831K |
| Reference: | [13] Lee, P.-Y.: Lanzhou Lectures on Henstock Integration.Series in Real Analysis 2. World Scientific, London (1989). Zbl 0699.26004, MR 1050957, 10.1142/0845 |
| Reference: | [14] Lee, P. Y., Výborný, R.: The Integral: An Easy Approach After Kurzweil and Henstock.Australian Mathematical Society Lecture Series 14. Cambridge University Press, Cambridge (2000). Zbl 0941.26003, MR 1756319 |
| Reference: | [15] Leng, N. W.: Nonabsolute Integration on Measure Spaces.Series in Real Analysis 14. World Scientific, Hackensack (2018). Zbl 1392.28001, MR 3752602, 10.1142/10489 |
| Reference: | [16] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability.Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge (1995). Zbl 0819.28004, MR 1333890, 10.1017/CBO9780511623813 |
| Reference: | [17] Mema, E.: Equiintegrability and controlled convergence for the Henstock-Kurzweil integral.Int. Math. Forum 8 (2013), 913-919. Zbl 1285.28017, MR 3069783, 10.12988/imf.2013.13097 |
| Reference: | [18] Park, J. M., Lee, D. H., Yoon, J. H., Kim, B. M.: The convergence theorems for ap-integral.J. Chung. Math. Soc. 12 (1999), 113-118. |
| Reference: | [19] Park, J. M., Oh, J. J., Park, C.-G., Lee, D. H.: The ap-Denjoy and ap-Henstock integrals.Czech. Math. J. 57 (2007), 689-696. Zbl 1174.26308, MR 2337623, 10.1007/s10587-007-0106-0 |
| Reference: | [20] Perron, O.: Über den Integralbegriff.Heidelb. Ak. Sitzungsber. 16 (1914), 1-16 German \99999JFM99999 45.0445.01. |
| Reference: | [21] Shin, K. C., Yoon, J. H.: Properties for AP-Henstock integral.Available at {\let \relax \brokenlink{https://www.researchgate.net/publication/349409139_PROPERTIES_FOR_THE_AP_}{-HENSTOCK_INTEGRAL}} (2021). |
| Reference: | [22] Skvortsov, V. A., Sworowska, T., Sworowski, P.: On approximately continuous integrals (a survey).Traditional and Present-Day Topics in Real Analysis {Ł}odź University Press, {Ł}odź (2013), 233-252. Zbl 1334.26012, MR 3204590 |
| Reference: | [23] Skvortsov, V. A., Sworowski, P.: The AP-Denjoy and AP-Henstock integrals revisited.Czech. Math. J. 62 (2012), 581-591. Zbl 1265.26019, MR 2984620, 10.1007/s10587-012-0050-5 |
| Reference: | [24] Soedijono, B., Lee, P. Y., Chew, T. S.: The Kubota Integral and Beyond.NUS Research Report 389. National University of Singapur, Singapur (1989). MR 1095739 |
| Reference: | [25] Wiener, N.: Generalized harmonic analysis.Acta Math. 55 (1930), 117-258 \99999JFM99999 56.0954.02. MR 1555316, 10.1007/BF02546511 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
