Variational Henstock integrability of Banach space valued functions
We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum\nolimits_{n=1}^{\infty} x_n\chi_{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets o...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2016-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/141/2/mb141_2_10.pdf |
| _version_ | 1818595575233249280 |
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| author | Luisa Di Piazza Valeria Marraffa Kazimierz Musiał |
| author_facet | Luisa Di Piazza Valeria Marraffa Kazimierz Musiał |
| author_sort | Luisa Di Piazza |
| collection | DOAJ |
| description | We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum\nolimits_{n=1}^{\infty} x_n\chi_{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum\nolimits_{n=1}^{\infty}x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a certain class of such functions. |
| first_indexed | 2024-12-16T11:18:12Z |
| format | Article |
| id | doaj.art-88efb5f085214ab6bc0c4486ba38c2de |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-16T11:18:12Z |
| publishDate | 2016-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-88efb5f085214ab6bc0c4486ba38c2de2022-12-21T22:33:32ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362016-07-01141228729610.21136/MB.2016.19MB.2016.19Variational Henstock integrability of Banach space valued functionsLuisa Di PiazzaValeria MarraffaKazimierz MusiałWe study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum\nolimits_{n=1}^{\infty} x_n\chi_{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum\nolimits_{n=1}^{\infty}x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a certain class of such functions.http://mb.math.cas.cz/full/141/2/mb141_2_10.pdf Kurzweil-Henstock integral variational Henstock integral Pettis integral |
| spellingShingle | Luisa Di Piazza Valeria Marraffa Kazimierz Musiał Variational Henstock integrability of Banach space valued functions Mathematica Bohemica Kurzweil-Henstock integral variational Henstock integral Pettis integral |
| title | Variational Henstock integrability of Banach space valued functions |
| title_full | Variational Henstock integrability of Banach space valued functions |
| title_fullStr | Variational Henstock integrability of Banach space valued functions |
| title_full_unstemmed | Variational Henstock integrability of Banach space valued functions |
| title_short | Variational Henstock integrability of Banach space valued functions |
| title_sort | variational henstock integrability of banach space valued functions |
| topic | Kurzweil-Henstock integral variational Henstock integral Pettis integral |
| url | http://mb.math.cas.cz/full/141/2/mb141_2_10.pdf |
| work_keys_str_mv | AT luisadipiazza variationalhenstockintegrabilityofbanachspacevaluedfunctions AT valeriamarraffa variationalhenstockintegrabilityofbanachspacevaluedfunctions AT kazimierzmusiał variationalhenstockintegrabilityofbanachspacevaluedfunctions |