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...
| Main Authors: | , , |
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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 |
| Summary: | 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. |
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| ISSN: | 0862-7959 2464-7136 |