Some full descriptive characterizations of the Henstock-Kurzweil integral in the Euclidean space

Lee Tuo-Yeong

Displaying similar documents to “Some full descriptive characterizations of the Henstock-Kurzweil integral in the Euclidean space”

A measure-theoretic characterization of the Henstock-Kurzweil integral revisited

Tuo-Yeong Lee (2008)

Czechoslovak Mathematical Journal

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In this paper we show that the measure generated by the indefinite Henstock-Kurzweil integral is $F_{σ δ}$ regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.

Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrability of strongly measurable functions

B. Bongiorno, Luisa Di Piazza, Kazimierz Musiał (2006)

Mathematica Bohemica

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We study the integrability of Banach valued strongly measurable functions defined on $[0, 1]$ . In case of functions $f$ given by $\sum_{n = 1}^{\infty} x_{n} χ_{E_{n}}$ , where $x_{n}$ belong to a Banach space and the sets $E_{n}$ are Lebesgue measurable and pairwise disjoint subsets of $[0, 1]$ , there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.

Banach-valued Henstock-Kurzweil integrable functions are McShane integrable on a portion

Tuo-Yeong Lee (2005)

Mathematica Bohemica

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It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$ -dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f {[0, 1]}^{2} ⟶ ℝ$ and a continuous function $F {[0, 1]}^{2} ⟶ ℝ$ such that $(¶) \int_{0}^{x} \{(¶) \int_{0}^{y} f (u, v) d v\} d u = (¶) \int_{0}^{y} \{(¶) \int_{0}^{x} f (u, v) d u\} d v = F (x, y)$ for all $(x, y) \in {[0, 1]}^{2}$ .

Some characterizations of the primitive of strong Henstock-Kurzweil integrable functions

Guoju Ye (2006)

Mathematica Bohemica

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In this paper we give some complete characterizations of the primitive of strongly Henstock-Kurzweil integrable functions which are defined on $ℝ^{m}$ with values in a Banach space.