A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from . The theory is recovered together with a few new results.
A short approach to the Kurzweil-Henstock integral is outlined, based on approximating a real function on a compact interval by suitable step-functions, and using filterbase convergence to define the integral. The properties of the integral are then easy to establish.
Let $[ a,b] $ be an interval in $\mathbb R$ and let $F$ be a real valued function defined at the endpoints of $[a,b]$ and with a certain number of discontinuities within $[ a,b] $. Assuming $F$ to be differentiable on a set $[ a,b] \backslash E$ to the derivative $f$, where $E$ is a subset of $[ a,b] $ at whose points $F$ can take values $\pm \infty $ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $\mathcal {KH}\hbox {\rm -vt}\int _a^bf=F( b) -F( a) $, where $\mathcal {KH}\hbox {\rm -vt}$ denotes the total value of the {\it Kurzweil-Henstock} integral. The paper ends with a few examples that illustrate the theory.
We study the integrability of Banach valued strongly measurable functions defined on [0, 1]. In case of functions f given by ∞∑ n=1 xnχEn , where xn belong to a Banach space and the sets En 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.
A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.
Some observations concerning McShane type integrals are collected. In particular, a simple construction of continuous major/minor functions for a McShane integrand in Rn is given.
Equiintegrability in a compact interval E may be defined as a uniform integrability property that involves both the integrand fn and the corresponding primitive Fn. The pointwise convergence of the integrands fn to some f and the equiintegrability of the functions fn together imply that f is also integrable with primitive F and that the primitives Fn converge uniformly to F. In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers E. Cabral and P. Y. Lee (2001/2002) is revisited. Under the assumption of pointwise convergence of the integrands fn, the three uniform integrability properties, namely equiintegrability and the two versions of the uniform double Lusin condition, are all equivalent. The first version of the double Lusin condition and its corresponding uniform double Lusin convergence theorem are also extended into the division space.
In this paper a full totalization is presented of the Kurzweil-Henstock integral in the multidimensional space. A residual function of the total Kurzweil-Henstock primitive is defined.
The McShane and Kurzweil-Henstock integrals for functions taking values in a locally convex space are defined and the relations with other integrals are studied. A characterization of locally convex spaces in which Henstock Lemma holds is given.