This paper presents a Komlós theorem that extends to the case of the set-valued Henstock-Kurzweil-Pettis integral a result obtained by Balder and Hess (in the integrably bounded case) and also a result of Hess and Ziat (in the Pettis integrability setting). As applications, a solution to a best approximation problem is given, weak compactness results are deduced and, finally, an existence theorem for an integral inclusion involving the Henstock-Kurzweil-Pettis set-valued integral is obtained.
In this paper we show that the measure generated by the indefinite Henstock-Kurzweil integral is $F_{\sigma \delta }$ regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.
It is shown that if g is of bounded variation in the sense of Hardy-Krause on ∏m i=1 [ai , bi ], then gχ ∏m i=1 (ai ,bi ) is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.
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
(P)∫ x0{ (P) ∫ y 0 f(u, v) dv } du = (P) ∫ y 0 { (P) ∫ x 0 f(u, v) du } dv = F(x, y) for all (x, y) ∈ [0, 1]2.
Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.
If f is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of f is || f || = sup I | f I f| where the supremum is taken over all intervals I ⊂ . Define the translation τx by τxf(y) = f(y − x). Then ||τxf − f || tends to 0 as x tends to 0, i.e., f is continuous in the Alexiewicz norm. For particular functions, ||τxf − f || can tend to 0 arbitrarily slowly. In general, ||τxf − f || ≥ osc f|x| as x → 0, where osc f is the oscillation of f. It is shown that if F is a primitive of f then ||τxF − F || || ≤ ||f || |x|. An example shows that the function y → τxF(y) − F(y) need not be in L 1 . However, if f ∈ L 1 then || τxF − Fk1 || ≤ || f ||1|x|. For a positive weight function w on the real line, necessary and sufficient conditions on w are given so that ||(τxf − f)w || → 0 as x → 0 whenever fw is Henstock-Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock-Kurzweil integrable functions as a subspace of Schwartz distributions.
In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.
In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.
When a real-valued function of one variable is approximated by its $n$th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock-Kurzweil integrable. When the only assumption is that $f^{(n)}~$ is Henstock-Kurzweil integrable then a modified form of the $n$th degree Taylor polynomial is used. When the only assumption is that $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.
This note contains a simple example which does clearly indicate the differences in the Henstock-Kurzweil, McShane and strong McShane integrals for Banach space valued functions.