A weak form of the Henstock Lemma for the PoU-integrable functions is given. This allows to prove the existence of a scalar Volterra derivative for the PoU-integral. Also the PoU-integrable functions are characterized by means of Pettis integrability and a condition involving finite pseudopartitions.
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.
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.
The McShane integral of functions f : I → ! defined on an m-dimensional interval I is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.
R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f\colon [0,1]\rightarrow X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C(K)$ spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces $K$, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from $[0,1]$ into $C(K)$ in McShane sense.
We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name $S^{*}$-integral. Our main result states that $S^{*}$-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).
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.
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.