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.
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.
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.
In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.
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.
We make some comments on the problem of how the Henstock-Kurzweil integral extends the McShane integral for vector-valued functions from the descriptive point of view.
In this paper we prove an existence theorem for the Cauchy problem \[ x^{\prime }(t) = f(t, x(t)), \quad x(0) = x_0, \quad t \in I_{\alpha } = [0, \alpha ] \] using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function $f$ are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function $f$ satisfies some conditions expressed in terms of measures of weak noncompactness.
We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a $\sigma $-finite outer regular quasi Radon measure space $(S,\Sigma ,\mathcal {T},\mu )$ into a Banach space $X$ and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function $f$ from $S$ into $X$ is weakly McShane integrable on each measurable subset of $S$ if and only if it is Pettis and weakly McShane integrable on $S$. On the other hand, we prove that if an $X$-valued function is weakly McShane integrable on $S$, then it is Pettis integrable on each member of an increasing sequence $(S_\ell )_{\ell \geq 1}$ of measurable sets of finite measure with union $S$. For weakly sequentially complete spaces or for spaces that do not contain a copy of $c_0$, a weakly McShane integrable function on $S$ is always Pettis integrable. A class of functions that are weakly McShane integrable on $S$ but not Pettis integrable is included.
We study the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions f given by ∑ ∞ n=1 xnχEn , where xn are points of a Banach space and the sets En 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 ∑∞ n=1 xn|En| 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.