We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, , where is a compact interval of , and are functions with values on and respectively, and and are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, , as well as to unbounded intervals .
The multiplier for the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014) is characterized.
In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.
The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in given by Kurzweil and...
We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a -finite outer regular quasi Radon measure space into a Banach space 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 from into is weakly McShane integrable on each measurable subset of if and only if...
We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R. P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.
We study the integrability of Banach space valued strongly measurable functions defined on . In the case of functions given by , where are points of a Banach space and the sets are Lebesgue measurable and pairwise disjoint subsets of , there are well known characterizations for Bochner and Pettis integrability of . The function is Bochner integrable if and only if the series is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of ....
We study properties of variational measures associated with certain conditionally convergent integrals in . In particular we give a full descriptive characterization of these integrals.
Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a -adic path system that defines a differentiation basis which does not possess Ward property.
In this paper, we prove the existence of continuous solutions of a Volterra integral inclusion involving the Henstock-Kurzweil-Pettis integral. Since this kind of integral is more general than the Bochner, Pettis and Henstock integrals, our result extends many of the results previously obtained in the single-valued setting or in the set-valued case.