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Kempisty's theorem for the integral product quasicontinuity

Zbigniew Grande (2006)

Colloquium Mathematicae

A function f: ℝⁿ → ℝ satisfies the condition Q i ( x ) (resp. Q s ( x ) , Q o ( x ) ) at a point x if for each real r > 0 and for each set U ∋ x open in the Euclidean topology of ℝⁿ (resp. strong density topology, ordinary density topology) there is an open set I such that I ∩ U ≠ ∅ and | ( 1 / μ ( U I ) ) U I f ( t ) d t - f ( x ) | < r . Kempisty’s theorem concerning the product quasicontinuity is investigated for the above notions.

Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrability of strongly measurable functions

B. Bongiorno, Luisa Di Piazza, Kazimierz Musiał (2006)

Mathematica Bohemica

We study the integrability of Banach valued strongly measurable functions defined on [ 0 , 1 ] . In case of functions f given by n = 1 x n χ E n , where x n belong to a Banach space and the sets E n 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.

Kurzweil-Henstock type integral on zero-dimensional group and some of its applications

Valentin Skvortsov, Francesco Tulone (2008)

Czechoslovak Mathematical Journal

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.

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