Kuelbs–Steadman Spaces for Banach Space-Valued Measures

Abstract

We introduce Kuelbs–Steadman-type spaces ( K S p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q -type spaces into K S p spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals.

Non-absolute integrable function spaces on metric measure spaces

“…We denote this spaces as KS 1 w (X ). One can see [2] for details. We will discuss a few fundamental results of KS p (X ) as follows:…”

Kluvánek-Lewis-Henstock integral in a Banach space

“…Definition 2.4. [4] We say that a Σ-measurable function f : T → R is Kluvánek-Lewis-Henstock-Kurzweil µ-integrable, shortly ( (HKL) µ-integrable) if the following properties hold:…”

“…The Henstock-Kurzweil (in short, (HK))-integral for real-valued functions, defined on abstract sets, with respect to (possibly infinite) non-negative measures, readers can see [1,3,10,18,19,22,28,36,37,38,42] and the references among them. Very recently, Boccuto et al [4] introduced the Kluvánek-Lewis ((KL)−)integral for Banach valued spaces. In [21] Gould introduced the integral over vector-valued measures.…”

Modular convergence in $H$-Orlicz spaces of Banach valued functions

“…In quantum theory and nonlinear analysis, HK-integrals are aid for highly oscillatory functions to integrate. Moreover, HK integrability encloses improper integrals (see [3,4,7,9]). Major drawback of Henstock-Kurzweil integrable function space is not complete with Alexiewicz norm.…”