In this paper, we define a $$-integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure $\mu$ is compatible with its topology in the sense that every open set is $\mu$-measurable. We prove that the $$-integral is equivalent to $\mu$-integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.

We give a Riemann-type definition of the double Denjoy integral of Chelidze and Djvarsheishvili using the new concept of $CD$ filtering.

A short approach to the Kurzweil-Henstock integral is outlined, based on approximating a real function on a compact interval by suitable step-functions, and using filterbase convergence to define the integral. The properties of the integral are then easy to establish.

Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in ${ℝ}^n$, we are led to an integration process over quite general sets $A\subseteq q{ℝ}^n$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [4]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In [3] Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig’s integral the Dushnik interior integral...

Using the concept of the ${\mathrm{H}}_1$-integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.

We introduce an axiomatic approach to the theory of non-absolutely convergent integrals. The definition of our ν-integral will be descriptive and depends mainly on characteristic null conditions. By specializing our concepts we will later obtain concrete theories of integration with natural properties and very general versions of the divergence theorem.

It is commonly known that absolute gauge integrability, or Henstock-Kurzweil (H-K) integrability implies Lebesgue integrability. In this article, we are going to present another proof of that fact which utilizes the basic definitions and properties of the Lebesgue and H-K integrals.

For a new Perron-type integral a concept of convergence is introduced such that the limit $f$ of a sequence of integrable functions $f_k$, $k\in \mathbb{N}$ is integrable and any integrable $f$ is the limit of a sequence of stepfunctions $g_k$, $k\in \mathbb{N}$.

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 $$\left(\P \right){\int }_0^x\left\{\left(\P \right){\int }_0^{y}f(u,v)\mathrm{d}v\right\}\mathrm{d}u=\left(\P \right){\int }_0^y\left\{\left(\P \right){\int }_0^{x}f(u,v)\mathrm{d}u\right\}\mathrm{d}v=F(x,y)$$ for all $(x,y)\in {[0,1]}^2$.

In the present paper, we investigate the existence of solutions to boundary value problems for the one-dimensional Schrödinger equation $-y^{\text{'}\text{'}}+qy=f$, where $q$ and $f$ are Henstock-Kurzweil integrable functions on $[a,b]$. Results presented in this article are generalizations of the classical results for the Lebesgue integral.

Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.

Let $[a,b]$ be an interval in $ℝ$ and let $F$ be a real valued function defined at the endpoints of $[a,b]$ and with a certain number of discontinuities within $[a,b]$. Assuming $F$ to be differentiable on a set $[a,b]\setminus E$ to the derivative $f$, where $E$ is a subset of $[a,b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $\mathrm{𝒦\mathscr{H}}\text{-vt}{\int }_a^{b}f=F\left(b\right)-F\left(a\right)$, where $\mathrm{𝒦\mathscr{H}}\text{-vt}$ denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate...