In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral ${\int }_a^{b}f\mathrm{d}g$ exists if $f\in {\mathrm{B}V}_{\phi }[a,b]$, $g\in {\mathrm{B}V}_{\psi }[a,b]$ and ${\sum }_{n=1}^{\infty }{\phi }^{-1}(1/n){\psi }^{-1}(1/n)<\infty$. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This...

This paper generalizes the results of papers which deal with the Kurzweil-Henstock construction of an integral in ordered spaces. The definition is given and some limit theorems for the integral of ordered group valued functions defined on a Hausdorff compact topological space $T$ with respect to an ordered group valued measure are proved in this paper.

Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities...

We propose an extended version of the Kurzweil integral which contains both the Young and the Kurzweil integral as special cases. The construction is based on a reduction of the class of $\delta$-fine partitions by excluding small sets.

The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive...

We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^r$-derivatives are integrable in this method.

A function $f:ℝ\to ℝ$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers ${\alpha }_0,...,{\alpha }_{n-1}$ such that $s^{n+1}{\int }_0^{\delta }{e}^{-st}[f(x+t)-{\sum }_{i=0}^{n-1}{\alpha }_i{t}^i/i!]dt$ converges as $s\to +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }\left(x\right)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized...

In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10],...

In this paper, we define the $M_{\alpha }$-integral of real-valued functions defined on an interval $[a,b]$ and investigate important properties of the $M_{\alpha }$-integral. In particular, we show that a function $f:[a,b]\to R$ is $M_{\alpha }$-integrable on $[a,b]$ if and only if there exists an $AC{G}_{\alpha }$ function $F$ such that $F^{\text{'}}=f$ almost everywhere on $[a,b]$. It can be seen easily that every McShane integrable function on $[a,b]$ is $M_{\alpha }$-integrable and every $M_{\alpha }$-integrable function on $[a,b]$ is Henstock integrable. In addition, we show that the $M_{\alpha }$-integral is equivalent to the $C$-integral....