This essay outlines a generalized Riemann approach to the analysis of random variation and illustrates it by a construction of Brownian motion in a new and simple manner.
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.
In this paper we give a representation theorem for the orthogonally additive functionals on the space BV in terms of a non-linear integral of the Henstock-Kurzweil-Stieltjes type.
In the paper, we show that the space of functions of bounded variation and the space of regulated functions are, in some sense, the dual space of each other, involving the Henstock-Kurzweil-Stieltjes integral.
In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral R b a f dg exists if f ∈ BVϕ[a, b], g ∈ BVψ[a, b] and ∑∞ n=1 ϕ −1 (1/n)ψ −1 (1/n) < ∞. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.