The Distributional Denjoy Integral

Abstract

Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F ′ = f (distributional derivative) then the distributional integral of f is defined as. It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock-Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, Hölder inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy's descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed. 2000 subject classification: 26A39, 46B42, 46E15, 46F05, 46G12

The distributional Henstock-Kurzweil integral and applications II

“…The distributional Henstock-Kurzweil integral is very wide and it includes the integrals of Riemann, Lebesgue, Henstock-Kurzweil, restricted and wide Denjoy (see [14,21,22]). …”

“…Under the Alexiewicz norm, D HK is a Banach space, see [21,Theorem 2]. In [5], the author first proved that the completion, under the Alexiewicz norm, of the family of all Henstock-Kurzweil integrable functions in [a, b], is the space …”

Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral

“…With this definition, this integral comprises Riemann, Lebesgue, Henstock-Kurzweil, Perron, Denjoy, and improper integrals as special cases ( [10,14,19,20,23]). …”

“…Integrals defined in the same way have also been proposed in other papers. For example, Ang et al [2] defined it in the plane and called it the G-integral, and Talvila [23] defined the A C -integral on the extended real line. In that case of integration over one-dimensional interval, these two integrals coincide.…”

The L p primitive integral

“…The distributions integrable in this sense are the weak derivative of L p functions but have many properties similar to L p functions. This approach was followed in [16] with the continuous primitive integral. The primitives were functions continuous on the extended real line.…”