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

A function on the real line is called regulated if it has a left limit and a right limit at each point. If f is a Schwartz distribution on the real line such that f = F ′ (distributional or weak derivative) for a regulated function F then the regulated primitive integral of f is (a,b) with similar definitions for other types of intervals. The space of integrable distributions is a Banach space and Banach lattice under the Alexiewicz norm. It contains the spaces of Lebesgue and Henstock-Kurzweil integrable functions as continuous embeddings. It is the completion of the space of signed Radon measures in the Alexiewicz norm. Functions of bounded variation form the dual space and the space of multipliers. The integrable distributions are a module over the functions of bounded variation. Properties such as integration by parts, change of variables, Hölder inequality, Taylor's theorem and convergence theorems are proved. 2000 subject classification: 26A39, 46E15, 46F05, 46G12∞ −∞ f φ defines a distribution. The differentiation formula T ′ , φ = − T, φ ′ ensures that all distributions have derivatives of all orders which are themselves distributions. This is known as the distributional derivative or weak derivative. The formula follows by mimicking integration by parts in the case of T f where f ∈ C 1 . We will usually denote distributional derivatives by F ′ and pointwise derivatives by F ′ (t). For T ∈ D ′ and t ∈ R the translation τ t is defined byMost of the results on distributions we use can be found in [11].It will be shown in Theorem 4 below that primitives are unique and that the spaces A R and B R are isometrically isomorphic, the integral constituting a

The Fourier transform is considered as a Henstock-Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann-Lebesgue lemma fails: Henstock-Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock-Kurzweil integrals and gives an estimate on the integral of a product. 2000 subject classification: 42A38, 26A39

In JR n , we prove uniqueness for the Dirichlet problem in the half space x n > 0, with continuous data, under the growth condition u = o(\x\sec Ί θ) as \x\ ->• oo (x n = |#|cos#, 7 G ffi). Under the natural integral condition for convergence of the Poisson integral with Dirichlet data, the Poisson integral will satisfy this growth condition with 7 = n -1. A PhragmenLindelδf principle is established under this same growth condition. We also consider the Dirichlet problem with data of higher order growth, including polynomial growth. In this case, if u = o(|#| N+1 sec 7 θ) (7 G M, N > 1), we prove solutions are unique up to the addition of a harmonic polynomial of degree N that vanishes when x n = 0.

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ𝒱, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.