If f is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of f is || f || = sup I | f I f| where the supremum is taken over all intervals I ⊂ . Define the translation τx by τxf(y) = f(y − x). Then ||τxf − f || tends to 0 as x tends to 0, i.e., f is continuous in the Alexiewicz norm. For particular functions, ||τxf − f || can tend to 0 arbitrarily slowly. In general, ||τxf − f || ≥ osc f|x| as x → 0, where osc f is the oscillation of f. It is shown that if F is a primitive of f then ||τxF − F || || ≤ ||f || |x|. An example shows that the function y → τxF(y) − F(y) need not be in L 1 . However, if f ∈ L 1 then || τxF − Fk1 || ≤ || f ||1|x|. For a positive weight function w on the real line, necessary and sufficient conditions on w are given so that ||(τxf − f)w || → 0 as x → 0 whenever fw is Henstock-Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock-Kurzweil integrable functions as a subspace of Schwartz distributions.
Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty $. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty $. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.