The regulated primitive integral

Abstract

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 $\int_{(a,b)}f=F(b-)-F(a+)$, 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.