Integrals and Banach spaces for finite order distributions

Let $\text{Bc}$ denote the real-valued functions continuous on the extended real line and vanishing at $-\mathrm{\infty }$. Let $\text{Br}$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\mathrm{\infty }$. Define $\text{acn}$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\text{Bc}$. Similarly with $\text{arn}$ from $\text{Br}$. A type of integral is defined on distributions in $\text{acn}$ and $\text{arn}$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb{N}$, the spaces $\text{acn}$ and $\text{arn}$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\text{Bc}$ and $\text{Br}$, 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 ${\text{ac}}^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space ${\text{ar}}^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: H"older inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.