  Let $\Bc$ denote the real-valued functions continuous on the extended real
line and vanishing at $-\infty$. Let $\Br$ denote the functions that are left
continuous, have a right limit at each point and vanish at $-\infty$. Define
$\acn$ to be the space of tempered distributions that are the $n$th
distributional derivative of a unique function in $\Bc$. Similarly with $\arn$
from $\Br$. A type of integral is defined on distributions in $\acn$ and
$\arn$. The multipliers are iterated integrals of functions of bounded
variation. For each $n\in\N$, the spaces $\acn$ and $\arn$ are Banach spaces,
Banach lattices and Banach algebras isometrically isomorphic to $\Bc$ and
$\Br$, 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
$\ac^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The
space $\ar^1$ 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.
