In this paper we provide a minimal constructive integration process of Riemann
type which includes the Lebesgue integral and also integrates the derivatives of
differentiable functions. We provide a new solution to the classical problem of
recovering a function from its derivative by integration, which, unlike the solution
provided by Denjoy, Perron and many others, does not possess the generality which
is not needed for this purpose.
The descriptive version of the problem was treated by A. M. Bruckner, R. J.
Fleissner and J. Foran in . Their approach was based on the trivial observation that
for the required minimal integral, a function F is the indefinite integral of f if and only
if F' = f almost everywhere and there exists a differentiable function H such that
F – H is absolutely continuous. They strengthen this definition by proving that F – H
can have arbitrary small variation. Nevertheless, their definition still needs a choice
of a differentiable function.