Functions with Bounded Variation and Absolutely Continuous Functions

The Riemann integral can be considered an evolution of Cauchy’s integral, in that certain functions that are not integrable according to Cauchy become integrable in Riemann’s theory. At the same time, alas, in the new framework integration is no longer the inverse operation to differentiation. Thus the fundamental theorem of calculus, in the version for continuous maps proved by Cauchy, loses its status of calculus’ highest pinnacle and becomes a mere special case of a much bigger picture. The existence of continuous maps with no derivative, of integrable functions whose integral map is not differentiable and the ensuing demise of the fundamental theorem of Cauchy’s integral calculus, persuaded many mathematicians, most notably Lebesgue, to investigate the relationship between integrals and primitives. In particular Lebesgue observed that the issues with integral calculus arise when the derivative f is not bounded. Lebesgue showed that for a function f to be summable the corresponding primitive F must have bounded variation. The idea of functions with bounded variation had in the meantime been elaborated by Jordan for others reasons. Yet, Lebesgue stopped short of saying $$|F(x)-F(a)|=\int _{[a, x]} |f(t)|\ell (dt)$$ for every \(x\in [a,b]\), because in that case the difference of the two sides would be a monotone map with bounded variation and zero derivative \(\ell \)-almost everywhere on [a, x]. He proved that among functions with bounded variation, the only ones satisfying $$F(x)-F(a)=\int _{[a, x]}f(t)\ell (dt)$$ are the absolutely continuous functions, as defined by Giuseppe Vitali. This Chapter is therefore devoted to the study of functions with bounded variation and of absolutely continuous functions.