ABSTRACT
In this chapter, the author discusses the kinds of properties a function needs to have so that the author have a Fundamental Theorem of Calculus type result in our setting of measures on the real line. He explores a relationship between the Lebesgue-Stieltjes outer measure and Lebesgue outer measure. The author builds a function which is continuous but differentiable nowhere. This is a messy construction, but well worth readers effort. It is also a nice review of series, uniform convergence in a classical analysis setting. The author now gets closer to our Fundamental Theorem of Calculus extension, and proves a very weak form of the Recapture Theorem in Riemann integration.
