Differentiation and Integration

In this chapter, I link integration, Radon–Nikodym derivative, and Fréchet derivative together that allow us to go smoothly from one thing to the other. I present generalized Carathéodory Extension Theorem 12.4 and introduce the concept of isomeasure. Produce measure spaces are studied in Sect. 12.3. Then, Tonelli’s theorem 12.29 and Fubini’s theorems are presented. I introduce the concept of a region Ω in ℝ m $$\mathbb {R}^m$$ and its principal P ( Ω ) $$\operatorname {\mathrm {P}}(\varOmega )$$ , which are required to define the functions of locally bounded variation or bounded variation. Theorem 12.50 and Propositions 12.51 and 12.52 establish relationship between the Banach space valued measure on ℝ m $$\mathbb {R}^m$$ and the Banach space valued cumulative distribution functions that are of locally bounded variation. Propositions 12.54 and 12.55 give conditions when a function of locally bounded variation or a cumulative distribution function of a Banach space valued measure is measurable on ℝ m $$\mathbb {R}^m$$ . This yields the definition of the Lebesgue–Stieltjes Integral. I then define absolutely continuity for Banach space valued functions of rectangles of ℝ m $$\mathbb {R}^m$$ . Then, I show that their link to absolutely continuity of Banach space valued measures with respect to the m-dimensional Borel measure μBm. Then, the generalized fundamental theorems of calculus I & II are presented together with Integration by Parts Theorem 12.89, the Change of Variable Theorem 12.91, and the Iterated Integral Theorem 12.127. The chapter ends with some preliminary results on the topic of manifolds of Banach space variant.