Math of QM: Appendix on Absolutely Continuous Functions, part 2

Absolutely continuous functions, singular functions, fundamental theorem of calculus for absolutely continuous functions.
This is the last part of the Appendix in the lecture series "Mathematical Aspects of Quantum Mechanics" of Roland Speicher in the winter term 2020/21 at Saarland University, Germany. The Appendix, which is split into 6 parts, provides some basic background information about Lebesgue-Stieltjes integration and absolutely continuous functions.