Mise en Scène: The Last Chapter

I have come to the end, but believe it is only the beginning; there is hard work ahead, and much to be discovered. Questions I could not answer have been scattered throughout the book, and in this chapter I look back, assess what has been done, and try to point to future lines. I will recall various notions and results from previous chapters, and expect readers (if there are any left…) to be familiar with them.

In §2, we outline rudiments of measure theory in fractional dimensions. Grothendieck inequality-type issues surface naturally in this context. The relevant chapters are IV, V, VI, VIII, IX, XII, and XIII.

In §3, combinatorial dimension – a basic gauge of interdependence – is cast in general topological and measurable settings. There are other notions of dimension, and questions regarding connections between these and combinatorial dimension lead to interesting problems. Relevant chapters are XII and XIII.

In §4, we reexamine basic structures underlying classical harmonic analysis. Standard textbook harmonic analysis starts and continues with Borel measures and their transforms, but going further one could start with finitely additive set-functions, and, in the spirit of §2, follow a course where the space of measures is but a first stop. Relevant chapters are VII, XII, and XIII.