The aim of this chapter is to present some basic mathematical tools on which many constructions in the subsequent chapters depend.

Thus we will often refer to what we call the ‘Cavalieri principle’. We try to revive this old familiar name because of the surprising frequency with which the transformations Cavalieri considered about 350 years ago occur in integral geometry.

No less useful will be the principles which we call ‘Lebesgue factorization’ and ‘Haar factorization’. The first is a rather simple corollary of a well-known fact that in ℝn there is only one (up to a constant factor) locally-finite measure which is invariant with respect to shifts of ℝn, namely the Lebesgue measure. Haar factorization is a similar corollary of a much more general theorem of uniqueness of Haar measures on topological groups. We use the two devices in the construction of Haar measures on groups starting from Haar measures on subgroups.

Integral geometry binds together such notions as metrics, convexity and measures, and these interconnections remain significant throughout the book; §§1.7 and 1.8 are introductory to this topic.

The Cavalieri principle

The classical Cavalieri principle in two dimensions can be formulated as follows.

Let D1 and D2 be two domains in a plane (see fig. 1.1.1).

If for each value of y the length of the chords X1 and X2 coincide, then the areas of D1 and D2 are equal.