The set FJ of all possible genealogies or graphs z of a multiplicative system produced from one particle of type i is here introduced as a fundamental concept in the theory of such systems (see I, II). This set possesses a natural intrinsic distance function d(z, z') under which it is a complete zero-dimensional metric space satisfying the second axiom of countability.
Simple axioms on (A) intervals and (B) measure of intervals are given for an abstract set from which the classical theory of completely additive measure is derived.
Intervals in the set J1 are defined intrinsically and shown to satisfy the axioms A. If now a particular multiplicative system with given generating transformation G(x) is given, the transition probabilities Pl (i;ji,.. ,jt) serve to define a measure for the intervals of 1; satisfying the axioms B. Proof of the latter is non-trivial due to non-local-compactness of the space 17.
With this mathematical structure at hand it becomes possible to state in a simple way some of the striking properties of multiplicative systems.
If x° = G(x° ) is the death-fix-point of G(x), then the set of graphs of Fi which terminate in death has measure xi .
If v = (vl,...,v 1) is the characteristic vector corresponding to the maximal positive characteristic root r > 1 of supercritical system, then the set of all graphs of 1 whose k-th generation population approaches the ratios v:v2:... :vi has measure I - A°. Thus, almost all graphs (genealogies) either terminate in death or approach the mode v as limit. These results are trivial for subcritical systems, in which by definition, 3z = 1.