The measure of diversity

The measure of diversity is developed from axiomatic foundations, independently of any statistical considerations, and various examples of such measures cited. It is shown that almost all the more usual measures which have been used in the past are closely related to measures which satisfy our axioms; thus the general theory of diversity is unified and comparisons between the various measures can be made. Brillouin's measure, which does satisfy our axioms, turns out to be the only one satisfying a further axiom. Shannon's function (from information theory) is shown to occupy a central position in our development as the asymptotic limit of average diversity measures. Finally we extend the notion of the measure to allow for various degrees of difference between classes in a set.