The symmetry properties of Haar measures and invariant measures on homogeneous spaces have been exploited in various branches of mathematics in the past. The title ‘Measures with symmetry properties’ of the book indicates, that this book treats measures which are invariant under group actions, i.e., more precisely, it allows measures which in general have a lower degree of symmetry than Haar measures or invariant measures on homogeneous spaces. Clearly such measures are no longer uniquely determined, but this weakening of the symmetry properties combined with a more general framework opens an amazingly wide range of applications, which is peculiar to this work. On the other hand it affords harder mathematical work going deeper as before. It should be said at once that the latter does not mean that this book addresses only to pure or applied mathematicians, but also to computer scientists, statisticians, and engineers. For this reason great care is given to make the book easy accessible to readers from a great variety of different backgrounds in keeping the mathematical prerequisites to a minimum and by making the exposition essentially self-contained. For a better understanding basic definitions and theorems are always illustrated by simple examples familiar from calculus.
The mathematical core of this book is condensed in one single chapter which can be skipped at a first reading by those interested only in applications. As the main part the chapter on applications occupies about two third of the whole treatise and considers problems on integration, stochastic simulation, and statistics. The examples range from computational geometry, Grassmannian manifolds, compact connected Lie groups, applied mathematics, computer aided graphic design to coding theory. All these topics are discussed in great detail and for an easy access all mathematical tools are summarized at the beginning of the chapter for better reference. Great care is given to illustrate how the fundamental mathematical results can be used to simplify and speed up concrete computations on the one side and how to make certain problems accessible at all. In this direction the chapter on applications is preceeded by a whole chapter discussing the significance, applicability, and the advantage of the methods developed in this book.
The main results of the mathematical part are taken from topological measure and integration theory. It is one of the most interesting features of this monograph that most applications can be subsumed under a mathematical analysis of the following standard situation. For second countable groups \(G\), \(S\), \(T\), \(H\), where \(T\) and \(H\) are locally compact, \(S\) Hausdorff, \(G\) compact acting on \(H\) and acting transitively on \(S\), there is given a Borel-measurable surjection \(\varphi\) from \(S\times T\) on \(H\), which is equivariant with respect to the action of \(G\), i.e., \(g\varphi(s, t)= \varphi(gs, t)\) for all \(g\in G\), all \(s\in S\), and all \(t\in T\). The assumptions imply the existence of a unique \(G\)-invariant probability \(\mu_{(S)}\) on \(S\), but the main interest is in the existence of \(G\)-invariant Borel measures on \(H\), while the space \(S\) and the mapping \(\varphi\) are considered in applications only as ancillary tools. The crucial problem to be solved for applications, is to find for given \(\sigma\)-finite Borel measure \(\nu\) on the group \(H\) a (not necessarily uniquely determined) Borel measure \(\tau\) on \(T\) such that \(\nu\) is the image measure of the product measure \(\mu_{(S)}\otimes\tau\) on \(S\times T\) under the map \(\varphi\). This problem is solved by reducing it to a measure extension problem. Some results obtained on these lines are of interest independent from applications. The advantage taken in applications lies in the fact, that \(\mu_{(S)}\) and \(\varphi\) can be considered as fixed and the individuality of \(\nu\) is encoded in \(\tau\) which may be considerably ‘simpler’. There are examples for Borel measures on \(H= \text{GL}(n)\) in which the dimension of the domain of integration shrinks from \(n^2\) to \(n\) due to reduction to \(\tau\). For conjugation-invariant measures there can be achieved savings in computing time for more than 74 per cent with simulation algorithms based on symmetry as compared to the ‘ordinary’ ones. Moreover, the simulation of i.i.d. \(\nu\)-distributed random variables can be decomposed in two independent simulations. Due to the symmetry of \(\mu_{(S)}\) this opens the way towards the existence of efficient algorithms for the generation of pseudorandom elements.