The measure spaces or integrals which actually arise in analysis frequently do so with additional elements of structure which are quite significant, and interact with the integration features, so that the additional features cannot really be considered separately. The simplest and most common such additional feature is a topology on the set under consideration, on which the measure in question is defined. Some indications have already been given earlier of the relations between the continuous and measurable functions which are then valid and useful. Another such feature is a group of transformations on the basic set S, leaving invariant the measure m. There are questions of the uniqueness of m, its existence when the transformations are given, and of the special relations and group action on the function spaces over S which then become relevant. The developments which emerge are extensive, important, and in some respects striking. It is to some of these developments that this chapter is devoted.
KeywordsInvariant Measure Topological Group Compact Group Compact Space Transformation Group
Unable to display preview. Download preview PDF.