Chapter 27 - Measures on Algebraic—Topological Structures

Abstract

This chapter discusses countably additive measures invariant under the groups of transformations. Invariant measures are present in many parts of mathematics, including harmonic analysis, ergodic theory, and topological dynamics. A typical example of such a measure is the Lebesgue measure. One of its key properties is that it assigns equal values to congruent sets so that the measure of a set does not depend on its position in the space. General necessary and sufficient conditions for the existence of finite and σ-finite invariant measures on arbitrary spaces are explained in the chapter. The invariant measures defined for all subsets of a given set are presented in the chapter. Results of invariant measures are parallel to the work of Tarski on the existence of finitely additive invariant measures. The existence of Borel measures on Polish groups is described in the chapter. In particular, standard facts about Haar measure in locally compact groups are summarized, and a survey of results connected with Sierpinski's problem is also presented in the chapter.