Transformation Groups and Invariant Measures

The following sections are included:

• Contents

• Preface

In this section of our book we shall fix the notation and present some elementary facts from set theory, general topology and measure theory. Clearly, we shall systematically use these facts in our further considerations.

In this section we discuss some purely algebraic properties of transformation groups. It will be shown below that these properties play an essential role in the general theory of quasiinvariant and invariant measures.

In this section we consider some general properties of quasiinvariant and invariant measures. Several applications of those properties will be discussed in the subsequent sections of the book.

In the previous section we were concerned with various spaces of type (E, G, μ) where E is a basic set, G is a group of transformations of E and μ is a σ-finite G-quasiinvariant (G-invariant) measure defined on a σ-algebra of subsets of E.

Here we discuss some analogues and generalizations of the classical Vitali construction producing a Lebesgue nonmeasurable set on the real line. Also, we consider some similar constructions of nonmeasurable sets for uncountable groups equipped with nonzero σ-finite quasiinvariant (invariant) measures.

Let E be a basic set, let G be a group of transformations of E and let μ be a nonzero σ-finite G-invariant (more generally, G-quasiinvariant) measure defined on some σ-algebra of subsets of E. We recall that the symbol I(μ) denotes the σ-ideal of subsets of E, generated by the family of all μ-measure zero sets. Obviously, I(μ) is a G-invariant class of subsets of E. Members of I(μ) are usually called negligible sets with respect to the given measure μ. Sometimes, they are also called small sets with respect to μ. The complements of small sets with respect to μ are sometimes called large sets with respect to μ.

Let E be an infinite basic set. We recall that the symbol Sym(E) denotes the family of all bijective mappings acting from E onto E, and that Sym(E) is a group with respect to the operation of the composition of mappings.

We begin this section with the following observation. Let (E, G) be a space equipped with a transformation group and let I be a G-invariant σ-ideal of subsets of E. We denote by S(I) the σ-algebra of subsets of E, generated by I. Then it is not difficult to see that there exists a complete G-invariant probability measure μ defined on S(I) and vanishing on all members from I.

At the present time, it is known that there exist various measures on the real line R which strictly extend the classical Lebesgue measure λ on R and are invariant under the group of all isometric transformations of R. One of the first works devoted to (countably additive) invariant extensions of the Lebesgue measure was the paper by Szpilrajn (Marczewski) [121] where several constructions of such extensions were considered. Also, we can point out another paper [120] by the same author, in which a list of important problems from measure theory is given and, in particular, certain invariant extensions of the Lebesgue measure are touched. Notice that, among other problems, Szpilrajn (Marczewski) formulates in [120] a problem of the existence of a nonseparable invariant extension of the measure λ. This problem was solved in 1950 when two papers - by Kakutani and Oxtoby [41] and by Kodaira and Kakutani [71] - appeared, in which two essentially different constructions of nonseparable invariant extensions of λ were presented. Afterwards, several questions concerning invariant extensions of λ became a subject of investigations of various authors (see, for instance, [12], [15], [17], [33], [38], [48], [63], [65], [66], [68], [69], [72], [100], [103], [132], [133], [134]). Also, some generalizations of the constructions obtained to the case of a Haar measure given on a locally compact topological group were discussed in a number of works (see, e.g., [35] and [39]).

This section is devoted to the uniqueness property of various invariant measures. Primarily, we discuss this property for the standard Lebesgue measure on the n-dimensional Euclidean space Rⁿ (sphere Sⁿ) and for the standard Borel measure on the same space (sphere), being the restriction of the Lebesgue measure to the Borel σ-algebra of Rⁿ (Sⁿ).

This final section of the book is devoted to σ-finite quasiinvariant Borel measures given on those topological groups which are usually called the standard ones. The precise definition can be formulated as follows. We say that a topological group G is standard if G is a Borel subgroup of some Polish group. Here we wish to present the fundamental Mackey-Weil theorem concerning a characterization of σ-finite quasiinvariant Borel measures on standard groups. This topic is discussed in the well-known textbook by Parthasarathy “Introduction to Probability and Measure”, 1980. However, our presentation is more thorough in details and contains some additional information.

The following sections are included:

• Appendix

• Bibliography

• Subject Index