Invariant measures and ideals on discrete groups.

*(English)*Zbl 0625.04009The present paper is both research and exposition in an area which touches a region common to set theory and measure theory. Chapter 1 surveys results surrounding the concept of measurable cardinal and the question of the existence of sigma-additive and sigma-finite measures vanishing on singletons and defined on the power set of the reals, i.e. on all subsets of reals. A measure defined on the power set of a set is termed universal.

Chapter II treats the question: given a group G and a subgroup H of G, does there exist a universal H-invariant measure on G? For example, the following result of Harazisvili is proved: For an arbitrary group G there are no invariant sigma-finite universal measures on G.

Closely connected with the existence of universal measures are problems of extending invariant measures. Results are presented in Chapter III related to the question: For a group G with subgroup H, does every non- universal H-invariant measure on G have a proper H-invariant extension?

Chapter IV contains results related to the set theoretic notion of saturation of an ideal. Ideals are in one-to-one correspondence with \(\{\) 0,1\(\}\)-valued measures. The saturation of an ideal is an index estimating the maximal size of a pairwise almost disjoint family of sets non-measurable with respect to the appropriate measure. An estimate is obtained of the saturation of sufficiently complete, invariant ideals on abelian groups.

Chapter II treats the question: given a group G and a subgroup H of G, does there exist a universal H-invariant measure on G? For example, the following result of Harazisvili is proved: For an arbitrary group G there are no invariant sigma-finite universal measures on G.

Closely connected with the existence of universal measures are problems of extending invariant measures. Results are presented in Chapter III related to the question: For a group G with subgroup H, does every non- universal H-invariant measure on G have a proper H-invariant extension?

Chapter IV contains results related to the set theoretic notion of saturation of an ideal. Ideals are in one-to-one correspondence with \(\{\) 0,1\(\}\)-valued measures. The saturation of an ideal is an index estimating the maximal size of a pairwise almost disjoint family of sets non-measurable with respect to the appropriate measure. An estimate is obtained of the saturation of sufficiently complete, invariant ideals on abelian groups.

Reviewer: B.B.Wells jun