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Measurable and Topological Rigidity

The concept of a rigid topological space has been the subject of a number of research papers. J. de Groot and R. Wille3 constructed a Peano curve i n the plane with no nontrivial topological automorphisms. Then, de Groot2 went on to demonstrate that every group is realizable as the group of topological automorphisms of some space. Further results were obtained in references 5 and 6 . It is natural to investigate analogous ideas in the category of measurable spaces and measurable automorphisms. Even in the class of separable metric spaces, such inquiries have yielded a number of interesting results. J. van Mill and I9 have defined the reduced (measurable) automorphism group of a space and have deduced a number of structural results for such groups. When the reduced automorphism group of a space is trivial, the space is “measurably rigid”. The existence of uncountable, separable metric, measurably rigid spaces was proved first in reference I (also see Droste4). In the first section of this paper, the relationship between the notions of topological and measurable rigidity is explored. In the second section, further structural results concerning the reduced automorphism group of a space are considered. In particular, it will be proved that if every element of such a group has finite order, then the elements of the group are of bounded order. Such bounds are always of the form,