Groups of measurable and measure preserving transformations

In chapters I and II, we show that the group G of invertible, non-singular transformations of a Lebesgue space is perfect, simple, and has no outer automorphisms. Some related results are obtained for the subgroup of measure preserving transformations and for the full group of an ergodic transformation. Further results are given with the underlying Lebesgue space replaced by a homogeneous measure algebra. It is also shown, in chapter III, that ergodic transformations are algebraically distinguishable from non-ergodics. Chapter IV introduces the notion of a fibered ergodic transformation. A fibered analogue of Dye's theorem is proved. In chapter V the family of transformations satisfying Dye's theorem for two fixed ergodics is shown to be dense in the coarse topology. Finally, in chapter VI, the concept of a triangle set in the unit square is introduced. Using this notion, a sufficiency condition for the ergodicity of T x S is obtained