1. On transformations without finite invariant measure
- Author
-
Ulrich Krengel and Lee K. Jones
- Subjects
Discrete mathematics ,Mathematics(all) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Disjoint sets ,01 natural sciences ,Measure (mathematics) ,law.invention ,010104 statistics & probability ,Invertible matrix ,law ,Countable set ,Partition (number theory) ,Invariant measure ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
We show that any invertible nonsingular transformation T of a finite measure space ( Ω , Ol , μ ) admits a countable partition of Ω into disjoint measurable sets Ω 0 , Ω 1 , Ω 2 ,… so that (a) Ω 0 and ∪ i ⩾1 Ω i are invariant under T , (b) T restricted to Ω 0 has a finite equivalent invariant measure, (c) each Ω i is an image under an integral power of T of each Ω i ( i, j ⩾ 1). If Ol is countably generated mod μ the sets Ω i ( i ⩾ 1) can be constructed with the additional property of being strongly generating in Ω / Ω 0 . We also give a streamlined introduction to some known results on existence of invariant measures and, thereby, make the paper completely self-contained.
- Published
- 1974