Abstract
The following theorem is proved: If T is a nonsingular invertible transformation in a separable probability space (Ω, F, μ) and there exists no T-invariant probability measure μo << μ, then the system of sets A for which ζ ={A,AC} is a strong generator (i.e. for which A, TA, T2A, … generates F mod μ) is dense in every increasing exhaustive subalgebra S of F (i.e. in every S ⊑ F with Tk S ↑ F). In particular for S=F it follows that nonsingular transformations without finite invariant measure have finite strong generators (of size 2).
The greatest difficulty appears in the case where a finite invariant measure exists on S but not on F. To show that this can happen we construct an ergodic Bernoulli shift (with nonidentical factor measures) for which no invariant probability measure μO << μ exists and for which the restriction of μ to an exhaustive S is invariant.
Keywords
- Invariant Measure
- Lebesgue Space
- Finite Measure
- Invariant Probability Measure
- Strong Generator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by the National Science Foundation, Grant GP-9354.
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Krengel, U. (1970). Transformations without finite invariant measure have finite strong generators. In: Contributions to Ergodic Theory and Probability. Lecture Notes in Mathematics, vol 160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0060652
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DOI: https://doi.org/10.1007/BFb0060652
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