Talk:Conservative system
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Inaccuracies and imprecisions
[edit]1) "A dynamical system (X, Σ, μ, τ) is a Borel space (X, Σ) equipped with a sigma-finite measure μ and a transformation τ."
A dynamical system need not have a measure. So this statement is false.
But a conservative system does need a measure, so this condition should be kept but formulated correctly.
2) "μ is a finite measure on the sigma-algebra, so that the triplet (X, Σ, μ) is a probability space."
A sigma-finite measure is not a finite measure on a sigma-algebra, not is a finite measure a probability measure.
Nor need a conservative system have a finite measure. Indeed, the non-wandering hypothesis is redundant for finite measure spaces.
3) "One is interested in invertible transformations"
Fine, but is it part of the formal definition or not?
2001:171B:2274:7C21:18AF:673B:6C41:91E2 (talk) 16:25, 24 October 2021 (UTC)
- I fixed 1) and 2). Concerning 3), see Omri Sarig, Lecture Notes on Ergodic Theory March 8, 2020, p. 32, Ex. 1.14., "Conservativity", where invertibility is not required, but measure-preserving is. It's at http://www.weizmann.ac.il/math/sarigo/sites/math.sarigo/files/uploads/ergodicnotes.pdf.