Theorems · Theorem · dynamical systems
Ergodic.iff_mem_extremePoints_measure_univ_eq
∀ {X : Type u_1} {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} {f : X → X} [MeasureTheory.IsFiniteMeasure μ],
Ergodic f μ ↔ μ ∈ Set.extremePoints ENNReal {ν | MeasureTheory.MeasurePreserving f ν ν ∧ ν Set.univ = μ Set.univ}- Defined in
- Mathlib.Dynamics.Ergodic.Extreme
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement · cited by 9,879
- Set.ofPredstatement · cited by 6,101
- Set.univstatement and proof · cited by 3,945
- MeasureTheory.IsFiniteMeasurestatement and proof · cited by 1,078
- MeasureTheory.MeasurePreservingstatement · cited by 259
- MeasureTheory.measure_ne_topproof · cited by 171
- Ergodicstatement · cited by 48
- Set.extremePointsstatement · cited by 43
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