Theorems · Theorem · dynamical systems
PreErgodic.ae_mem_or_ae_notMem
∀ {α : Type u_1} {m : MeasurableSpace α} {s : Set α} {f : α → α} {μ : MeasureTheory.Measure α},
PreErgodic f μ → MeasurableSet s → f ⁻¹' s = s → (∀ᵐ (x : α) ∂μ, x ∈ s) ∨ ∀ᵐ (x : α) ∂μ, x ∉ s- Defined in
- Mathlib.Dynamics.Ergodic.Ergodic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- Set.preimagestatement and proof · cited by 4,946
- Filter.Eventuallystatement · cited by 3,134
- MeasurableSetstatement and proof · cited by 3,075
- MeasureTheory.aestatement · cited by 2,352
- PreErgodicstatement and proof · cited by 19
- PreErgodic.aeconst_setproof · cited by 7
- Filter.eventuallyConst_setproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- PreErgodic.ae_eq_const_of_ae_eq_compproof · cited by 0