Documentation

Mathlib.MeasureTheory.Group.AddCircle

Measure-theoretic results about the additive circle #

The file is a place to collect measure-theoretic results about the additive circle.

Main definitions: #

theorem AddCircle.closedBall_ae_eq_ball {T : } [hT : Fact (0 < T)] {x : AddCircle T} {ε : } :
Metric.closedBall x ε =ᵐ[MeasureTheory.volume] Metric.ball x ε
theorem AddCircle.isAddFundamentalDomain_of_ae_ball {T : } [hT : Fact (0 < T)] (I : Set (AddCircle T)) (u : AddCircle T) (x : AddCircle T) (hu : IsOfFinAddOrder u) (hI : I =ᵐ[MeasureTheory.volume] Metric.ball x (T / (2 * (addOrderOf u)))) :

Let G be the subgroup of AddCircle T generated by a point u of finite order n : ℕ. Then any set I that is almost equal to a ball of radius T / 2n is a fundamental domain for the action of G on AddCircle T by left addition.

theorem AddCircle.volume_of_add_preimage_eq {T : } [hT : Fact (0 < T)] (s : Set (AddCircle T)) (I : Set (AddCircle T)) (u : AddCircle T) (x : AddCircle T) (hu : IsOfFinAddOrder u) (hs : u +ᵥ s =ᵐ[MeasureTheory.volume] s) (hI : I =ᵐ[MeasureTheory.volume] Metric.ball x (T / (2 * (addOrderOf u)))) :
MeasureTheory.volume s = addOrderOf u MeasureTheory.volume (s I)