| import OAI.Analysis.CircleDomains.Probability.ArcNullImage |
| import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic |
|
|
| namespace OAI |
|
|
| /-! |
| # Null images of local circle-valued traces |
|
|
| Projection modulo one preserves real outer-null sets. A countable family |
| of null real branches therefore gives a null image in the circle with |
| its Haar measure of total mass one. |
| -/ |
|
|
| open Set MeasureTheory |
|
|
| namespace CircleDomainRigidity |
|
|
| theorem volume_addCircle_image_eq_zero (hA : volume A = 0) : |
| volume (((↑) : ℝ → AddCircle (1 : ℝ)) '' A) = 0 := by |
| have hpiece (a : ℝ) : |
| volume (((↑) : ℝ → AddCircle (1 : ℝ)) '' (A ∩ Ioc a (a + 1))) = 0 := by |
| let e := AddCircle.measurableEquivIoc (1 : ℝ) a |
| let S : Set (Ioc a (a + 1)) := Subtype.val ⁻¹' A |
| have he : MeasurePreserving e volume (volume.comap Subtype.val) := |
| AddCircle.measurePreserving_equivIoc (1 : ℝ) |
| have hset : e ⁻¹' S = ((↑) : ℝ → AddCircle (1 : ℝ)) '' (A ∩ Ioc a (a + 1)) := by |
| ext y |
| constructor |
| · intro hy |
| refine ⟨(e y).val, ⟨hy, (e y).property⟩, ?_⟩ |
| exact AddCircle.coe_equivIoc |
| · rintro ⟨x, hx, rfl⟩ |
| change (AddCircle.equivIoc (1 : ℝ) a (x : AddCircle (1 : ℝ))).val ∈ A |
| rw [AddCircle.equivIoc_coe_eq hx.2] |
| exact hx.1 |
| rw [← hset, he.measure_preimage_equiv S, |
| (MeasurableEmbedding.subtype_coe measurableSet_Ioc).comap_apply] |
| apply measure_mono_null _ hA |
| rintro _ ⟨x, hx, rfl⟩ |
| exact hx |
| have hcover : ((↑) : ℝ → AddCircle (1 : ℝ)) '' A ⊆ |
| ⋃ k : ℤ, ((↑) : ℝ → AddCircle (1 : ℝ)) '' |
| (A ∩ Ioc ((k : ℝ) - 1) ((k : ℝ) - 1 + 1)) := by |
| rintro y ⟨x, hx, rfl⟩ |
| apply mem_iUnion.mpr |
| refine ⟨⌈x⌉, x, ⟨hx, ?_⟩, rfl⟩ |
| have hlo := Int.ceil_lt_add_one x |
| have hhi := Int.le_ceil x |
| constructor <;> linarith |
| apply measure_mono_null hcover |
| exact measure_iUnion_null (fun k => hpiece ((k : ℝ) - 1)) |
|
|
| /-- A circle-valued image covered by countably many null real branch |
| images is Haar-null, with no measurability assumption on the image. -/ |
| theorem volume_eq_zero_of_countable_real_branches |
| (F : Set X) (w : X → AddCircle (1 : ℝ)) |
| (V : ℕ → Set X) (v : ℕ → X → ℝ) |
| (hcover : F ⊆ ⋃ n, V n) |
| (hbranch : ∀ n, ∀ x ∈ F ∩ V n, (v n x : AddCircle (1 : ℝ)) = w x) |
| (hnull : ∀ n, volume (v n '' (F ∩ V n)) = 0) : |
| volume (w '' F) = 0 := by |
| have himage : w '' F ⊆ ⋃ n, |
| ((↑) : ℝ → AddCircle (1 : ℝ)) '' (v n '' (F ∩ V n)) := by |
| rintro y ⟨x, hx, rfl⟩ |
| obtain ⟨n, hn⟩ := mem_iUnion.mp (hcover hx) |
| exact mem_iUnion.mpr ⟨n, v n x, ⟨x, ⟨hx, hn⟩, rfl⟩, hbranch n x ⟨hx, hn⟩⟩ |
| exact measure_mono_null himage |
| (measure_iUnion_null (fun n => volume_addCircle_image_eq_zero (hnull n))) |
|
|
| end CircleDomainRigidity |
|
|
| end OAI |
|
|
