| import OAI.MeasureTheory.Falconer.Model | |
| import Mathlib.Analysis.InnerProductSpace.PiL2 | |
| import Mathlib.Topology.MetricSpace.HausdorffDimension | |
| import Mathlib.MeasureTheory.Measure.Lebesgue.Basic | |
| import Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | |
| namespace OAI | |
| namespace PlanarFalconer | |
| open MeasureTheory Set | |
| theorem distanceSet_eq_image (E : Set Plane) : | |
| distanceSet E = (fun z : Plane × Plane => dist z.1 z.2) '' (E ×ˢ E) := by | |
| ext r | |
| constructor | |
| · rintro ⟨x,hx,y,hy,hr⟩ | |
| exact ⟨(x,y), ⟨hx,hy⟩, hr⟩ | |
| · rintro ⟨⟨x,y⟩, ⟨hx,hy⟩,hr⟩ | |
| exact ⟨x,hx,y,hy,hr⟩ | |
| theorem distanceSet_isCompact {E : Set Plane} (hE : IsCompact E) : | |
| IsCompact (distanceSet E) := by | |
| rw [distanceSet_eq_image] | |
| exact (hE.prod hE).image (continuous_fst.dist continuous_snd) | |
| theorem distanceSet_measurable {E : Set Plane} (hE : IsCompact E) : | |
| MeasurableSet (distanceSet E) := | |
| (distanceSet_isCompact hE).measurableSet | |
| theorem distanceSet_nonnegative {E : Set Plane} {r : ℝ} (hr : r ∈ distanceSet E) : | |
| 0 ≤ r := by | |
| obtain ⟨x,_,y,_,rfl⟩ := hr | |
| exact dist_nonneg | |
| theorem zero_mem_distanceSet {E : Set Plane} (hE : E.Nonempty) : | |
| 0 ∈ distanceSet E := by | |
| obtain ⟨x,hx⟩ := hE | |
| exact ⟨x,hx,x,hx,dist_self x⟩ | |
| theorem positive_volume_of_carried_absolutelyContinuous | |
| (D : Set ℝ) (τ : Measure ℝ) (hmass : 0 < τ univ) | |
| (hcarried : τ Dᶜ = 0) (hac : τ ≪ volume) : 0 < volume D := by | |
| by_contra h | |
| have hzero : volume D = 0 := le_antisymm (not_lt.mp h) bot_le | |
| have hτ : τ D = 0 := hac hzero | |
| have hz := measure_union_null hτ hcarried | |
| have : τ univ = 0 := by simpa only [union_compl_self] using hz | |
| exact (ne_of_gt hmass) this | |
| theorem distanceSet_positive_of_measure (E : Set Plane) (τ : Measure ℝ) | |
| (hmass : 0 < τ univ) (hcarried : τ (distanceSet E)ᶜ = 0) (hac : τ ≪ volume) : | |
| 0 < volume (distanceSet E) := | |
| positive_volume_of_carried_absolutelyContinuous (distanceSet E) τ hmass hcarried hac | |
| end PlanarFalconer | |
| end OAI | |