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b17ead8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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
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