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d4317d7 | 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 59 60 61 | import OAI.MeasureTheory.Falconer.Geometry.ProjectionExceptional
import OAI.MeasureTheory.Falconer.Geometry.ConditionalProjectionBuffer
namespace OAI
/-! # Literal occupied-anchor projection tests -/
namespace Falconer
open Set MeasureTheory
open scoped ENNReal
variable {d : ℕ}
theorem projectionGoodSet_mono_width (μ : Measure (Space d)) {W W' T : ℝ} (hW : W ≤ W') :
projectionGoodSet μ W' T ⊆ projectionGoodSet μ W T := by
intro w hw
apply le_trans (measure_mono ?_) hw
intro z hz
exact hz.trans hW
namespace Regularization
noncomputable def conditionalProjectionGoodSet (Q : DyadicCube d) (ρ : Measure (Space d))
(P : CellPath d) (W T : ℝ) : Set (Sphere d) :=
if 0 < ρ (Q.descendant P).carrier then
projectionGoodSet (ProbabilityTheory.cond ρ (Q.descendant P).carrier) W T else ∅
theorem conditionalProjectionGoodSet_measurable (Q : DyadicCube d)
(ρ : Measure (Space d)) [IsFiniteMeasure ρ] (P : CellPath d) (W T : ℝ) :
MeasurableSet (conditionalProjectionGoodSet Q ρ P W T) := by
classical
unfold conditionalProjectionGoodSet
split_ifs
· exact measurableSet_projectionGoodSet _ _ _
· exact MeasurableSet.empty
theorem conditionalProjectionGoodSet_buffer (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) [IsFiniteMeasure ρ] (P : CellPath d)
{δ W W' T : ℝ} (hδ : 0 ≤ δ) {v w : Sphere d} (hvw : dist v w ≤ δ)
(hwidth : W+(Real.sqrt d/(2:ℝ)^P.length)*δ ≤ W')
(hw : w ∈ conditionalProjectionGoodSet Q ρ P W' T) :
v ∈ conditionalProjectionGoodSet Q ρ P W T := by
classical
unfold conditionalProjectionGoodSet at hw ⊢
split_ifs at hw ⊢ with hocc
· exact conditional_projection_buffer Q hQ ρ P hδ hvw hwidth hw
· exact False.elim hw
theorem conditionalProjectionGoodSet_mono_width (Q : DyadicCube d) (ρ : Measure (Space d))
(P : CellPath d) {W W' T : ℝ} (hW : W ≤ W') :
conditionalProjectionGoodSet Q ρ P W' T ⊆ conditionalProjectionGoodSet Q ρ P W T := by
classical
unfold conditionalProjectionGoodSet
split_ifs
· exact projectionGoodSet_mono_width _ hW
· exact Subset.rfl
end Regularization
end Falconer
end OAI
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