| 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 |
|
|
