| import OAI.MeasureTheory.Falconer.Measures.OriginalConditionalFrostman |
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| namespace OAI |
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| /-! |
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| namespace Falconer.Regularization |
| open Set MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
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| theorem conditional_cell_pair_geometry (Q : DyadicCube d) (hQ : Q.side = 1) |
| (ρ : Measure (Space d)) (P : CellPath d) {g : ℕ} (hP : P.length = g) |
| {D : ℝ} (hD : Real.sqrt d ≤ D) |
| (G : Set (Space d × Space d)) |
| (hpin : ∀ z x, (z,x) ∈ G → ∀ y ∈ (Q.descendant P).carrier, ‖y-x‖ ≤ D) : |
| ∀ᵐ z ∂ProbabilityTheory.cond ρ (Q.descendant P).carrier, |
| ∀ᵐ y ∂ProbabilityTheory.cond ρ (Q.descendant P).carrier, |
| ‖y-z‖ ≤ D/(2:ℝ)^g ∧ ∀ x, (z,x) ∈ G → ‖y-x‖ ≤ D := by |
| apply ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant P).measurable |
| intro z hz |
| apply ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant P).measurable |
| intro y hy |
| refine ⟨?_,fun x hx => hpin z x hx y hy⟩ |
| have hside : 0 ≤ (Q.descendant P).side := by rw [Q.descendant_side,hQ]; positivity |
| have hh := (Q.descendant P).dist_le hside hy hz |
| rw [dist_eq_norm,Q.descendant_side,hQ,hP] at hh |
| apply hh.trans |
| simpa only [mul_one_div] using div_le_div_of_nonneg_right hD |
| (by positivity : (0:ℝ) ≤ 2^g) |
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| end Falconer.Regularization |
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| end OAI |
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