| import OAI.MeasureTheory.Falconer.Recursion.ProfileBalls |
| import Mathlib.Probability.ConditionalProbability |
|
|
| namespace OAI |
|
|
| /-! |
| # Full conditional class laws, normalized by the actual cell mass |
|
|
| The profile is only a comparison for that mass. The conditional measure |
| is the original restricted class law divided by its own cell mass. |
| -/ |
|
|
| namespace Falconer.Regularization |
| open Set MeasureTheory Metric |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| theorem conditional_real_le (ρ : Measure (Space d)) [IsFiniteMeasure ρ] |
| (A T : Set (Space d)) : |
| (ProbabilityTheory.cond ρ A T).toReal ≤ (ρ T).toReal/(ρ A).toReal := by |
| unfold ProbabilityTheory.cond |
| simp only [Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul,ENNReal.toReal_inv] |
| have hh : ((ρ.restrict A) T).toReal ≤ (ρ T).toReal := |
| ENNReal.toReal_mono (measure_ne_top _ _) (Measure.restrict_apply_le _ _) |
| simpa only [div_eq_mul_inv,mul_comm] using |
| mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr (show 0 ≤ (ρ A).toReal from ENNReal.toReal_nonneg)) |
|
|
| theorem sourceClass_conditional_probability (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] (N B : ℕ) |
| {p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves Q μ N) (P : CellPath d) |
| (hPpos : 0 < sourceClass Q μ N B p₀ (Q.descendant P).carrier) : |
| IsProbabilityMeasure (ProbabilityTheory.cond (sourceClass Q μ N B p₀) (Q.descendant P).carrier) := by |
| have : IsProbabilityMeasure (sourceClass Q μ N B p₀) := |
| regularization_class_probability Q hQ μ N B hp₀ |
| exact ProbabilityTheory.cond_isProbabilityMeasure hPpos.ne' |
|
|
| theorem sourceClass_conditional_ball_upper (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B) |
| {lam : ℝ} (hwidth : (B:ℝ) ≤ lam*(N:ℝ)) |
| (hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam) |
| {p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves Q μ N) |
| {p n : ℕ} (hpN : p ≤ N) (hn : n ≤ N) {P : CellPath d} (hP : P.length = p) |
| (hPpos : 0 < sourceClass Q μ N B p₀ (Q.descendant P).carrier) |
| (x : Space d) {C : ℝ} (hC : 0 ≤ C) : |
| (ProbabilityTheory.cond (sourceClass Q μ N B p₀) (Q.descendant P).carrier |
| (ball x (C*(Q.side/(2:ℝ)^n)))).toReal ≤ |
| ((2*Nat.ceil (2*C+4*Real.sqrt d)+1)^d:ℕ)*((2:ℝ)^N)^ |
| (sourceProfile Q μ N B p₀ p-sourceProfile Q μ N B p₀ n+2*((d:ℝ)+1)*lam) := by |
| have : IsProbabilityMeasure (sourceClass Q μ N B p₀) := |
| regularization_class_probability Q hQ μ N B hp₀ |
| let R : ℝ := (2:ℝ)^N |
| let c : ℝ := ((d:ℝ)+1)*lam |
| let mp := sourceProfile Q μ N B p₀ p |
| let mn := sourceProfile Q μ N B p₀ n |
| let D : ℝ := ((2*Nat.ceil (2*C+4*Real.sqrt d)+1)^d:ℕ) |
| have hRp : 0 < R := by dsimp [R]; positivity |
| have hb := sourceClass_ball_upper Q hQ μ hN hB hwidth hlarge hp₀ hn x hC |
| have hl := (sourceClass_occupied_comparison Q hQ μ hN hB hwidth hlarge hp₀ hpN hP hPpos).1 |
| have hc := conditional_real_le (sourceClass Q μ N B p₀) (Q.descendant P).carrier |
| (ball x (C*(Q.side/(2:ℝ)^n))) |
| have hratio := div_le_div₀ (show 0 ≤ D*(R^c*R^(-mn)) by dsimp [D]; positivity) |
| hb (show 0 < R^(-c)*R^(-mp) by positivity) (show R^(-c)*R^(-mp) ≤ |
| (sourceClass Q μ N B p₀ (Q.descendant P).carrier).toReal by |
| simpa only [c,mp,R,neg_mul] using hl) |
| apply (hc.trans hratio).trans_eq |
| change (D*(R^c*R^(-mn)))/(R^(-c)*R^(-mp)) = D*R^(mp-mn+2*((d:ℝ)+1)*lam) |
| rw [mul_div_assoc,← Real.rpow_add hRp,← Real.rpow_add hRp,← Real.rpow_sub hRp] |
| congr 2 |
| dsimp [c] |
| ring |
|
|
| end Falconer.Regularization |
|
|
| end OAI |
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|
