| import Mathlib.MeasureTheory.Function.L2Space |
| import Mathlib.MeasureTheory.Integral.Bochner.Set |
| import Mathlib.MeasureTheory.Measure.WithDensity |
| import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
| open scoped BigOperators |
|
|
| namespace Invariant233.DigitAveraging |
|
|
| theorem sq_integral_le_mass_mul_integral_sq |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
| {u : Ω → ℝ} (hu : Integrable u μ) |
| (hu2 : Integrable (fun x => u x ^ 2) μ) (hM : 0 < μ.real univ) : |
| (∫ x, u x ∂μ) ^ 2 ≤ μ.real univ * ∫ x, u x ^ 2 ∂μ := by |
| let M := μ.real univ |
| let A := ∫ x, u x ∂μ |
| let a := A / M |
| have ha : a * M = A := div_mul_cancel₀ A (ne_of_gt hM) |
| have he : (∫ x, (u x - a) ^ 2 ∂μ) = |
| (∫ x, u x ^ 2 ∂μ) - 2 * a * A + a ^ 2 * M := by |
| have hlin : Integrable (fun x => 2 * a * u x) μ := hu.const_mul _ |
| have hconst : Integrable (fun _ : Ω => a ^ 2) μ := integrable_const _ |
| have hpoint : (fun x => (u x - a) ^ 2) = |
| (fun x => u x ^ 2 - 2 * a * u x + a ^ 2) := by |
| funext x |
| ring |
| rw [hpoint, integral_add (show Integrable (fun x => u x ^ 2 - 2 * a * u x) μ from hu2.sub hlin) hconst, integral_sub hu2 hlin, |
| integral_const_mul, integral_const] |
| simp only [smul_eq_mul, Measure.real, M, A] |
| ring |
| have hnonneg : 0 ≤ ∫ x, (u x - a) ^ 2 ∂μ := integral_nonneg (fun _ => sq_nonneg _) |
| rw [he, ← ha] at hnonneg |
| have hb : a ^ 2 * M ≤ ∫ x, u x ^ 2 ∂μ := by nlinarith |
| change A ^ 2 ≤ M * ∫ x, u x ^ 2 ∂μ |
| calc |
| A ^ 2 = (a * M) ^ 2 := congrArg (fun z : ℝ => z ^ 2) ha.symm |
| _ = M * (a ^ 2 * M) := by ring |
| _ ≤ M * ∫ x, u x ^ 2 ∂μ := mul_le_mul_of_nonneg_left hb hM.le |
|
|
| theorem sum_integral_fibers {Ω α : Type*} [MeasurableSpace Ω] |
| [MeasurableSpace α] [MeasurableSingletonClass α] [Fintype α] |
| {μ : Measure Ω} {f : Ω → α} (hf : Measurable f) |
| {u : Ω → ℝ} (hu : Integrable u μ) : |
| (∑ a : α, ∫ x in {x | f x = a}, u x ∂μ) = ∫ x, u x ∂μ := by |
| have hm : ∀ a : α, MeasurableSet {x | f x = a} := |
| fun a => hf (measurableSet_singleton a) |
| have hd : Pairwise (fun a b : α => Disjoint {x | f x = a} {x | f x = b}) := by |
| intro a b hab |
| rw [Set.disjoint_left] |
| intro x hxa hxb |
| exact hab (hxa.symm.trans hxb) |
| have hcover : (⋃ a : α, {x | f x = a}) = univ := by |
| ext x |
| simp |
| rw [← integral_iUnion_fintype hm hd (fun _ => hu.integrableOn), hcover, |
| Measure.restrict_univ] |
|
|
| theorem digit_averaging_energy_le |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {q : ℕ} (hq : 0 < q) {f : Ω → Fin q} (hf : Measurable f) |
| (hmass : ∀ a : Fin q, μ.real {x | f x = a} = 1 / (q : ℝ)) |
| {h : Ω → ℝ} (hh : MemLp h 2 μ) : |
| (1 / (q : ℝ)) * ∑ a : Fin q, |
| ((q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) - 1) ^ 2 ≤ |
| ∫ x, (h x - 1) ^ 2 ∂μ := by |
| have hqR : 0 < (q : ℝ) := by exact_mod_cast hq |
| have hq0 : (q : ℝ) ≠ 0 := ne_of_gt hqR |
| have hhInt : Integrable h μ := hh.integrable (by norm_num) |
| have hu : MemLp (fun x => h x - 1) 2 μ := hh.sub (memLp_const 1) |
| have huInt : Integrable (fun x => h x - 1) μ := hu.integrable (by norm_num) |
| have hu2Int := hu.integrable_sq |
| have hlocal : ∀ a : Fin q, |
| (1 / (q : ℝ)) * ((q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) - 1) ^ 2 ≤ |
| ∫ x in {x | f x = a}, (h x - 1) ^ 2 ∂μ := by |
| intro a |
| have hpos : 0 < (μ.restrict {x | f x = a}).real univ := by |
| rw [measureReal_restrict_apply_univ, hmass] |
| positivity |
| have hcs := sq_integral_le_mass_mul_integral_sq |
| huInt.restrict hu2Int.restrict hpos |
| rw [measureReal_restrict_apply_univ, hmass] at hcs |
| have he : (q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) - 1 = |
| (q : ℝ) * (∫ x in {x | f x = a}, h x - 1 ∂μ) := by |
| rw [integral_sub hhInt.restrict (integrable_const 1), setIntegral_const] |
| simp only [smul_eq_mul, mul_one, hmass] |
| field_simp |
| rw [he] |
| calc |
| (1 / (q : ℝ)) * ((q : ℝ) * (∫ x in {x | f x = a}, h x - 1 ∂μ)) ^ 2 = |
| (q : ℝ) * (∫ x in {x | f x = a}, h x - 1 ∂μ) ^ 2 := by field_simp |
| _ ≤ (q : ℝ) * ((1 / (q : ℝ)) * |
| (∫ x in {x | f x = a}, (h x - 1) ^ 2 ∂μ)) := |
| mul_le_mul_of_nonneg_left hcs hqR.le |
| _ = ∫ x in {x | f x = a}, (h x - 1) ^ 2 ∂μ := by field_simp |
| rw [Finset.mul_sum] |
| exact (Finset.sum_le_sum (fun a _ => hlocal a)).trans_eq |
| (sum_integral_fibers hf hu2Int) |
|
|
| theorem digit_average_nonneg |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| {q : ℕ} {f : Ω → Fin q} {h : Ω → ℝ} |
| (hh : ∀ᵐ x ∂μ, 0 ≤ h x) (a : Fin q) : |
| 0 ≤ (q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) := |
| mul_nonneg (Nat.cast_nonneg q) (integral_nonneg_of_ae (ae_restrict_of_ae hh)) |
|
|
| theorem digit_average_mean |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| {q : ℕ} (hq : 0 < q) {f : Ω → Fin q} (hf : Measurable f) |
| {h : Ω → ℝ} (hh : Integrable h μ) : |
| (1 / (q : ℝ)) * ∑ a : Fin q, |
| (q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) = ∫ x, h x ∂μ := by |
| rw [← Finset.mul_sum, sum_integral_fibers hf hh] |
| have hq0 : (q : ℝ) ≠ 0 := by positivity |
| field_simp |
|
|
| theorem digit_average_le |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {q : ℕ} (hq : 0 < q) {f : Ω → Fin q} |
| (hmass : ∀ a : Fin q, μ.real {x | f x = a} = 1 / (q : ℝ)) |
| {h : Ω → ℝ} (hh : Integrable h μ) {B : ℝ} |
| (hB : ∀ᵐ x ∂μ, h x ≤ B) (a : Fin q) : |
| (q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) ≤ B := by |
| have hqR : 0 < (q : ℝ) := by exact_mod_cast hq |
| have hq0 : (q : ℝ) ≠ 0 := hqR.ne' |
| calc |
| (q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) ≤ |
| (q : ℝ) * (∫ _x in {x | f x = a}, B ∂μ) := |
| mul_le_mul_of_nonneg_left |
| (integral_mono_ae hh.restrict (integrable_const B) (ae_restrict_of_ae hB)) hqR.le |
| _ = B := by |
| rw [setIntegral_const, smul_eq_mul, hmass] |
| field_simp |
| |
| theorem map_withDensity_eq_digit_density |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {q : ℕ} (hq : 0 < q) {f : Ω → Fin q} (hf : Measurable f) |
| (hmass : ∀ a : Fin q, μ.real {x | f x = a} = 1 / (q : ℝ)) |
| {h : Ω → ℝ} (hh : Integrable h μ) (hnonneg : ∀ᵐ x ∂μ, 0 ≤ h x) : |
| (μ.withDensity (fun x => ENNReal.ofReal (h x))).map f = |
| (μ.map f).withDensity (fun a => ENNReal.ofReal |
| ((q : ℝ) * ∫ x in {x | f x = a}, h x ∂μ)) := by |
| have hqR : 0 < (q : ℝ) := by exact_mod_cast hq |
| have hq0 : (q : ℝ) ≠ 0 := hqR.ne' |
| apply Measure.ext_of_singleton |
| intro a |
| have hfiber : f ⁻¹' {a} = {x | f x = a} := by ext x; simp |
| have hmeas : MeasurableSet {x | f x = a} := hf (measurableSet_singleton a) |
| have hmapa : (μ.map f) {a} = ENNReal.ofReal (1 / (q : ℝ)) := by |
| rw [Measure.map_apply hf (measurableSet_singleton a), hfiber] |
| rw [← ENNReal.ofReal_toReal (measure_ne_top μ {x | f x = a})] |
| exact congrArg ENNReal.ofReal (hmass a) |
| rw [Measure.map_apply hf (measurableSet_singleton a), hfiber, |
| withDensity_apply _ hmeas, |
| withDensity_apply _ (measurableSet_singleton a), lintegral_singleton, hmapa, |
| ← ofReal_integral_eq_lintegral_ofReal hh.restrict (ae_restrict_of_ae hnonneg)] |
| rw [← ENNReal.ofReal_mul (digit_average_nonneg hnonneg a)] |
| congr 1 |
| field_simp |
| |
| theorem integral_map_eq_uniform_sum |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {q : ℕ} {f : Ω → Fin q} (hf : Measurable f) |
| (hmass : ∀ a : Fin q, μ.real {x | f x = a} = 1 / (q : ℝ)) |
| (g : Fin q → ℝ) : |
| (∫ a, g a ∂μ.map f) = (1 / (q : ℝ)) * ∑ a : Fin q, g a := by |
| have hg : Integrable g (μ.map f) := by |
| exact Integrable.of_finite |
| rw [integral_fintype hg, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro a _ |
| have hmap : (μ.map f).real {a} = μ.real {x | f x = a} := by |
| unfold Measure.real |
| rw [Measure.map_apply hf (measurableSet_singleton a)] |
| congr 1 |
| rw [smul_eq_mul, hmap, hmass] |
| |
| theorem integral_digit_energy_le |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {q : ℕ} (hq : 0 < q) {f : Ω → Fin q} (hf : Measurable f) |
| (hmass : ∀ a : Fin q, μ.real {x | f x = a} = 1 / (q : ℝ)) |
| {h : Ω → ℝ} (hh : MemLp h 2 μ) : |
| (∫ a, ((q : ℝ) * (∫ x in {x | f x = a}, h x ∂μ) - 1) ^ 2 ∂μ.map f) ≤ |
| ∫ x, (h x - 1) ^ 2 ∂μ := by |
| rw [integral_map_eq_uniform_sum hf hmass] |
| exact digit_averaging_energy_le hq hf hmass hh |
| |
| end Invariant233.DigitAveraging |
| |
| end OAI |
| |
