| import Mathlib.Probability.Independence.Conditional |
| import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator |
| import Mathlib.MeasureTheory.Function.ConditionalExpectation.Real |
| import Mathlib.Analysis.SpecialFunctions.Exp |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Classical |
|
|
| namespace Invariant233.ConditionalAvoidance |
|
|
| variable {Ω ι : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] |
| variable {μ : Measure Ω} [IsFiniteMeasure μ] |
|
|
| omit mΩ [IsFiniteMeasure μ] in |
|
|
| theorem measurableSet_filter_selection_eq (I S : Finset ι) (P : ι → Ω → Prop) |
| (hP : ∀ i ∈ I, MeasurableSet[m] {ω | P i ω}) : |
| MeasurableSet[m] {ω | I.filter (fun i => P i ω) = S} := by |
| classical |
| by_cases hSI : S ⊆ I |
| · have heq : {ω | I.filter (fun i => P i ω) = S} = |
| ⋂ i ∈ I, {ω | P i ω ↔ i ∈ S} := by |
| ext ω |
| simp only [Set.mem_ofPred_eq, Set.mem_iInter] |
| constructor |
| · intro h i hi |
| have := Finset.ext_iff.mp h i |
| simpa only [Finset.mem_filter, hi, true_and] using this |
| · intro h |
| ext i |
| by_cases hi : i ∈ I |
| · simpa only [Finset.mem_filter, hi, true_and] using h i hi |
| · simp only [Finset.mem_filter, hi, false_and, false_iff] |
| exact fun his => hi (hSI his) |
| rw [heq] |
| apply MeasurableSet.biInter I.countable_toSet |
| intro i hi |
| by_cases his : i ∈ S |
| · simpa only [his, iff_true] using hP i hi |
| · simpa only [his, iff_false, ← Set.compl_ofPred] using (hP i hi).compl |
| · have heq : {ω | I.filter (fun i => P i ω) = S} = (∅ : Set Ω) := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro ω hω |
| exact hSI (hω ▸ Finset.filter_subset _ _) |
| rw [heq] |
| exact MeasurableSet.empty |
|
|
| theorem condProb_compl (hm : m ≤ mΩ) {A : Set Ω} (hA : MeasurableSet A) : |
| μ⟦Aᶜ | m⟧ =ᵐ[μ] fun ω => 1 - (μ⟦A | m⟧) ω := by |
| have heq : Aᶜ.indicator (fun _ : Ω => (1 : ℝ)) = |
| (fun _ => (1 : ℝ)) - A.indicator (fun _ => (1 : ℝ)) := by |
| ext ω |
| by_cases hω : ω ∈ A <;> simp [hω] |
| change μ[Aᶜ.indicator (fun _ => (1 : ℝ)) | m] =ᵐ[μ] _ |
| rw [heq] |
| have hsub := condExp_sub (μ := μ) (integrable_const (1 : ℝ)) |
| ((integrable_const (1 : ℝ)).indicator hA) m |
| filter_upwards [hsub] with ω hω |
| simpa only [Pi.sub_apply, condExp_const hm] using hω |
|
|
| omit [IsFiniteMeasure μ] in |
|
|
| theorem condProb_nonneg (A : Set Ω) : |
| ∀ᵐ ω ∂μ, 0 ≤ (μ⟦A | m⟧) ω := by |
| apply condExp_nonneg |
| exact Eventually.of_forall fun ω => by |
| by_cases hω : ω ∈ A <;> simp [hω] |
|
|
| theorem condProb_le_on_of_ae_inter_subset (hm : m ≤ mΩ) |
| {H K B : Set Ω} (hH : MeasurableSet H) (hK : MeasurableSet[m] K) |
| (hB : MeasurableSet B) (hsub : (H ∩ K : Set Ω) ≤ᵐ[μ] B) : |
| ∀ᵐ ω ∂μ, ω ∈ K → (μ⟦H | m⟧) ω ≤ (μ⟦B | m⟧) ω := by |
| have hpull := condExp_indicator (μ := μ) |
| ((integrable_const (1 : ℝ)).indicator hH) hK |
| have hmono := condExp_mono (μ := μ) (m := m) |
| (((integrable_const (1 : ℝ)).indicator hH).indicator (hm _ hK)) |
| ((integrable_const (1 : ℝ)).indicator hB) |
| (hsub.mono fun ω hsubω => show |
| K.indicator (H.indicator (fun _ => (1 : ℝ))) ω ≤ |
| B.indicator (fun _ => (1 : ℝ)) ω from by |
| by_cases hωK : ω ∈ K |
| · by_cases hωH : ω ∈ H |
| · have hωB : ω ∈ B := hsubω ⟨hωH, hωK⟩ |
| simp [hωK, hωH, hωB] |
| · simp [hωK, hωH, Set.indicator_nonneg] |
| · simp [hωK, Set.indicator_nonneg]) |
| filter_upwards [hpull, hmono] with ω hp hmω hωK |
| simpa only [hp, Set.indicator_of_mem hωK] using hmω |
|
|
| theorem condProb_le_on_of_inter_subset (hm : m ≤ mΩ) |
| {H K B : Set Ω} (hH : MeasurableSet H) (hK : MeasurableSet[m] K) |
| (hB : MeasurableSet B) (hsub : H ∩ K ⊆ B) : |
| ∀ᵐ ω ∂μ, ω ∈ K → (μ⟦H | m⟧) ω ≤ (μ⟦B | m⟧) ω := |
| condProb_le_on_of_ae_inter_subset hm hH hK hB (Eventually.of_forall hsub) |
|
|
| theorem prod_one_sub_le_exp (S : Finset ι) (q : ι → ℝ) {p : ℝ} |
| (hq : ∀ i ∈ S, 0 ≤ 1 - q i) (hp : ∀ i ∈ S, p ≤ q i) : |
| (∏ i ∈ S, (1 - q i)) ≤ Real.exp (-p * S.card) := by |
| calc |
| (∏ i ∈ S, (1 - q i)) ≤ ∏ _i ∈ S, Real.exp (-p) := by |
| apply Finset.prod_le_prod₀ hq |
| intro i hi |
| calc |
| 1 - q i ≤ 1 - p := sub_le_sub_left (hp i hi) 1 |
| _ ≤ Real.exp (-p) := by linarith [Real.add_one_le_exp (-p)] |
| _ = Real.exp (-p) ^ S.card := by simp |
| _ = Real.exp (-p * S.card) := by rw [← Real.exp_nat_mul]; congr 1; ring |
|
|
| section ConditionalIndependence |
|
|
| variable [StandardBorelSpace Ω] |
|
|
| theorem condProb_avoid_eq_prod (hm : m ≤ mΩ) (A : ι → Set Ω) |
| (hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ) |
| (S : Finset ι) : |
| μ⟦⋂ i ∈ S, (A i)ᶜ | m⟧ =ᵐ[μ] |
| fun ω => ∏ i ∈ S, (1 - (μ⟦A i | m⟧) ω) := by |
| have hmA : ∀ i, MeasurableSpace.generateFrom {A i} ≤ mΩ := by |
| intro i |
| apply MeasurableSpace.generateFrom_le |
| intro s hs |
| rcases Set.mem_singleton_iff.mp hs with rfl |
| exact hA i |
| have hi := (iCondIndepSet_iff_iCondIndep m hm A μ).mp hind |
| have hprod := (iCondIndep_iff m hm _ hmA μ).mp hi S |
| (f := fun i => (A i)ᶜ) (fun i _ => |
| (MeasurableSpace.measurableSet_generateFrom (Set.mem_singleton (A i))).compl) |
| have hcomp : ∀ᵐ ω ∂μ, ∀ i ∈ S, |
| (μ⟦(A i)ᶜ | m⟧) ω = 1 - (μ⟦A i | m⟧) ω := by |
| exact (ae_ball_iff S.countable_toSet).mpr (fun i _ => condProb_compl hm (hA i)) |
| filter_upwards [hprod, hcomp] with ω hω hc |
| rw [hω, Finset.prod_apply] |
| exact Finset.prod_congr rfl hc |
|
|
| theorem condProb_avoid_random_selection_le_exp_ae |
| (hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω) |
| (hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ) |
| (J : Ω → Finset ι) |
| (hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S}) |
| (hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H) |
| {p r : ℝ} (hp : 0 ≤ p) |
| (hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → p ≤ (μ⟦A i | m⟧) ω) |
| (hcard : ∀ᵐ ω ∂μ, ω ∈ H → r ≤ ((J ω).card : ℝ)) |
| (havoid : ∀ᵐ ω ∂μ, ω ∈ H → ∀ i ∈ J ω, ω ∉ A i) : |
| ∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-p * r) := by |
| classical |
| have hlocal : ∀ S ∈ I.powerset, ∀ᵐ ω ∂μ, |
| J ω = S → (μ⟦H | m⟧) ω ≤ Real.exp (-p * r) := by |
| intro S hS |
| have hSI : S ⊆ I := Finset.mem_powerset.mp hS |
| by_cases hSr : r ≤ (S.card : ℝ) |
| · have hsub : (H ∩ {ω | J ω = S} : Set Ω) ≤ᵐ[μ] (⋂ i ∈ S, (A i)ᶜ : Set Ω) := by |
| filter_upwards [havoid] with ω ha hω |
| change ω ∈ (⋂ i ∈ S, (A i)ᶜ : Set Ω) |
| simp only [Set.mem_iInter, Set.mem_compl_iff] |
| intro i hi |
| exact ha hω.1 i (hω.2.symm ▸ hi) |
| have hloc := condProb_le_on_of_ae_inter_subset (μ := μ) hm hH (hJ S) |
| (MeasurableSet.biInter S.countable_toSet (fun i _ => (hA i).compl)) hsub |
| have hprod := condProb_avoid_eq_prod (μ := μ) hm A hA hind S |
| have hpost : ∀ᵐ ω ∂μ, ∀ i ∈ S, |
| i ∈ J ω → p ≤ (μ⟦A i | m⟧) ω := |
| (ae_ball_iff S.countable_toSet).mpr (fun i hi => hposterior i (hSI hi)) |
| have hnonneg : ∀ᵐ ω ∂μ, ∀ i ∈ S, 0 ≤ 1 - (μ⟦A i | m⟧) ω := by |
| apply (ae_ball_iff S.countable_toSet).mpr |
| intro i _ |
| filter_upwards [condProb_nonneg (μ := μ) (m := m) (A i)ᶜ, |
| condProb_compl hm (hA i)] with ω hn hc |
| simpa only [hc] using hn |
| filter_upwards [hloc, hprod, hpost, hnonneg] with ω hl he hpo hnn hωS |
| calc |
| (μ⟦H | m⟧) ω ≤ (μ⟦⋂ i ∈ S, (A i)ᶜ | m⟧) ω := hl hωS |
| _ = ∏ i ∈ S, (1 - (μ⟦A i | m⟧) ω) := he |
| _ ≤ Real.exp (-p * S.card) := prod_one_sub_le_exp S _ hnn |
| (fun i hi => hpo i hi (hωS.symm ▸ hi)) |
| _ ≤ Real.exp (-p * r) := Real.exp_le_exp.mpr (by nlinarith) |
| · have hsub : (H ∩ {ω | J ω = S} : Set Ω) ≤ᵐ[μ] (∅ : Set Ω) := by |
| filter_upwards [hcard] with ω hc hω |
| have heq : J ω = S := hω.2 |
| exact (hSr (by simpa only [heq] using hc hω.1)).elim |
| have hloc := condProb_le_on_of_ae_inter_subset (μ := μ) hm hH (hJ S) |
| MeasurableSet.empty hsub |
| filter_upwards [hloc] with ω hl hωS |
| have hz : (μ⟦H | m⟧) ω ≤ 0 := by simpa using hl hωS |
| exact hz.trans (Real.exp_pos _).le |
| have hall := (ae_ball_iff I.powerset.countable_toSet).mpr hlocal |
| filter_upwards [hall] with ω hω |
| exact hω (J ω) (Finset.mem_powerset.mpr (hJI ω)) rfl |
|
|
| theorem condProb_avoid_random_selection_le_exp |
| (hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω) |
| (hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ) |
| (J : Ω → Finset ι) |
| (hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S}) |
| (hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H) |
| {p r : ℝ} (hp : 0 ≤ p) |
| (hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → p ≤ (μ⟦A i | m⟧) ω) |
| (hcard : ∀ ω ∈ H, r ≤ ((J ω).card : ℝ)) |
| (havoid : ∀ ω ∈ H, ∀ i ∈ J ω, ω ∉ A i) : |
| ∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-p * r) := |
| condProb_avoid_random_selection_le_exp_ae hm I A hA hind J hJ hJI hH hp |
| hposterior (Eventually.of_forall hcard) (Eventually.of_forall havoid) |
|
|
| theorem condProb_half_selection_le_exp |
| (hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω) |
| (hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ) |
| (J : Ω → Finset ι) |
| (hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S}) |
| (hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H) |
| {ρ M : ℝ} (hρ : 0 ≤ ρ) (hI : M / 20 ≤ (I.card : ℝ)) |
| (hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → ρ / 10 ≤ (μ⟦A i | m⟧) ω) |
| (hcard : ∀ ω ∈ H, (I.card : ℝ) / 2 ≤ ((J ω).card : ℝ)) |
| (havoid : ∀ ω ∈ H, ∀ i ∈ J ω, ω ∉ A i) : |
| ∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-M * ρ / 400) := by |
| have h := condProb_avoid_random_selection_le_exp hm I A hA hind J hJ hJI hH |
| (div_nonneg hρ (by norm_num : (0 : ℝ) ≤ 10)) hposterior hcard havoid |
| filter_upwards [h] with ω hω |
| apply hω.trans |
| apply Real.exp_le_exp.mpr |
| nlinarith |
|
|
| theorem condProb_avoid_intensity_le_exp_ae |
| (hm : m ≤ mΩ) (I : Finset ι) (A : ι → Set Ω) |
| (hA : ∀ i, MeasurableSet (A i)) (hind : iCondIndepSet m hm A μ) |
| (J : Ω → Finset ι) |
| (hJ : ∀ S : Finset ι, MeasurableSet[m] {ω | J ω = S}) |
| (hJI : ∀ ω, J ω ⊆ I) {H : Set Ω} (hH : MeasurableSet H) |
| {ρ M L : ℝ} (hρ : 0 ≤ ρ) (hL : L ≤ M * ρ) |
| (hI : M / 20 ≤ (I.card : ℝ)) |
| (hposterior : ∀ i ∈ I, ∀ᵐ ω ∂μ, i ∈ J ω → ρ / 10 ≤ (μ⟦A i | m⟧) ω) |
| (hcard : ∀ᵐ ω ∂μ, ω ∈ H → (I.card : ℝ) / 2 ≤ ((J ω).card : ℝ)) |
| (havoid : ∀ᵐ ω ∂μ, ω ∈ H → ∀ i ∈ J ω, ω ∉ A i) : |
| ∀ᵐ ω ∂μ, (μ⟦H | m⟧) ω ≤ Real.exp (-L / 400) := by |
| have h := condProb_avoid_random_selection_le_exp_ae hm I A hA hind J hJ hJI hH |
| (div_nonneg hρ (by norm_num : (0 : ℝ) ≤ 10)) hposterior hcard havoid |
| filter_upwards [h] with ω hω |
| apply hω.trans |
| apply Real.exp_le_exp.mpr |
| have hi := mul_le_mul_of_nonneg_right hI hρ |
| nlinarith |
|
|
| theorem iCondIndepSet_preimages {β : ι → Type*} |
| [mβ : ∀ i, MeasurableSpace (β i)] |
| (hm : m ≤ mΩ) (f : ∀ i, Ω → β i) (hf : ∀ i, Measurable (f i)) |
| (hind : iCondIndepFun m hm f μ) (C : ∀ i, Set (β i)) |
| (hC : ∀ i, MeasurableSet (C i)) : |
| iCondIndepSet m hm (fun i => f i ⁻¹' C i) μ := by |
| rw [iCondIndepSet_iff m hm _ (fun i => (hf i) (hC i)) μ] |
| intro S |
| exact (iCondIndepFun_iff_condExp_inter_preimage_eq_mul mβ f hf).mp hind S |
| (fun i _ => hC i) |
|
|
| end ConditionalIndependence |
| end Invariant233.ConditionalAvoidance |
|
|
| end |
|
|
| end OAI |
|
|
