openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalEventLocalization.lean
| import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | |
| import Mathlib.Probability.Notation | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ProbabilityTheory | |
| namespace Invariant233.ConditionalEventLocalization | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsFiniteMeasure μ] {m : MeasurableSpace Ω} | |
| theorem condProb_mono_on (hm : m ≤ mΩ) {A B S : Set Ω} | |
| (hA : MeasurableSet[mΩ] A) (hB : MeasurableSet[mΩ] B) (hS : MeasurableSet[m] S) | |
| (hsub : ∀ᵐ ω ∂μ, ω ∈ S → ω ∈ A → ω ∈ B) : | |
| ∀ᵐ ω ∂μ, ω ∈ S → (μ⟦A | m⟧) ω ≤ (μ⟦B | m⟧) ω := by | |
| have hAi : Integrable (A.indicator (fun _ => (1 : ℝ))) μ := | |
| (integrable_const _).indicator hA | |
| have hBi : Integrable (B.indicator (fun _ => (1 : ℝ))) μ := | |
| (integrable_const _).indicator hB | |
| have hle : S.indicator (A.indicator (fun _ => (1 : ℝ))) ≤ᵐ[μ] | |
| S.indicator (B.indicator (fun _ => (1 : ℝ))) := by | |
| filter_upwards [hsub] with ω hω | |
| by_cases hs : ω ∈ S | |
| · simp only [indicator_of_mem hs] | |
| by_cases ha : ω ∈ A | |
| · simp only [indicator_of_mem ha, indicator_of_mem (hω hs ha), le_refl] | |
| · simp only [indicator_of_notMem ha] | |
| exact indicator_nonneg (fun _ _ => by norm_num) _ | |
| · simp only [indicator_of_notMem hs, le_refl] | |
| have hc := condExp_mono (m := m) (hAi.indicator (hm _ hS)) | |
| (hBi.indicator (hm _ hS)) hle | |
| filter_upwards [hc, condExp_indicator hAi hS, condExp_indicator hBi hS] | |
| with ω hle hleft hright | |
| intro hs | |
| rw [hleft, hright, indicator_of_mem hs, indicator_of_mem hs] at hle | |
| exact hle | |
| theorem condProb_lower_on_of_forcing (hm : m ≤ mΩ) {A B S : Set Ω} | |
| (hA : MeasurableSet[mΩ] A) (hB : MeasurableSet[mΩ] B) (hS : MeasurableSet[m] S) | |
| (hsub : ∀ᵐ ω ∂μ, ω ∈ S → ω ∈ A → ω ∈ B) | |
| (r : Ω → ℝ) (hlower : ∀ᵐ ω ∂μ, ω ∈ S → r ω ≤ (μ⟦A | m⟧) ω) : | |
| ∀ᵐ ω ∂μ, ω ∈ S → r ω ≤ (μ⟦B | m⟧) ω := by | |
| filter_upwards [ Ω mΩ μ _ m hm A B S hA hB hS hsub, hlower] with ω hmono hlo | |
| exact fun hs => (hlo hs).trans (hmono hs) | |
| end Invariant233.ConditionalEventLocalization | |
| end | |
| end OAI | |