openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /ConditionalKernelSupport.lean
| import Mathlib.Probability.Kernel.Condexp | |
| import Mathlib.Tactic | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory | |
| namespace Invariant233.CrossingAvoidance | |
| variable {Ω β : Type*} | |
| theorem ae_ae_condExpKernel_of_ae (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} | |
| [StandardBorelSpace Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ] | |
| {P : Ω → Prop} (hP : ∀ᵐ ω ∂μ, P ω) : | |
| ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, P ω := by | |
| apply ae_of_ae_trim hm' | |
| apply Measure.ae_ae_of_ae_comp | |
| rwa [condExpKernel_comp_trim hm'] | |
| theorem condExpKernel_ae_eq_of_measurable (m' : MeasurableSpace Ω) | |
| {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] | |
| [mβ : MeasurableSpace β] [MeasurableEq β] | |
| (hm' : m' ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (f : Ω → β) (hf : Measurable[m'] f) : | |
| ∀ᵐ x ∂μ, ∀ᵐ ω ∂condExpKernel μ m' x, f ω = f x := by | |
| let : MeasurableSpace (Ω × Ω) := m'.prod mΩ | |
| have hf1 : @Measurable (Ω × Ω) β (m'.prod mΩ) mβ (fun z => f z.1) := | |
| hf.comp (@measurable_fst Ω Ω m' mΩ) | |
| have hf2 : @Measurable (Ω × Ω) β (m'.prod mΩ) mβ (fun z => f z.2) := | |
| (hf.mono hm' le_rfl).comp (@measurable_snd Ω Ω m' mΩ) | |
| have hdiag : @Measurable Ω (Ω × Ω) mΩ (m'.prod mΩ) (fun x => (x, x)) := | |
| (measurable_id'' hm').prodMk measurable_id | |
| have hprod : ∀ᵐ z ∂(μ.trim hm') ⊗ₘ condExpKernel μ m', f z.2 = f z.1 := by | |
| rw [compProd_trim_condExpKernel hm'] | |
| exact (ae_map_iff hdiag.aemeasurable (measurableSet_eq_fun hf2 hf1)).mpr | |
| (Filter.Eventually.of_forall (fun _ => rfl)) | |
| exact ae_of_ae_trim hm' (Measure.ae_ae_of_ae_compProd hprod) | |
| end Invariant233.CrossingAvoidance | |
| end OAI | |