| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| namespace Problem367 | |
| theorem integral_of_conditional_versions | |
| {T Ω : Type*} [mT : MeasurableSpace T] (m : MeasurableSpace Ω) | |
| [mΩ : MeasurableSpace Ω] (ν : Measure T) (μ : Measure Ω) [SFinite ν] [SFinite μ] | |
| (hm : m ≤ mΩ) [SigmaFinite (μ.trim hm)] | |
| {F G : T × Ω → ℝ} | |
| (hF : Integrable F (ν.prod μ)) (hG : Integrable G (ν.prod μ)) | |
| (hGm : StronglyMeasurable[@MeasurableSpace.prod T Ω mT m] G) | |
| (hcond : ∀ᵐ t ∂ν, | |
| (fun ω => G (t, ω)) =ᵐ[μ] μ[(fun ω => F (t, ω)) | m]) : | |
| (fun ω => ∫ t, G (t, ω) ∂ν) =ᵐ[μ] | |
| μ[(fun ω => ∫ t, F (t, ω) ∂ν) | m] := by | |
| apply ae_eq_condExp_of_forall_setIntegral_eq hm hF.integral_prod_right | |
| · intro s hs hfin | |
| exact hG.integral_prod_right.integrableOn | |
| · intro s hs hfin | |
| have hFrestrict : Integrable F (ν.prod (μ.restrict s)) := | |
| hF.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self) | |
| have hGrestrict : Integrable G (ν.prod (μ.restrict s)) := | |
| hG.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self) | |
| calc | |
| (∫ ω in s, ∫ t, G (t, ω) ∂ν ∂μ) = | |
| ∫ t, ∫ ω in s, G (t, ω) ∂μ ∂ν := | |
| (integral_integral_swap (f := fun t ω => G (t, ω)) hGrestrict).symm | |
| _ = ∫ t, ∫ ω in s, F (t, ω) ∂μ ∂ν := by | |
| apply integral_congr_ae | |
| filter_upwards [hcond, hF.prod_right_ae] with t ht hFt | |
| exact (setIntegral_congr_ae (hm s hs) (ht.mono (fun ω hω _ => hω))).trans | |
| (setIntegral_condExp hm hFt hs) | |
| _ = ∫ ω in s, ∫ t, F (t, ω) ∂ν ∂μ := | |
| integral_integral_swap (f := fun t ω => F (t, ω)) hFrestrict | |
| · have hgm : StronglyMeasurable[m] (fun ω => ∫ t, G (t, ω) ∂ν) := by | |
| let : MeasurableSpace Ω := m | |
| exact hGm.integral_prod_left' | |
| exact hgm.aestronglyMeasurable | |
| theorem condExp_setIntegral_martingale | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (ℱ : Filtration ℝ mΩ) {f : ℝ → Ω → ℝ} (hf : Martingale f ℱ μ) | |
| {s t : ℝ} (hst : s ≤ t) | |
| (hInt : Integrable (fun p : ℝ × Ω => f p.1 p.2) | |
| ((volume.restrict (Set.Ioc s t)).prod μ)) : | |
| μ[(fun ω => ∫ u in Set.Ioc s t, f u ω) | ℱ s] =ᵐ[μ] | |
| (fun ω => (t - s) * f s ω) := by | |
| let ν : Measure ℝ := volume.restrict (Set.Ioc s t) | |
| have hGm : StronglyMeasurable[@MeasurableSpace.prod ℝ Ω inferInstance (ℱ s)] | |
| (fun p : ℝ × Ω => f s p.2) := by | |
| let : MeasurableSpace Ω := ℱ s | |
| exact (hf.stronglyMeasurable s).comp_measurable measurable_snd | |
| have hcond : ∀ᵐ u ∂ν, | |
| (fun ω => f s ω) =ᵐ[μ] μ[(fun ω => f u ω) | ℱ s] := by | |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu | |
| exact (hf.condExp_ae_eq hu.1.le).symm | |
| have h := integral_of_conditional_versions (ℱ s) ν μ (ℱ.le s) | |
| hInt ((hf.integrable s).comp_snd ν) hGm hcond | |
| simpa [ν, Real.volume_real_Ioc_of_le hst] using h.symm | |
| theorem condExp_intervalIntegral_martingale | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (ℱ : Filtration ℝ mΩ) {f : ℝ → Ω → ℝ} (hf : Martingale f ℱ μ) | |
| {s t : ℝ} (hst : s ≤ t) | |
| (hInt : Integrable (fun p : ℝ × Ω => f p.1 p.2) | |
| ((volume.restrict (Set.Ioc s t)).prod μ)) : | |
| μ[(fun ω => ∫ u in s..t, f u ω) | ℱ s] =ᵐ[μ] | |
| (fun ω => (t - s) * f s ω) := by | |
| simpa only [intervalIntegral.integral_of_le hst] using | |
| condExp_setIntegral_martingale μ ℱ hf hst hInt | |
| end Problem367 | |
| end | |
| end OAI | |