| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| namespace LogConcaveSampling |
|
|
| lemma lintegral_fin_nat_prod {n : ℕ} {E : Type*} |
| [MeasurableSpace E] {μ : Fin n → Measure E} |
| [∀ i, SigmaFinite (μ i)] (f : Fin n → E → ℝ≥0∞) |
| (hf : ∀ i, Measurable (f i)) : |
| ∫⁻ x : Fin n → E, ∏ i, f i (x i) ∂Measure.pi μ = |
| ∏ i, ∫⁻ x, f i x ∂μ i := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| calc |
| _ = ∫⁻ x : E × (Fin n → E), |
| f 0 x.1 * ∏ i : Fin n, f (Fin.succ i) (x.2 i) |
| ∂((μ 0).prod (Measure.pi (fun i ↦ μ i.succ))) := by |
| rw [((measurePreserving_piFinSuccAbove μ 0).symm).lintegral_map_equiv] |
| simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv, |
| Fin.prod_univ_succ, Fin.insertNth_zero, Equiv.coe_fn_mk, Fin.cons_succ, |
| Fin.zero_succAbove, cast_eq, Fin.cons_zero] |
| _ = (∫⁻ x, f 0 x ∂μ 0) * ∏ i : Fin n, ∫⁻ x, f (Fin.succ i) x ∂μ i.succ := by |
| rw [lintegral_prod_mul (g := fun x : Fin n → E => ∏ i, f i.succ (x i)) |
| (hf 0).aemeasurable |
| (Finset.measurable_prod _ (fun i _ => (hf i.succ).comp (measurable_pi_apply i))).aemeasurable, ih _ (fun i => hf _)] |
| _ = _ := by rw [Fin.prod_univ_succ] |
|
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| theorem pi_withDensity_fin {n : ℕ} {E : Type*} |
| [MeasurableSpace E] (μ : Fin n → Measure E) |
| [∀ i, SigmaFinite (μ i)] (f : Fin n → E → ℝ≥0∞) |
| (hf : ∀ i, Measurable (f i)) [∀ i, SigmaFinite ((μ i).withDensity (f i))] : |
| Measure.pi (fun i => (μ i).withDensity (f i)) = |
| (Measure.pi μ).withDensity (fun x => ∏ i, f i (x i)) := by |
| apply Measure.pi_eq |
| intro s hs |
| rw [withDensity_apply _ (MeasurableSet.univ_pi hs)] |
| rw [Measure.restrict_pi_pi] |
| rw [lintegral_fin_nat_prod _ hf] |
| apply Finset.prod_congr rfl |
| intro i _ |
| exact (withDensity_apply _ (hs i)).symm |
|
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| end LogConcaveSampling |
|
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| end UpperProof |
| end |
| end |
| end |
|
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| end OAI |
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|
