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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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]
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
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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