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a2ee19a | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import OAI.Probability.RadialTransition.ClusterConvolution
namespace OAI
universe uX uY uIndex
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
lemma clusterConvolution_add {X : Type uX} (F G : Multiset X → ℝ≥0∞) (s u : Multiset X) :
clusterConvolution F G (s+u) =
(s.antidiagonal.map (fun p => clusterConvolution (fun v => F (p.1+v)) (fun v => G (p.2+v)) u)).sum := by
simp [clusterConvolution, Multiset.antidiagonal_add, Multiset.map_bind, Multiset.sum_bind,
Multiset.map_map]
section Measurable
variable {X : Type uX} [MeasurableSpace X]
lemma measurable_configuration_subtype {F : Multiset X → ℝ≥0∞} (hF : MeasurableConfiguration F)
(s : Multiset X) {ι : Type} [Fintype ι] (p : ι → Prop) [DecidablePred p] :
Measurable (fun x : ι → X => F (s+configurationMultiset (fun i : Subtype p => x i))) := by
exact (hF s (Subtype p)).comp (measurable_pi_iff.mpr (fun i => measurable_pi_apply i.val))
lemma measurable_configuration_restrict {F : Multiset X → ℝ≥0∞} (hF : MeasurableConfiguration F)
(s : Multiset X) {ι : Type} [Fintype ι] (T : Finset ι) :
Measurable (fun x : ι → X => F (s+configurationMultiset (fun i : T => x i))) := by
apply (hF s T).comp
exact measurable_pi_iff.mpr (fun i => measurable_pi_apply i.val)
lemma measurable_multiset_sum {Y : Type uY} [MeasurableSpace Y] {ι : Type uIndex}
(s : Multiset ι) (f : ι → Y → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) :
Measurable (fun x => (s.map (fun i => f i x)).sum) := by
induction s using Multiset.induction_on with
| empty => simp
| cons a s ih =>
simp only [Multiset.map_cons, Multiset.sum_cons]
exact measurable_add.comp ((hf a).prodMk ih)
lemma measurable_clusterConvolution {F G : Multiset X → ℝ≥0∞}
(hF : MeasurableConfiguration F) (hG : MeasurableConfiguration G) :
MeasurableConfiguration (clusterConvolution F G) := by
classical
intro s ι _
simp_rw [clusterConvolution_add, clusterConvolution_configuration]
apply measurable_multiset_sum
intro p
apply Finset.measurable_sum
intro T _
exact (measurable_configuration_subtype hF p.1 (fun i => i ∉ T)).mul
(measurable_configuration_restrict hG p.2 T)
lemma configuration_integral_card (F : Multiset X → ℝ≥0∞) (μ : Measure X) [SigmaFinite μ]
(ι : Type) [Fintype ι] :
(∫⁻ x : ι → X, F (configurationMultiset x) ∂Measure.pi (fun _ => μ)) =
∫⁻ x : Fin (Fintype.card ι) → X, F (configurationMultiset x) ∂Measure.pi (fun _ => μ) := by
exact functional_integral_equiv (fun _ι _ x => F (configurationMultiset x))
(fun _ι _κ _ _ e x => congrArg F (configurationMultiset_equiv e x)) (Fintype.equivFin ι) (fun _ => μ)
lemma configuration_split_integral {F G : Multiset X → ℝ≥0∞}
(hF : MeasurableConfiguration F) (hG : MeasurableConfiguration G)
(μ : Measure X) [SigmaFinite μ] {ι : Type} [Fintype ι] [DecidableEq ι] (T : Finset ι) :
(∫⁻ x : ι → X, F (configurationMultiset (fun i : {i // i ∉ T} => x i)) *
G (configurationMultiset (fun i : T => x i)) ∂Measure.pi (fun _ => μ)) =
(∫⁻ x : {i // i ∉ T} → X, F (configurationMultiset x) ∂Measure.pi (fun _ => μ))*
(∫⁻ x : T → X, G (configurationMultiset x) ∂Measure.pi (fun _ => μ)) := by
have hp := measurePreserving_piEquivPiSubtypeProd (fun _ : ι => μ) (fun i => i ∈ T)
have hG0 : Measurable (fun x : T → X => G (configurationMultiset x)) := by
simpa only [zero_add] using hG 0 T
have hF0 : Measurable (fun x : {i // i ∉ T} → X => F (configurationMultiset x)) := by
simpa only [zero_add] using hF 0 {i // i ∉ T}
have hmeas : Measurable (fun p : (T → X) × ({i // i ∉ T} → X) =>
G (configurationMultiset p.1)*F (configurationMultiset p.2)) := by
exact (hG0.comp measurable_fst).mul (hF0.comp measurable_snd)
have hh := hp.lintegral_comp hmeas
simp only [MeasurableEquiv.piEquivPiSubtypeProd_apply] at hh
have hi : Subtype.fintype (fun i : ι => i ∈ T) = Finset.Subtype.fintype T := Subsingleton.elim _ _
rw [hi] at hh
have hent := lintegral_prod_mul (μ := Measure.pi (fun _ : T => μ))
(ν := Measure.pi (fun _ : {i // i ∉ T} => μ)) hG0.aemeasurable hF0.aemeasurable
have hh := hh.trans hent
simpa only [mul_comm] using hh
end Measurable
end RadialTransition
end
end OAI
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