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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 91 92 93 94 95 96 97 98 | import OAI.Probability.RadialTransition.ProductConfigurations
namespace OAI
universe uX uY
noncomputable section
open Filter MeasureTheory Set
open scoped ContDiff NNReal BigOperators ENNReal
namespace RadialTransition
section MapConfigurations
variable {X : Type uX} {Y : Type uY} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
[SecondCountableTopology X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y]
[SecondCountableTopology Y]
def mapParticle (f : X → Y) (x : ParticleState X) : ParticleState Y := ⟨x.1, f ∘ x.2⟩
lemma measurable_mapParticle {f : X → Y} (hf : Measurable f) : Measurable (mapParticle f) := by
apply measurable_particle_iff.mpr
intro n
exact (measurable_particle_mk n).comp (measurable_pi_iff.mpr (fun i => hf.comp (measurable_pi_apply i)))
lemma particleBase_map (μ : Measure X) [IsFiniteMeasure μ] (z : ℝ)
{f : X → Y} (hf : Measurable f) :
Measure.map (mapParticle f) (particleBase μ z) = particleBase (Measure.map f μ) z := by
unfold particleBase
rw [Measure.map_sum (measurable_mapParticle hf).aemeasurable]
congr 1
funext n
rw [Measure.map_smul _ (measurable_mapParticle hf).aemeasurable,
Measure.map_map (measurable_mapParticle hf) (measurable_particle_mk n)]
congr 1
rw [← Measure.pi_map_pi (fun _ : Fin n => hf.aemeasurable)]
exact (Measure.map_map (measurable_particle_mk n)
(show Measurable (fun x : Fin n → X => fun i => f (x i)) from
measurable_pi_iff.mpr (fun i => hf.comp (measurable_pi_apply i)))).symm
end MapConfigurations
section FiniteFock
variable {X : Type uX} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
[SecondCountableTopology X]
variable (F : Multiset X → ℝ≥0∞) (hF : ContinuousConfiguration F)
include hF
lemma particleBase_zero_observable (z : ℝ) (s : Multiset X) :
(∫⁻ x, F (s+particleMultiset x) ∂particleBase (0 : Measure X) z) = F s := by
rw [lintegral_particleBase _ _ (continuous_particleObservable F hF s).measurable]
rw [tsum_eq_single 0]
· simp [particleMultiset, configurationMultiset]
· intro n hn
have hμ : Measure.pi (fun _ : Fin n => (0 : Measure X)) = 0 := by
apply Measure.measure_univ_eq_zero.mp
rw [Measure.pi_univ]
simp [hn]
rw [hμ, lintegral_zero_measure, mul_zero]
lemma particleBase_fin_sum_observable (n : ℕ) (μ : Fin n → Measure X)
[∀ i, IsFiniteMeasure (μ i)] (z : ℝ) (s : Multiset X) :
(∫⁻ x, F (s+particleMultiset x) ∂particleBase (∑ i, μ i) z) =
∫⁻ x, F (s+multiParticleMultiset x) ∂Measure.pi (fun i => particleBase (μ i) z) := by
induction n generalizing s with
| zero =>
simp only [Fin.sum_univ_zero]
rw [particleBase_zero_observable F hF]
simp [multiParticleMultiset]
| succ n ih =>
rw [Fin.sum_univ_succ, particleBase_add_observable F hF]
simp_rw [ih]
let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) => ParticleState X) 0
have hp := MeasurePreserving.symm e
(measurePreserving_piFinSuccAbove (fun i => particleBase (μ i) z) 0)
have hc := (continuous_multiParticleObservable F hF s (I := Fin (n+1))).measurable
rw [← hp.lintegral_comp hc]
change _ = ∫⁻ p : ParticleState X × (Fin n → ParticleState X),
F (s+multiParticleMultiset (e.symm p))
∂(particleBase (μ 0) z).prod (Measure.pi (fun i => particleBase (μ i.succ) z))
have hpmeas : Measurable (fun p : ParticleState X × (Fin n → ParticleState X) =>
F (s+multiParticleMultiset (e.symm p))) := hc.comp e.symm.measurable
rw [lintegral_prod _ hpmeas.aemeasurable]
apply lintegral_congr
intro x
apply lintegral_congr
intro y
congr 1
simp [multiParticleMultiset, Fin.sum_univ_succ, e,
MeasurableEquiv.piFinSuccAbove_symm_apply, add_assoc]
end FiniteFock
end RadialTransition
end
end OAI
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