lean/OAI/Geometry/PinnedDistances/ProductLimits.lean

import OAI.Geometry.PinnedDistances.VarianceRates

namespace OAI

universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12
universe u13 u14

section
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators BoundedContinuousFunction
namespace WeakPinned.Bounded

variable {E : Type u1} {X : Type u2} {J : Type u3} [MeasurableSpace E] [MeasurableSpace X]
  [TopologicalSpace E] [TopologicalSpace X] [BorelSpace E] [BorelSpace X]
  [SecondCountableTopology E] [SecondCountableTopology X]
  [HasOuterApproxClosed E] [HasOuterApproxClosed X] [Fintype J]

omit [SecondCountableTopology E] in
lemma weak_product_limit
    (Rn : ℕ → ProbabilityMeasure (E × (J → X))) (R : ProbabilityMeasure (E × (J → X)))
    (Qn : ℕ → ProbabilityMeasure E) (Q : ProbabilityMeasure E)
    (αn : ℕ → J → ProbabilityMeasure X) (β : J → ProbabilityMeasure X)
    (hR : Tendsto Rn atTop (𝓝 R)) (hQ : Tendsto Qn atTop (𝓝 Q))
    (hα : ∀ j,Tendsto (fun n => αn n j) atTop (𝓝 (β j)))
    (herr : ∀ (f : E →ᵇ ℝ) (g : J → X →ᵇ ℝ),
      Tendsto (fun n => (∫ p,f p.1*(∏ j,g j (p.2 j)) ∂ (Rn n : Measure (E × (J → X))))-
        (∫ e,f e ∂ (Qn n : Measure E))*(∏ j,∫ x,g j x ∂ (αn n j : Measure X))) atTop (𝓝 0)) :
    (R : Measure (E × (J → X)))=(Q : Measure E).prod (Measure.pi (fun j => (β j : Measure X))) := by
  apply Measure.eq_prod_of_integral_mul_prod_boundedContinuousFunction
  intro f g
  let F : E × (J → X) →ᵇ ℝ :=
    f.compContinuous ⟨Prod.fst,continuous_fst⟩ *
      ∏ j,(g j).compContinuous ⟨fun p : E × (J → X) => p.2 j,by fun_prop⟩
  have hRf := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hR F
  have hQf := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hQ f
  have hβg (j) := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp (hα j) (g j)
  have ht := hRf.sub (hQf.mul (tendsto_finsetProd Finset.univ (fun j _ => hβg j)))
  have hF (p : E × (J → X)) : F p=f p.1 * ∏ j,g j (p.2 j) := by simp [F]
  simp only [hF] at ht
  have he := tendsto_nhds_unique ht (herr f g)
  rw [integral_fintype_prod_eq_prod]
  exact sub_eq_zero.mp he

end WeakPinned.Bounded

end
end
section
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators BoundedContinuousFunction
namespace WeakPinned.Bounded

lemma weak_pair_product_limit {X : Type u4} {Y : Type u5} [MeasurableSpace X] [MeasurableSpace Y]
    [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y]
    [HasOuterApproxClosed X] [HasOuterApproxClosed Y]
    [SecondCountableTopology X] [SecondCountableTopology Y]
    (Rn : ℕ → ProbabilityMeasure (X × Y)) (R : ProbabilityMeasure (X × Y))
    (μn : ℕ → ProbabilityMeasure X) (μ : ProbabilityMeasure X)
    (νn : ℕ → ProbabilityMeasure Y) (ν : ProbabilityMeasure Y)
    (hR : Tendsto Rn atTop (𝓝 R)) (hμ : Tendsto μn atTop (𝓝 μ)) (hν : Tendsto νn atTop (𝓝 ν))
    (herr : ∀ (f : X →ᵇ ℝ) (g : Y →ᵇ ℝ),
      Tendsto (fun n => (∫ p,f p.1*g p.2 ∂ (Rn n : Measure (X × Y)))-
        (∫ x,f x ∂ (μn n : Measure X))*(∫ y,g y ∂ (νn n : Measure Y))) atTop (𝓝 0)) :
    (R : Measure (X × Y))=(μ : Measure X).prod (ν : Measure Y) := by
  apply Measure.eq_prod_of_integral_mul_boundedContinuousFunction
  intro f g
  let F : X × Y →ᵇ ℝ := f.compContinuous ⟨Prod.fst,continuous_fst⟩ *
    g.compContinuous ⟨Prod.snd,continuous_snd⟩
  have ht := (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hR F).sub
    ((ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f).mul
      (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hν g))
  exact sub_eq_zero.mp (tendsto_nhds_unique ht (herr f g))

lemma pair_of_pi_map {T : Type u6} {E : Type u7} {X : Type u8} [MeasurableSpace T] [MeasurableSpace E] [MeasurableSpace X]
    (R : Measure T) (Q : Measure E) (μ : Measure X) [SFinite Q] [IsFiniteMeasure μ]
    (e : T → E) (z w : T → X) (he : Measurable e) (hz : Measurable z) (hw : Measurable w)
    (hm : R.map (fun r => (e r,![z r,w r]))=Q.prod (Measure.pi (fun _ : Fin 2 => μ))) :
    R.map (fun r => (e r,(z r,w r)))=Q.prod (μ.prod μ) := by
  have hp : Measurable (fun r => (e r,![z r,w r])) := by
    apply he.prodMk
    apply Measurable.of_eval
    intro j
    fin_cases j
    · exact hz
    · exact hw
  let hme := MeasurableEquiv.finTwoArrow (α:=X)
  have hm2 : Measurable (Prod.map (id : E → E) hme) := measurable_id.prodMap hme.measurable
  have hh := congrArg (fun m : Measure (E × (Fin 2 → X)) => m.map (Prod.map id hme)) hm
  rw [Measure.map_map hm2 hp,← Measure.map_prod_map _ _ measurable_id hme.measurable,
    Measure.map_id,(measurePreserving_finTwoArrow μ).map_eq] at hh
  exact hh

end WeakPinned.Bounded

end
end
section
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators BoundedContinuousFunction
namespace WeakPinned.Bounded

variable {E : Type u9} {X : Type u10} {T : Type u11} [MeasurableSpace E] [MeasurableSpace X] [MeasurableSpace T]
  [TopologicalSpace E] [TopologicalSpace X] [TopologicalSpace T]
  [BorelSpace E] [BorelSpace X] [BorelSpace T]
  [SecondCountableTopology E] [SecondCountableTopology X]
  [HasOuterApproxClosed E] [HasOuterApproxClosed X]

lemma weak_double_product_limit (Rn : ℕ → ProbabilityMeasure T) (R : ProbabilityMeasure T)
    (Qn : ℕ → ProbabilityMeasure E) (Q : ProbabilityMeasure E)
    (αn : ℕ → ProbabilityMeasure X) (β : ProbabilityMeasure X)
    (e : T → E) (z w : T → X) (he : Continuous e) (hz : Continuous z) (hw : Continuous w)
    (hR : Tendsto Rn atTop (𝓝 R)) (hQ : Tendsto Qn atTop (𝓝 Q)) (hα : Tendsto αn atTop (𝓝 β))
    (herr : ∀ (f : E →ᵇ ℝ) (g h : X →ᵇ ℝ),
      Tendsto (fun n => (∫ r,f (e r)*g (z r)*h (w r) ∂ (Rn n : Measure T))-
        (∫ q,f q ∂ (Qn n : Measure E))*(∫ x,g x ∂ (αn n : Measure X))*
          (∫ x,h x ∂ (αn n : Measure X))) atTop (𝓝 0)) :
    (R:Measure T).map (fun r => (e r,(z r,w r)))=(Q:Measure E).prod ((β:Measure X).prod β) := by
  let k : T → E × (Fin 2 → X) := fun r => (e r,![z r,w r])
  have hk : Continuous k := by
    apply he.prodMk
    apply continuous_pi
    intro i
    fin_cases i
    · exact hz
    · exact hw
  have hRm := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Rn R hR hk
  have hh := weak_product_limit (fun n => (Rn n).map k)
    (R.map k) Qn Q (fun n _ => αn n) (fun _ : Fin 2 => β) hRm hQ (fun _ => hα) ?_
  · exact pair_of_pi_map (R:Measure T) (Q:Measure E) (β:Measure X) e z w he.measurable hz.measurable hw.measurable hh
  · intro f g
    have hl := herr f (g 0) (g 1)
    convert hl using 1
    funext n
    rw [show ((Rn n).map k : Measure (E × (Fin 2 → X)))=(Rn n:Measure T).map k from rfl]
    rw [integral_map_of_stronglyMeasurable hk.measurable (by fun_prop)]
    simp only [Fin.prod_univ_two,k,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one,mul_assoc]

end WeakPinned.Bounded

end
end
section
noncomputable section
open MeasureTheory Filter
open scoped Topology ENNReal NNReal BoundedContinuousFunction
namespace WeakPinned.Bounded

variable {X : Type u12} [MeasurableSpace X] [TopologicalSpace X] [BorelSpace X]
  [TopologicalSpace.PseudoMetrizableSpace X]

lemma measure_le_of_continuous_lintegral (μ ν : Measure X) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
    (h : ∀ f : X →ᵇ ℝ≥0,(∫⁻ x,f x ∂ μ)≤∫⁻ x,f x ∂ ν) : μ≤ν := by
  have hclosed (K : Set X) (hK : IsClosed K) : μ K≤ν K := by
    exact le_of_tendsto_of_tendsto
      (HasOuterApproxClosed.tendsto_lintegral_apprSeq hK μ)
      (HasOuterApproxClosed.tendsto_lintegral_apprSeq hK ν)
      (Eventually.of_forall (fun n => h (hK.apprSeq n)))
  apply Measure.le_iff.mpr
  intro s hs
  rw [hs.measure_eq_iSup_isClosed_of_ne_top (measure_ne_top μ s)]
  exact iSup_le (fun K => iSup_le (fun hKs => iSup_le (fun hK => (hclosed K hK).trans (measure_mono hKs))))

lemma weak_domination (μn νn : ℕ → ProbabilityMeasure X) (μ ν : ProbabilityMeasure X)
    (hμ : Tendsto μn atTop (𝓝 μ)) (hν : Tendsto νn atTop (𝓝 ν)) (c : ℝ≥0)
    (hle : ∀ n,(μn n : Measure X)≤(c:ℝ≥0∞) • (νn n : Measure X)) :
    (μ : Measure X)≤(c:ℝ≥0∞) • (ν : Measure X) := by
  have : IsFiniteMeasure ((c:ℝ≥0∞) • (ν:Measure X)) := (ν:Measure X).smul_finite ENNReal.coe_ne_top
  apply measure_le_of_continuous_lintegral
  intro f
  have hμf := ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mp hμ f
  have hνf := ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mp hν f
  rw [lintegral_smul_measure]
  apply le_of_tendsto_of_tendsto hμf
    (ENNReal.Tendsto.const_mul hνf (Or.inr ENNReal.coe_ne_top))
  filter_upwards [] with n
  simpa only [lintegral_smul_measure,smul_eq_mul] using lintegral_mono' (hle n) (le_refl (fun x => (f x:ℝ≥0∞)))

end WeakPinned.Bounded

end
end
section
noncomputable section
open MeasureTheory Filter
open scoped Topology NNReal ENNReal
namespace WeakPinned.Bounded
variable {X : Type u13} {Y : Type u14} [MeasurableSpace X] [MeasurableSpace Y]
  [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y]
  [HasOuterApproxClosed Y]

lemma weak_map_eq (αn : ℕ → ProbabilityMeasure X) (α : ProbabilityMeasure X)
    (βn : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y)
    (f : X → Y) (hf : Continuous f) (hα : Tendsto αn atTop (𝓝 α))
    (hβ : Tendsto βn atTop (𝓝 β)) (hm : ∀ n,(αn n:Measure X).map f=(βn n:Measure Y)) :
    (α:Measure X).map f=(β:Measure Y) := by
  have he (n) : (αn n).map f=βn n := ProbabilityMeasure.toMeasure_injective (hm n)
  have hl := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous αn α hα hf
  simp only [he] at hl
  exact congrArg ProbabilityMeasure.toMeasure (tendsto_nhds_unique hl hβ)

omit [HasOuterApproxClosed Y] in
lemma weak_map_domination [TopologicalSpace.PseudoMetrizableSpace Y]
    (αn : ℕ → ProbabilityMeasure X) (α : ProbabilityMeasure X)
    (βn : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y)
    (f : X → Y) (hf : Continuous f) (hα : Tendsto αn atTop (𝓝 α))
    (hβ : Tendsto βn atTop (𝓝 β)) (c : ℝ≥0)
    (hm : ∀ n,(αn n:Measure X).map f≤(c:ℝ≥0∞) • (βn n:Measure Y)) :
    (α:Measure X).map f≪(β:Measure Y) := by
  have hl := weak_domination (fun n => (αn n).map f)
    βn (α.map f) β
    (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous αn α hα hf) hβ c hm
  exact Measure.absolutelyContinuous_of_le_smul hl

end WeakPinned.Bounded

end
end

end OAI