import OAI.Dynamics.MultipleMixing.Entropy namespace OAI noncomputable section namespace Rokhlin section open scoped InnerProductSpace CStarAlgebra open ContinuousLinearMap open MeasureTheory Filter open scoped BigOperators Topology variable {Ω : Type*} [MeasurableSpace Ω] def timeMap (T : Ω ≃ᵐ Ω) (n : ℤ) : Ω → Ω := ⇑((T.toEquiv : Equiv.Perm Ω) ^ n) def IsMixing (μ : Measure Ω) (T : Ω ≃ᵐ Ω) : Prop := ∀ A B : Set Ω, MeasurableSet A → MeasurableSet B → Tendsto (fun n : ℤ => μ (A ∩ timeMap T n ⁻¹' B)) (comap Int.natAbs atTop) (𝓝 (μ A * μ B)) def layoutTime {k : ℕ} (gaps : Fin (k - 1) → ℕ+) (i : Fin k) : ℕ := ∑ j : Fin (k - 1), if j.val < i.val then (gaps j : ℕ) else 0 def MixingOfOrder (μ : Measure Ω) (T : Ω ≃ᵐ Ω) (k : ℕ) : Prop := ∀ A : Fin k → Set Ω, (∀ i, MeasurableSet (A i)) → Tendsto (fun gaps : Fin (k - 1) → ℕ+ => μ (⋂ i : Fin k, timeMap T (layoutTime gaps i : ℤ) ⁻¹' A i)) atTop (𝓝 (∏ i : Fin k, μ (A i))) section Rectangle variable {H₁ H₂ H₃ H₄ : Type*} [NormedAddCommGroup H₁] [InnerProductSpace ℂ H₁] [CompleteSpace H₁] [NormedAddCommGroup H₂] [InnerProductSpace ℂ H₂] [CompleteSpace H₂] [NormedAddCommGroup H₃] [InnerProductSpace ℂ H₃] [CompleteSpace H₃] [NormedAddCommGroup H₄] [InnerProductSpace ℂ H₄] [CompleteSpace H₄] open ContinuousLinearMap theorem rectangle_square_identities (F : H₂ →L[ℂ] H₁) (G : H₄ →L[ℂ] H₃) (K : H₃ →L[ℂ] H₁) (N : H₄ →L[ℂ] H₂) (h₁ : F ∘L N = K ∘L G) (h₂ : F.adjoint ∘L K = N ∘L G.adjoint) : (K ∘L K.adjoint) ∘L F = F ∘L (N ∘L N.adjoint) ∧ (K.adjoint ∘L K) ∘L G = G ∘L (N.adjoint ∘L N) ∧ (F.adjoint ∘L F) ∘L N = N ∘L (G.adjoint ∘L G) ∧ (F ∘L F.adjoint) ∘L K = K ∘L (G ∘L G.adjoint) := by have h₂' : K.adjoint ∘L F = G ∘L N.adjoint := by simpa only [adjoint_comp, adjoint_adjoint] using congrArg adjoint h₂ refine ⟨?_, ?_, ?_, ?_⟩ · calc (K ∘L K.adjoint) ∘L F = K ∘L (K.adjoint ∘L F) := comp_assoc _ _ _ _ = K ∘L (G ∘L N.adjoint) := congrArg (K ∘L ·) h₂' _ = (K ∘L G) ∘L N.adjoint := (comp_assoc _ _ _).symm _ = (F ∘L N) ∘L N.adjoint := congrArg (· ∘L N.adjoint) h₁.symm _ = F ∘L (N ∘L N.adjoint) := comp_assoc _ _ _ · calc (K.adjoint ∘L K) ∘L G = K.adjoint ∘L (K ∘L G) := comp_assoc _ _ _ _ = K.adjoint ∘L (F ∘L N) := congrArg (K.adjoint ∘L ·) h₁.symm _ = (K.adjoint ∘L F) ∘L N := (comp_assoc _ _ _).symm _ = (G ∘L N.adjoint) ∘L N := congrArg (· ∘L N) h₂' _ = G ∘L (N.adjoint ∘L N) := comp_assoc _ _ _ · calc (F.adjoint ∘L F) ∘L N = F.adjoint ∘L (F ∘L N) := comp_assoc _ _ _ _ = F.adjoint ∘L (K ∘L G) := congrArg (F.adjoint ∘L ·) h₁ _ = (F.adjoint ∘L K) ∘L G := (comp_assoc _ _ _).symm _ = (N ∘L G.adjoint) ∘L G := congrArg (· ∘L G) h₂ _ = N ∘L (G.adjoint ∘L G) := comp_assoc _ _ _ · calc (F ∘L F.adjoint) ∘L K = F ∘L (F.adjoint ∘L K) := comp_assoc _ _ _ _ = F ∘L (N ∘L G.adjoint) := congrArg (F ∘L ·) h₂ _ = (F ∘L N) ∘L G.adjoint := (comp_assoc _ _ _).symm _ = (K ∘L G) ∘L G.adjoint := congrArg (· ∘L G.adjoint) h₁ _ = K ∘L (G ∘L G.adjoint) := comp_assoc _ _ _ end Rectangle section EssentialRanges variable {X E H J : Type*} [MeasurableSpace X] [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedAddCommGroup H] [NormedSpace ℂ H] [NormedAddCommGroup J] [NormedSpace ℂ J] def HasEssentialTotalRange (μ : Measure X) (F : X → E →L[ℂ] H) : Prop := ∀ p : Submodule ℂ H, IsClosed (p : Set H) → (∀ᵐ x ∂μ, LinearMap.range (F x).toLinearMap ≤ p) → p = ⊤ theorem HasEssentialTotalRange.cancel_left {μ : Measure X} {F : X → E →L[ℂ] H} (hF : HasEssentialTotalRange μ F) {A B : H →L[ℂ] J} (h : ∀ᵐ x ∂μ, A ∘L F x = B ∘L F x) : A = B := by have hker : (A - B).ker = ⊤ := by apply hF (A - B).ker (A - B).isClosed_ker filter_upwards [h] with x hx rintro y ⟨v, rfl⟩ change A (F x v) - B (F x v) = 0 exact sub_eq_zero.mpr (DFunLike.congr_fun hx v) apply sub_eq_zero.mp ext v have hv : v ∈ (A - B).ker := by rw [hker]; trivial exact hv theorem essentially_constant_of_intertwining {C D : Type*} [MeasurableSpace C] [MeasurableSpace D] (μC : Measure C) (μX : Measure X) (μD : Measure D) [IsProbabilityMeasure μC] [IsProbabilityMeasure μX] [IsProbabilityMeasure μD] (F : X → E →L[ℂ] H) (A : C → H →L[ℂ] H) (B : D → E →L[ℂ] E) (hF : HasEssentialTotalRange μX F) (h : ∀ᵐ p ∂μC.prod (μX.prod μD), A p.1 ∘L F p.2.1 = F p.2.1 ∘L B p.2.2) : ∃ Q : H →L[ℂ] H, ∀ᵐ c ∂μC, A c = Q := by have h' : ∀ᵐ c ∂μC, ∀ᵐ x ∂μX, ∀ᵐ d ∂μD, A c ∘L F x = F x ∘L B d := (Measure.ae_ae_of_ae_prod h).mono fun _ hc => Measure.ae_ae_of_ae_prod hc obtain ⟨c₀, hc₀⟩ := h'.exists refine ⟨A c₀, h'.mono fun c hc => ?_⟩ apply hF.cancel_left filter_upwards [hc, hc₀] with x hx hx₀ obtain ⟨d, hd, hd₀⟩ := (hx.and hx₀).exists exact hd.trans hd₀.symm end EssentialRanges section HilbertEssentialRanges variable {X E H : Type*} [MeasurableSpace X] [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] theorem HasEssentialTotalRange.eq_zero_of_adjoint_ae_eq_zero {μ : Measure X} {F : X → E →L[ℂ] H} (hF : HasEssentialTotalRange μ F) {v : H} (hv : ∀ᵐ x ∂μ, (F x).adjoint v = 0) : v = 0 := by have hh : innerSL ℂ v = (0 : H →L[ℂ] ℂ) := by apply hF.cancel_left filter_upwards [hv] with x hx ext w change inner ℂ v (F x w) = 0 rw [← ContinuousLinearMap.adjoint_inner_left, hx, inner_zero_left] exact inner_self_eq_zero.mp (DFunLike.congr_fun hh v) theorem fixed_source_square_injective {μ : Measure X} (F : X → E →L[ℂ] H) (hF : HasEssentialTotalRange μ (fun x => (F x).adjoint)) (Q : E →L[ℂ] E) (hQ : ∀ᵐ x ∂μ, (F x).adjoint ∘L F x = Q) : Function.Injective Q := by apply LinearMap.ker_eq_bot.mp apply LinearMap.ker_eq_bot'.mpr intro v hv apply hF.eq_zero_of_adjoint_ae_eq_zero filter_upwards [hQ] with x hx rw [ContinuousLinearMap.adjoint_adjoint] apply (inner_self_eq_zero (𝕜 := ℂ)).mp rw [← ContinuousLinearMap.adjoint_inner_right] have hz : (F x).adjoint (F x v) = 0 := by change ((F x).adjoint ∘L F x) v = 0 rw [hx] exact hv rw [hz, inner_zero_right] theorem essentially_constant_right_of_intertwining {C D : Type*} [MeasurableSpace C] [MeasurableSpace D] (μC : Measure C) (μX : Measure X) (μD : Measure D) [IsProbabilityMeasure μC] [IsProbabilityMeasure μX] [IsProbabilityMeasure μD] (F : X → E →L[ℂ] H) (A : C → H →L[ℂ] H) (B : D → E →L[ℂ] E) (hF : HasEssentialTotalRange μX (fun x => (F x).adjoint)) (h : ∀ᵐ p ∂μC.prod (μX.prod μD), A p.1 ∘L F p.2.1 = F p.2.1 ∘L B p.2.2) : ∃ Q : E →L[ℂ] E, ∀ᵐ d ∂μD, B d = Q := by obtain ⟨c₀, hc₀⟩ := (Measure.ae_ae_of_ae_prod h).exists have hh : ∀ᵐ p ∂μD.prod μX, A c₀ ∘L F p.2 = F p.2 ∘L B p.1 := by have hh' : ∀ᵐ p ∂(μD.prod μX).map Prod.swap, A c₀ ∘L F p.1 = F p.1 ∘L B p.2 := by rw [Measure.prod_swap] exact hc₀ exact ae_of_ae_map measurable_swap.aemeasurable hh' have h' := Measure.ae_ae_of_ae_prod hh obtain ⟨d₀, hd₀⟩ := h'.exists refine ⟨B d₀, h'.mono fun d hd => ?_⟩ have heq : (B d).adjoint = (B d₀).adjoint := by apply hF.cancel_left filter_upwards [hd, hd₀] with x hx hx₀ have hh := congrArg adjoint (hx.symm.trans hx₀) simpa only [adjoint_comp] using hh simpa only [adjoint_adjoint] using congrArg adjoint heq theorem fixed_ends_of_product_intertwining {C D : Type*} [MeasurableSpace C] [MeasurableSpace D] (μC : Measure C) (μX : Measure X) (μD : Measure D) [IsProbabilityMeasure μC] [IsProbabilityMeasure μX] [IsProbabilityMeasure μD] (F : X → E →L[ℂ] H) (A : C → H →L[ℂ] H) (B : D → E →L[ℂ] E) (hF : HasEssentialTotalRange μX F) (hF' : HasEssentialTotalRange μX (fun x => (F x).adjoint)) (h : ∀ᵐ p ∂μC.prod (μX.prod μD), A p.1 ∘L F p.2.1 = F p.2.1 ∘L B p.2.2) : (∃ Q : H →L[ℂ] H, ∀ᵐ c ∂μC, A c = Q) ∧ (∃ Q : E →L[ℂ] E, ∀ᵐ d ∂μD, B d = Q) := ⟨essentially_constant_of_intertwining μC μX μD F A B hF h, essentially_constant_right_of_intertwining μC μX μD F A B hF' h⟩ end HilbertEssentialRanges section Polar variable {E H : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] theorem denseRange_iff_adjoint_injective (T : E →L[ℂ] H) : DenseRange T ↔ Function.Injective T.adjoint := by change Dense (LinearMap.range T.toLinearMap : Set H) ↔ _ rw [Submodule.dense_iff_topologicalClosure_eq_top, Submodule.topologicalClosure_eq_top_iff, T.orthogonal_range, LinearMap.ker_eq_bot] rfl theorem exists_unitary_of_selfAdjoint_square (T : E →L[ℂ] H) (A : E →L[ℂ] E) (hA : IsSelfAdjoint A) (hAi : Function.Injective A) (hT : DenseRange T) (hAT : A ∘L A = T.adjoint ∘L T) : ∃ U : E ≃ₗᵢ[ℂ] H, ∀ x, U (A x) = T x := by have hAd : DenseRange A := (denseRange_iff_adjoint_injective A).mpr (by rw [hA.adjoint_eq]; exact hAi) have hn : ∀ x, ‖T x‖ = ‖A x‖ := by intro x have hsq : ‖T x‖ ^ 2 = ‖A x‖ ^ 2 := by rw [T.apply_norm_sq_eq_inner_adjoint_right, A.apply_norm_sq_eq_inner_adjoint_right, hA.adjoint_eq, hAT] nlinarith [norm_nonneg (T x), norm_nonneg (A x)] refine ⟨(LinearEquiv.refl ℂ E).extendOfIsometry A.toLinearMap T.toLinearMap hAd hT hn, ?_⟩ intro x exact LinearEquiv.extendOfIsometry_eq _ _ _ hAd hT hn x noncomputable def operatorAbs (T : E →L[ℂ] H) : E →L[ℂ] E := CFC.sqrt (T.adjoint ∘L T) theorem operatorAbs_nonneg (T : E →L[ℂ] H) : 0 ≤ operatorAbs T := CFC.sqrt_nonneg _ theorem operatorAbs_sq (T : E →L[ℂ] H) : operatorAbs T ∘L operatorAbs T = T.adjoint ∘L T := by exact CFC.sqrt_mul_sqrt_self _ ((nonneg_iff_isPositive).mpr (isPositive_adjoint_comp_self T)) theorem operatorAbs_injective (T : E →L[ℂ] H) (hT : Function.Injective T) : Function.Injective (operatorAbs T) := by have hTT : Function.Injective (T.adjoint ∘L T) := T.adjoint_comp_self_injective_iff.mpr hT rw [← operatorAbs_sq] at hTT intro x y hxy exact hTT (congrArg (operatorAbs T) hxy) theorem exists_unitary_polar (T : E →L[ℂ] H) (hTi : Function.Injective T) (hTd : DenseRange T) : ∃ U : E ≃ₗᵢ[ℂ] H, ∀ x, U (operatorAbs T x) = T x := by exact exists_unitary_of_selfAdjoint_square T (operatorAbs T) (IsSelfAdjoint.of_nonneg (operatorAbs_nonneg T)) (operatorAbs_injective T hTi) hTd (operatorAbs_sq T) end Polar section variable {E H : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] def intertwinerAlgebra (N : E →L[ℂ] H) : StarSubalgebra ℝ ((H →L[ℂ] H) × (E →L[ℂ] E)) where carrier := {p | p.1 ∘L N = N ∘L p.2 ∧ star p.1 ∘L N = N ∘L star p.2} add_mem' := by rintro ⟨A,B⟩ ⟨C,D⟩ ⟨h₁,h₂⟩ ⟨h₃,h₄⟩ change (A+C) ∘L N = N ∘L (B+D) ∧ star (A+C) ∘L N = N ∘L star (B+D) simp only [add_comp, comp_add, star_add, h₁, h₂, h₃, h₄, and_self] mul_mem' := by rintro ⟨A,B⟩ ⟨C,D⟩ ⟨h₁,h₂⟩ ⟨h₃,h₄⟩ change (A ∘L C) ∘L N = N ∘L (B ∘L D) ∧ star (A*C) ∘L N = N ∘L star (B*D) constructor · rw [comp_assoc, h₃, ← comp_assoc, h₁, comp_assoc] · simp only [star_mul] change (star C ∘L star A) ∘L N = N ∘L (star D ∘L star B) rw [comp_assoc, h₂, ← comp_assoc, h₄, comp_assoc] algebraMap_mem' := by intro r constructor <;> ext x <;> simp [← algebraMap_star_comm] star_mem' := by rintro ⟨A,B⟩ ⟨h₁,h₂⟩ exact ⟨h₂, by simpa using h₁⟩ instance intertwinerAlgebra_isClosed (N : E →L[ℂ] H) : IsClosed (intertwinerAlgebra N : Set ((H →L[ℂ] H) × (E →L[ℂ] E))) := by change IsClosed {p : (H →L[ℂ] H) × (E →L[ℂ] E) | p.1 ∘L N = N ∘L p.2 ∧ star p.1 ∘L N = N ∘L star p.2} apply IsClosed.inter · exact isClosed_eq (by fun_prop) (by fun_prop) · exact isClosed_eq (by fun_prop) (by fun_prop) theorem sqrt_intertwines (A : H →L[ℂ] H) (B : E →L[ℂ] E) (N : E →L[ℂ] H) (hA : 0 ≤ A) (hB : 0 ≤ B) (h : A ∘L N = N ∘L B) : CFC.sqrt A ∘L N = N ∘L CFC.sqrt B := by have hm : (A,B) ∈ intertwinerAlgebra N := by refine ⟨h, ?_⟩ rw [(IsSelfAdjoint.of_nonneg hA).star_eq, (IsSelfAdjoint.of_nonneg hB).star_eq] exact h have hh := cfcₙ_mem (p := IsSelfAdjoint) (𝕜 := ℝ) (𝕜' := ℝ) (s := intertwinerAlgebra N) (fun x : ℝ => Real.sqrt x) hm rw [← CFC.sqrt_eq_real_sqrt (A,B) (show 0 ≤ (A,B) from ⟨hA,hB⟩), CFC.sqrt_map_prod hA hB] at hh exact hh.1 end section PolarRectangle variable {H₁ H₂ H₃ H₄ : Type*} [NormedAddCommGroup H₁] [InnerProductSpace ℂ H₁] [CompleteSpace H₁] [NormedAddCommGroup H₂] [InnerProductSpace ℂ H₂] [CompleteSpace H₂] [NormedAddCommGroup H₃] [InnerProductSpace ℂ H₃] [CompleteSpace H₃] [NormedAddCommGroup H₄] [InnerProductSpace ℂ H₄] [CompleteSpace H₄] omit [CompleteSpace H₁] [CompleteSpace H₂] [CompleteSpace H₃] in theorem eq_of_comp_eq_of_denseRange (A B : H₂ →L[ℂ] H₁) (C : H₃ →L[ℂ] H₂) (hC : DenseRange C) (h : A ∘L C = B ∘L C) : A = B := by apply DFunLike.coe_injective exact hC.equalizer A.continuous B.continuous (by funext x exact congrArg (fun S : H₃ →L[ℂ] H₁ => S x) h) theorem operatorAbs_denseRange (T : H₂ →L[ℂ] H₁) (hT : Function.Injective T) : DenseRange (operatorAbs T) := by apply (denseRange_iff_adjoint_injective _).mpr rw [(IsSelfAdjoint.of_nonneg (operatorAbs_nonneg T)).adjoint_eq] exact operatorAbs_injective T hT theorem polar_rectangle (F : H₂ →L[ℂ] H₁) (G : H₄ →L[ℂ] H₃) (K : H₃ →L[ℂ] H₁) (N : H₄ →L[ℂ] H₂) (h₁ : F ∘L N = K ∘L G) (h₂ : F.adjoint ∘L K = N ∘L G.adjoint) (hGi : Function.Injective G) (hNi : Function.Injective N) (UF : H₂ ≃ₗᵢ[ℂ] H₁) (UG : H₄ ≃ₗᵢ[ℂ] H₃) (UK : H₃ ≃ₗᵢ[ℂ] H₁) (UN : H₄ ≃ₗᵢ[ℂ] H₂) (hF : (UF : H₂ →L[ℂ] H₁) ∘L operatorAbs F = F) (hG : (UG : H₄ →L[ℂ] H₃) ∘L operatorAbs G = G) (hK : (UK : H₃ →L[ℂ] H₁) ∘L operatorAbs K = K) (hN : (UN : H₄ →L[ℂ] H₂) ∘L operatorAbs N = N) : (UF : H₂ →L[ℂ] H₁) ∘L (UN : H₄ →L[ℂ] H₂) = (UK : H₃ →L[ℂ] H₁) ∘L (UG : H₄ →L[ℂ] H₃) := by have hsq := (rectangle_square_identities F G K N h₁ h₂).2.2.1 have hroot : operatorAbs F ∘L N = N ∘L operatorAbs G := sqrt_intertwines _ _ N ((nonneg_iff_isPositive).mpr (isPositive_adjoint_comp_self F)) ((nonneg_iff_isPositive).mpr (isPositive_adjoint_comp_self G)) hsq have hmid : (UF : H₂ →L[ℂ] H₁) ∘L N = K ∘L (UG : H₄ →L[ℂ] H₃) := by apply eq_of_comp_eq_of_denseRange _ _ (operatorAbs G) (operatorAbs_denseRange G hGi) calc ((UF : H₂ →L[ℂ] H₁) ∘L N) ∘L operatorAbs G = (UF : H₂ →L[ℂ] H₁) ∘L (operatorAbs F ∘L N) := by rw [comp_assoc, hroot] _ = F ∘L N := by rw [← comp_assoc, hF] _ = K ∘L G := h₁ _ = (K ∘L (UG : H₄ →L[ℂ] H₃)) ∘L operatorAbs G := by rw [comp_assoc, hG] have hss : N.adjoint ∘L N = (UG.symm : H₃ →L[ℂ] H₄) ∘L (K.adjoint ∘L K) ∘L (UG : H₄ →L[ℂ] H₃) := by have hh := congrArg (fun S : H₄ →L[ℂ] H₁ => S.adjoint ∘L S) hmid ext x simpa [adjoint_comp] using congrArg (fun S : H₄ →L[ℂ] H₄ => S x) hh have hsq' : (K.adjoint ∘L K) ∘L (UG : H₄ →L[ℂ] H₃) = (UG : H₄ →L[ℂ] H₃) ∘L (N.adjoint ∘L N) := by rw [hss] ext x simp have hroot' : operatorAbs K ∘L (UG : H₄ →L[ℂ] H₃) = (UG : H₄ →L[ℂ] H₃) ∘L operatorAbs N := sqrt_intertwines _ _ (UG : H₄ →L[ℂ] H₃) ((nonneg_iff_isPositive).mpr (isPositive_adjoint_comp_self K)) ((nonneg_iff_isPositive).mpr (isPositive_adjoint_comp_self N)) hsq' apply eq_of_comp_eq_of_denseRange _ _ (operatorAbs N) (operatorAbs_denseRange N hNi) calc ((UF : H₂ →L[ℂ] H₁) ∘L (UN : H₄ →L[ℂ] H₂)) ∘L operatorAbs N = (UF : H₂ →L[ℂ] H₁) ∘L N := by rw [comp_assoc, hN] _ = K ∘L (UG : H₄ →L[ℂ] H₃) := hmid _ = (UK : H₃ →L[ℂ] H₁) ∘L (operatorAbs K ∘L (UG : H₄ →L[ℂ] H₃)) := by rw [← comp_assoc, hK] _ = ((UK : H₃ →L[ℂ] H₁) ∘L (UG : H₄ →L[ℂ] H₃)) ∘L operatorAbs N := by rw [hroot', comp_assoc] end PolarRectangle variable {X Y C D H₁ H₂ H₃ H₄ : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace C] [MeasurableSpace D] [NormedAddCommGroup H₁] [InnerProductSpace ℂ H₁] [CompleteSpace H₁] [NormedAddCommGroup H₂] [InnerProductSpace ℂ H₂] [CompleteSpace H₂] [NormedAddCommGroup H₃] [InnerProductSpace ℂ H₃] [CompleteSpace H₃] [NormedAddCommGroup H₄] [InnerProductSpace ℂ H₄] [CompleteSpace H₄] def HasFixedInjectiveSquares (μ : Measure X) (F : X → H₂ →L[ℂ] H₁) : Prop := ∃ P : H₂ →L[ℂ] H₂, ∃ Q : H₁ →L[ℂ] H₁, Function.Injective P ∧ Function.Injective Q ∧ (∀ᵐ x ∂μ, (F x).adjoint ∘L F x = P) ∧ (∀ᵐ x ∂μ, F x ∘L (F x).adjoint = Q) theorem hasFixedInjectiveSquares_of_fixed (μ : Measure X) (F : X → H₂ →L[ℂ] H₁) (hF : HasEssentialTotalRange μ F) (hFa : HasEssentialTotalRange μ (fun x => (F x).adjoint)) (hP : ∃ P, ∀ᵐ x ∂μ, (F x).adjoint ∘L F x = P) (hQ : ∃ Q, ∀ᵐ x ∂μ, F x ∘L (F x).adjoint = Q) : HasFixedInjectiveSquares μ F := by obtain ⟨P, hP⟩ := hP obtain ⟨Q, hQ⟩ := hQ refine ⟨P, Q, fixed_source_square_injective F hFa P hP, ?_, hP, hQ⟩ apply fixed_source_square_injective (fun x => (F x).adjoint) (by simpa using hF) Q simpa using hQ theorem fixed_squares_of_product_identities (μX : Measure X) (μY : Measure Y) (μC : Measure C) (μD : Measure D) [IsProbabilityMeasure μX] [IsProbabilityMeasure μY] [IsProbabilityMeasure μC] [IsProbabilityMeasure μD] (F : X → H₂ →L[ℂ] H₁) (G : Y → H₄ →L[ℂ] H₃) (K : C → H₃ →L[ℂ] H₁) (N : D → H₄ →L[ℂ] H₂) (hF : HasEssentialTotalRange μX F) (hFa : HasEssentialTotalRange μX (fun x => (F x).adjoint)) (hG : HasEssentialTotalRange μY G) (hGa : HasEssentialTotalRange μY (fun y => (G y).adjoint)) (hK : HasEssentialTotalRange μC K) (hKa : HasEssentialTotalRange μC (fun c => (K c).adjoint)) (hN : HasEssentialTotalRange μD N) (hNa : HasEssentialTotalRange μD (fun d => (N d).adjoint)) (h₁ : ∀ᵐ p ∂μC.prod (μX.prod μD), (K p.1 ∘L (K p.1).adjoint) ∘L F p.2.1 = F p.2.1 ∘L (N p.2.2 ∘L (N p.2.2).adjoint)) (h₂ : ∀ᵐ p ∂μC.prod (μY.prod μD), ((K p.1).adjoint ∘L K p.1) ∘L G p.2.1 = G p.2.1 ∘L ((N p.2.2).adjoint ∘L N p.2.2)) (h₃ : ∀ᵐ p ∂μX.prod (μD.prod μY), ((F p.1).adjoint ∘L F p.1) ∘L N p.2.1 = N p.2.1 ∘L ((G p.2.2).adjoint ∘L G p.2.2)) (h₄ : ∀ᵐ p ∂μX.prod (μC.prod μY), (F p.1 ∘L (F p.1).adjoint) ∘L K p.2.1 = K p.2.1 ∘L (G p.2.2 ∘L (G p.2.2).adjoint)) : HasFixedInjectiveSquares μX F ∧ HasFixedInjectiveSquares μY G ∧ HasFixedInjectiveSquares μC K ∧ HasFixedInjectiveSquares μD N := by have hKK := essentially_constant_of_intertwining μC μX μD F (fun c => K c ∘L (K c).adjoint) (fun d => N d ∘L (N d).adjoint) hF h₁ have hNN := essentially_constant_right_of_intertwining μC μX μD F (fun c => K c ∘L (K c).adjoint) (fun d => N d ∘L (N d).adjoint) hFa h₁ have hKK' := essentially_constant_of_intertwining μC μY μD G (fun c => (K c).adjoint ∘L K c) (fun d => (N d).adjoint ∘L N d) hG h₂ have hNN' := essentially_constant_right_of_intertwining μC μY μD G (fun c => (K c).adjoint ∘L K c) (fun d => (N d).adjoint ∘L N d) hGa h₂ have hFF' := essentially_constant_of_intertwining μX μD μY N (fun x => (F x).adjoint ∘L F x) (fun y => (G y).adjoint ∘L G y) hN h₃ have hGG' := essentially_constant_right_of_intertwining μX μD μY N (fun x => (F x).adjoint ∘L F x) (fun y => (G y).adjoint ∘L G y) hNa h₃ have hFF := essentially_constant_of_intertwining μX μC μY K (fun x => F x ∘L (F x).adjoint) (fun y => G y ∘L (G y).adjoint) hK h₄ have hGG := essentially_constant_right_of_intertwining μX μC μY K (fun x => F x ∘L (F x).adjoint) (fun y => G y ∘L (G y).adjoint) hKa h₄ exact ⟨hasFixedInjectiveSquares_of_fixed μX F hF hFa hFF' hFF, hasFixedInjectiveSquares_of_fixed μY G hG hGa hGG' hGG, hasFixedInjectiveSquares_of_fixed μC K hK hKa hKK' hKK, hasFixedInjectiveSquares_of_fixed μD N hN hNa hNN' hNN⟩ end section open MeasureTheory Filter variable {Ω : Type*} [MeasurableSpace Ω] lemma timeMap_zero (T : Ω ≃ᵐ Ω) : timeMap T 0 = id := by simp [timeMap] lemma timeMap_add (T : Ω ≃ᵐ Ω) (m n : ℤ) : timeMap T (m+n) = timeMap T m ∘ timeMap T n := by simp only [timeMap,zpow_add,Equiv.Perm.coe_mul] lemma timeMap_nat (T : Ω ≃ᵐ Ω) (n : ℕ) : timeMap T n = (T : Ω → Ω)^[n] := by simp only [timeMap,zpow_natCast,Equiv.Perm.coe_pow,MeasurableEquiv.coe_toEquiv] lemma timeMap_neg_nat (T : Ω ≃ᵐ Ω) (n : ℕ) : timeMap T (-(n:ℤ)) = (T.symm : Ω → Ω)^[n] := by simp only [timeMap,zpow_neg,zpow_natCast,←inv_pow,Equiv.Perm.coe_pow] rfl lemma timeMap_preserving (μ : Measure Ω) (T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ) (n : ℤ) : MeasurePreserving (timeMap T n) μ μ := by induction n using Int.negInduction with | nat n => rw [timeMap_nat]; exact hT.iterate n | neg _ n => rw [timeMap_neg_nat]; exact (hT.symm T).iterate n lemma timeMap_measurable (T : Ω ≃ᵐ Ω) (n : ℤ) : Measurable (timeMap T n) := by induction n using Int.negInduction with | nat n => rw [timeMap_nat]; exact T.measurable.iterate n | neg _ n => rw [timeMap_neg_nat]; exact T.symm.measurable.iterate n lemma timeMap_comm (T : Ω ≃ᵐ Ω) (m n : ℤ) (x : Ω) : timeMap T m (timeMap T n x) = timeMap T n (timeMap T m x) := by change (timeMap T m ∘ timeMap T n) x = (timeMap T n ∘ timeMap T m) x rw [←timeMap_add,←timeMap_add,add_comm] lemma timeMap_inv_apply (T : Ω ≃ᵐ Ω) (n : ℤ) (x : Ω) : timeMap T (-n) (timeMap T n x) = x := by change (timeMap T (-n) ∘ timeMap T n) x = x rw [←timeMap_add,neg_add_cancel,timeMap_zero] rfl end section open MeasureTheory Filter open scoped Topology RealInnerProductSpace variable {Ω : Type*} [m₀ : MeasurableSpace Ω] (μ : Measure Ω) [retained6 : IsProbabilityMeasure μ] include retained6 in lemma reverse_tail_pairing_along (m : ℕ → MeasurableSpace Ω) (hm : ∀ n, m n ≤ m₀) (hanti : Antitone m) (c : ℕ → ℕ) (hc : Tendsto c atTop atTop) (f : Lp ℝ 2 μ) (g : ℕ → Lp ℝ 2 μ) (hg : ∀ n, AEStronglyMeasurable[m (c n)] (g n) μ) (hb : ∀ n, ‖g n‖ ≤ 1) : Tendsto (fun n => ⟪f - (condExpL2 ℝ ℝ (show (⨅ n, completedFactor (m₀ := m₀) μ (m n)) ≤ m₀ from (iInf_le _ 0).trans (completedFactor_le μ (m 0))) f : Lp ℝ 2 μ), g n⟫) atTop (𝓝 0) := by let _ := retained6 let ft : Lp ℝ 2 μ := condExpL2 ℝ ℝ (show (⨅ n, completedFactor (m₀ := m₀) μ (m n)) ≤ m₀ from (iInf_le _ 0).trans (completedFactor_le μ (m 0))) f have he n : ⟪f-ft,g n⟫ = ⟪(condExpL2 ℝ ℝ (hm (c n)) f : Lp ℝ 2 μ)-ft,g n⟫ := by simp only [inner_sub_left,inner_condExpL2_eq_inner_fun (hm (c n)) f (g n) (hg n)] have hbound n : |⟪f-ft,g n⟫| ≤ ‖(condExpL2 ℝ ℝ (hm (c n)) f : Lp ℝ 2 μ)-ft‖ := by rw [he] exact (abs_real_inner_le_norm _ _).trans (by nlinarith [norm_nonneg ((condExpL2 ℝ ℝ (hm (c n)) f : Lp ℝ 2 μ)-ft),hb n]) have ht := (reverse_conditional_L2 μ m hm hanti f).comp hc have ht' : Tendsto (fun n => ‖(condExpL2 ℝ ℝ (hm (c n)) f : Lp ℝ 2 μ)-ft‖) atTop (𝓝 0) := by simpa only [ft,sub_self,norm_zero,Function.comp_apply] using (ht.sub (tendsto_const_nhds (x := ft))).norm exact squeeze_zero_norm (fun n => by simpa only [Real.norm_eq_abs] using hbound n) ht' lemma integral_tail_replacement (m : ℕ → MeasurableSpace Ω) (hm : ∀ n, m n ≤ m₀) (hanti : Antitone m) (c : ℕ → ℕ) (hc : Tendsto c atTop atTop) (f : Ω → ℝ) (hf : MemLp f 2 μ) (g : ℕ → Ω → ℝ) (hg : ∀ n, Measurable[m (c n)] (g n)) (hb : ∀ n x, |g n x| ≤ 1) : Tendsto (fun n => ∫ x, (f x-μ[f | ⨅ n, completedFactor (m₀ := m₀) μ (m n)] x)*g n x ∂μ) atTop (𝓝 0) := by have hgm n : MemLp (g n) 2 μ := MemLp.of_bound ((hg n).mono (hm (c n)) le_rfl).aestronglyMeasurable 1 (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hb n x) let G n : Lp ℝ 2 μ := (hgm n).toLp (g n) let ht : (⨅ n, completedFactor (m₀ := m₀) μ (m n)) ≤ m₀ := (iInf_le _ 0).trans (completedFactor_le μ (m 0)) let F : Lp ℝ 2 μ := hf.toLp f let FT : Lp ℝ 2 μ := condExpL2 ℝ ℝ ht F have hG n : AEStronglyMeasurable[m (c n)] (G n) μ := (hg n).stronglyMeasurable.aestronglyMeasurable.congr (hgm n).coeFn_toLp.symm have hGb n : ‖G n‖ ≤ 1 := by have hb' : ∀ᵐ x ∂μ, ‖G n x‖ ≤ (1:ℝ) := by filter_upwards [(hgm n).coeFn_toLp] with x hx simpa only [G,hx,Real.norm_eq_abs] using hb n x simpa only [measureUnivNNReal,measure_univ,ENNReal.toNNReal_one,NNReal.coe_one,Real.one_rpow,one_mul] using Lp.norm_le_of_ae_bound (f := G n) (by norm_num : (0:ℝ) ≤ 1) hb' have hh := reverse_tail_pairing_along μ m hm hanti c hc F G hG hGb change Tendsto (fun n => ⟪F-FT,G n⟫) atTop (𝓝 0) at hh convert hh using 1 funext n rw [L2.inner_def] apply integral_congr_ae filter_upwards [hf.coeFn_toLp,(hgm n).coeFn_toLp, hf.condExpL2_ae_eq_condExp (𝕜 := ℝ) ht,Lp.coeFn_sub F FT] with x hx hy hz hw change F x = f x at hx change G n x = g n x at hy change FT x = μ[f | ⨅ n, completedFactor (m₀ := m₀) μ (m n)] x at hz rw [hw,Pi.sub_apply,hx,hy,hz] simp [mul_comm] end section open MeasureTheory MeasurableSpace open scoped ENNReal variable {Y : Type*} [m : MeasurableSpace Y] [CountablyGenerated Y] local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ def hilbertTestAlgebra : Set (Set Y) := generateSetAlgebra (Set.range (natGeneratingSequence Y)) lemma hilbertTestAlgebra_algebra : IsSetAlgebra (hilbertTestAlgebra (Y := Y)) := isSetAlgebra_generateSetAlgebra lemma hilbertTestAlgebra_countable : (hilbertTestAlgebra (Y := Y)).Countable := countable_generateSetAlgebra (Set.countable_range _) lemma hilbertTestAlgebra_generates : m = generateFrom (hilbertTestAlgebra (Y := Y)) := by rw [hilbertTestAlgebra,generateFrom_generateSetAlgebra_eq,generateFrom_natGeneratingSequence] lemma hilbertTestAlgebra_measurable {s : Set Y} (hs : s ∈ hilbertTestAlgebra) : MeasurableSet s := by rw [hilbertTestAlgebra_generates (Y := Y)] exact measurableSet_generateFrom hs lemma hilbertTestAlgebra_dense (μ : Measure Y) [IsFiniteMeasure μ] : (Submodule.span ℂ (Set.range (fun s : hilbertTestAlgebra (Y := Y) => indicatorConstLp 2 (hilbertTestAlgebra_measurable s.property) (measure_ne_top μ s) (1 : ℂ)))).topologicalClosure = ⊤ := by let D := (Submodule.span ℂ (Set.range (fun s : hilbertTestAlgebra (Y := Y) => indicatorConstLp 2 (hilbertTestAlgebra_measurable s.property) (measure_ne_top μ s) (1 : ℂ)))).topologicalClosure have hd := Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite μ hilbertTestAlgebra_algebra hilbertTestAlgebra_generates have hunit (s : Set Y) (hs : s ∈ hilbertTestAlgebra) (hμs : μ s ≠ ∞) (c : ℂ) : indicatorConstLp 2 (hd.measurable s hs) hμs c ∈ D := by have he : indicatorConstLp 2 (hd.measurable s hs) hμs c = c • indicatorConstLp 2 (hilbertTestAlgebra_measurable hs) (measure_ne_top μ s) (1 : ℂ) := by apply Lp.ext filter_upwards [indicatorConstLp_coeFn (p := 2) (hs := hd.measurable s hs) (hμs := hμs) (c := c), Lp.coeFn_smul c (indicatorConstLp 2 (hilbertTestAlgebra_measurable hs) (measure_ne_top μ s) (1 : ℂ)), indicatorConstLp_coeFn (p := 2) (hs := hilbertTestAlgebra_measurable hs) (hμs := measure_ne_top μ s) (c := (1 : ℂ))] with y hy hsm h1 rw [hy,hsm] change s.indicator (fun _ => c) y = c • indicatorConstLp 2 (hilbertTestAlgebra_measurable hs) (measure_ne_top μ s) (1 : ℂ) y rw [h1] by_cases hys : y ∈ s <;> simp [hys] rw [he] exact D.smul_mem c (Submodule.le_topologicalClosure _ (Submodule.subset_span ⟨⟨s,hs⟩,rfl⟩)) apply top_unique intro v _ change v ∈ D apply Lp.induction (by norm_num : (2 : ℝ≥0∞) ≠ ∞) (fun v => v ∈ D) · intro c s hs hμs have hm := hd.indicatorConstLp_subset_closure (p := 2) c ⟨s,hs,hμs.ne,rfl⟩ exact (closure_minimal (by rintro _ ⟨s,hs,hμs,rfl⟩; exact hunit s hs hμs c) (Submodule.isClosed_topologicalClosure _)) hm · intro f g hf hg hfg hF hG exact D.add_mem hF hG · exact (Submodule.isClosed_topologicalClosure _) end section open MeasureTheory MeasurableSpace Topology open scoped ENNReal variable {Y : Type*} [MeasurableSpace Y] [CountablyGenerated Y] local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ abbrev HilbertTestKey (Y : Type*) [MeasurableSpace Y] [CountablyGenerated Y] := List ((ℚ × ℚ) × hilbertTestAlgebra (Y := Y)) instance : Countable (HilbertTestKey Y) := by have : Countable (hilbertTestAlgebra (Y := Y)) := hilbertTestAlgebra_countable.to_subtype infer_instance def hilbertTestFunction (t : HilbertTestKey Y) : Y → ℂ := fun y => (t.map (fun p => ((p.1.1 : ℂ) + p.1.2 * Complex.I) * (p.2.val.indicator (fun _ => (1 : ℂ))) y)).sum lemma hilbertTestFunction_measurable (t : HilbertTestKey Y) : Measurable (hilbertTestFunction t) := by induction t with | nil => change Measurable (fun _ : Y => (0 : ℂ)) exact measurable_const | cons p t ih => change Measurable (fun y => ((p.1.1 : ℂ) + p.1.2 * Complex.I) * (p.2.val.indicator (fun _ => (1 : ℂ))) y + hilbertTestFunction t y) exact (measurable_const.mul (measurable_const.indicator (hilbertTestAlgebra_measurable p.2.property))).add ih def hilbertTestVector (μ : Measure Y) [IsFiniteMeasure μ] (t : HilbertTestKey Y) : Lp ℂ 2 μ := (t.map (fun p => ((p.1.1 : ℂ) + p.1.2 * Complex.I) • indicatorConstLp 2 (hilbertTestAlgebra_measurable p.2.property) (measure_ne_top μ p.2.val) (1 : ℂ))).sum lemma hilbertTestVector_append (μ : Measure Y) [IsFiniteMeasure μ] (s t : HilbertTestKey Y) : hilbertTestVector μ (s ++ t) = hilbertTestVector μ s + hilbertTestVector μ t := by simp [hilbertTestVector] lemma rationalComplex_dense : DenseRange (fun p : ℚ × ℚ => (p.1 : ℂ) + p.2 * Complex.I) := by have h := Rat.denseRange_cast (𝕜 := ℝ) have hh := h.prodMap h have hsurj : Function.Surjective (fun p : ℝ × ℝ => (p.1 : ℂ) + p.2 * Complex.I) := by intro z exact ⟨(z.re,z.im),Complex.re_add_im z⟩ convert hsurj.denseRange.comp hh (by fun_prop) using 1 ext p simp lemma hilbertTestVector_dense (μ : Measure Y) [IsFiniteMeasure μ] : DenseRange (hilbertTestVector μ) := by let S : AddSubmonoid (Lp ℂ 2 μ) := { carrier := Set.range (hilbertTestVector μ) zero_mem' := ⟨[],by simp [hilbertTestVector]⟩ add_mem' := by rintro _ _ ⟨s,rfl⟩ ⟨t,rfl⟩ exact ⟨s ++ t,hilbertTestVector_append μ s t⟩ } let D := S.topologicalClosure have hd : IsClosed (D : Set (Lp ℂ 2 μ)) := AddSubmonoid.isClosed_topologicalClosure S have hsingle (t : hilbertTestAlgebra (Y := Y)) (c : ℂ) : c • indicatorConstLp 2 (hilbertTestAlgebra_measurable t.property) (measure_ne_top μ t.val) (1 : ℂ) ∈ D := by refine rationalComplex_dense.induction_on c (hd.preimage (continuous_id.smul continuous_const)) ?_ intro q apply (show (S : Set (Lp ℂ 2 μ)) ⊆ D from subset_closure) exact ⟨[(q,t)],by simp [hilbertTestVector]⟩ have hspan : (Submodule.span ℂ (Set.range (fun t : hilbertTestAlgebra (Y := Y) => indicatorConstLp 2 (hilbertTestAlgebra_measurable t.property) (measure_ne_top μ t.val) (1 : ℂ))) : Set (Lp ℂ 2 μ)) ⊆ D := by intro v hv have h : ∀ c : ℂ, c • v ∈ D := by induction hv using Submodule.span_induction with | mem x hx => obtain ⟨t,rfl⟩ := hx exact hsingle t | zero => intro c; simp only [smul_zero]; exact D.zero_mem | add x y hx hy ihx ihy => intro c simpa only [smul_add] using D.add_mem (ihx c) (ihy c) | smul a x hx ih => intro c; simpa only [smul_smul] using ih (c*a) simpa using h 1 intro v have hall : (Submodule.span ℂ (Set.range (fun t : hilbertTestAlgebra (Y := Y) => indicatorConstLp 2 (hilbertTestAlgebra_measurable t.property) (measure_ne_top μ t.val) (1 : ℂ)))).topologicalClosure = ⊤ := hilbertTestAlgebra_dense μ have hv : v ∈ (Submodule.span ℂ (Set.range (fun t : hilbertTestAlgebra (Y := Y) => indicatorConstLp 2 (hilbertTestAlgebra_measurable t.property) (measure_ne_top μ t.val) (1 : ℂ)))).topologicalClosure := by rw [hall]; trivial exact closure_minimal hspan hd hv end section open Filter Topology MeasureTheory def IsDistanceCode (d : ℕ → ℕ → ℝ) (z : ℕ → ℝ) : Prop := (∀ n, 0 ≤ z n) ∧ (∀ n m, z n ≤ d n m + z m) ∧ (∀ n m, d n m ≤ z n + z m) ∧ ∀ m : ℕ, ∃ n, z n < (1/2 : ℝ)^m variable {H : Type*} [MetricSpace H] [CompleteSpace H] omit [CompleteSpace H] in lemma distance_isCode {a : ℕ → H} (ha : DenseRange a) (v : H) : IsDistanceCode (fun n m => dist (a n) (a m)) (fun n => dist v (a n)) := by refine ⟨fun n => dist_nonneg, ?_, ?_, ?_⟩ · intro n m simpa [dist_comm,add_comm] using dist_triangle v (a m) (a n) · intro n m simpa [dist_comm] using dist_triangle (a n) v (a m) · intro m exact ha.exists_dist_lt v (by positivity : 0 < (1/2 : ℝ)^m) lemma IsDistanceCode.exists_eq {a : ℕ → H} {z : ℕ → ℝ} (hz : IsDistanceCode (fun n m => dist (a n) (a m)) z) : ∃ v : H, ∀ n, dist v (a n) = z n := by choose b hb using hz.2.2.2 have ht : Tendsto (fun m => z (b m)) atTop (𝓝 0) := squeeze_zero' (Eventually.of_forall fun _ => hz.1 _) (Eventually.of_forall fun m => (hb m).le) (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)) have hc : CauchySeq (fun m => a (b m)) := by rw [Metric.cauchySeq_iff] intro ε hε obtain ⟨N,hN⟩ := eventually_atTop.mp (ht.eventually (gt_mem_nhds (half_pos hε))) refine ⟨N, fun m hm n hn => ?_⟩ have h := hz.2.2.1 (b m) (b n) have := hN m hm have := hN n hn linarith obtain ⟨v,hv⟩ := cauchySeq_tendsto_of_complete hc refine ⟨v,fun n => le_antisymm ?_ ?_⟩ · have h := le_of_tendsto_of_tendsto (hv.dist tendsto_const_nhds) (ht.add_const (z n)) (Eventually.of_forall fun m => hz.2.2.1 (b m) n) simpa using h · have h := le_of_tendsto_of_tendsto tendsto_const_nhds ((tendsto_const_nhds.dist hv).add ht) (Eventually.of_forall fun m => hz.2.1 n (b m)) simpa [dist_comm] using h omit [CompleteSpace H] in lemma distance_code_injective {a : ℕ → H} (ha : DenseRange a) : Function.Injective (fun v : H => fun n => dist v (a n)) := by intro v w he have hh : ∀ b : H, dist v b = dist w b := fun b => ha.induction_on b (isClosed_eq (continuous_const.dist continuous_id) (continuous_const.dist continuous_id)) (fun n => congrFun he n) simpa [dist_comm] using (hh v).symm lemma isDistanceCode_iff {a : ℕ → H} (ha : DenseRange a) {z : ℕ → ℝ} : IsDistanceCode (fun n m => dist (a n) (a m)) z ↔ ∃! v : H, ∀ n, dist v (a n) = z n := by constructor · intro hz obtain ⟨v,hv⟩ := hz.exists_eq exact ⟨v,hv,fun w hw => distance_code_injective ha (funext fun n => (hw n).trans (hv n).symm)⟩ · rintro ⟨v,hv,_⟩ have he : (fun n => dist v (a n)) = z := funext hv rw [←he] exact distance_isCode ha v variable {U : Type*} [MeasurableSpace U] lemma measurableSet_distanceCode (d : U → ℕ → ℕ → ℝ) (hd : ∀ n m, Measurable (fun u => d u n m)) : MeasurableSet {p : U × (ℕ → ℝ) | IsDistanceCode (d p.1) p.2} := by have hz (n : ℕ) : Measurable (fun p : U × (ℕ → ℝ) => p.2 n) := (measurable_pi_apply n).comp measurable_snd have hd' (n m : ℕ) : Measurable (fun p : U × (ℕ → ℝ) => d p.1 n m) := (hd n m).comp measurable_fst simp only [IsDistanceCode, Set.ofPred_and, Set.ofPred_forall, Set.ofPred_exists] exact (MeasurableSet.iInter fun n => measurableSet_le measurable_const (hz n)).inter ((MeasurableSet.iInter fun n => MeasurableSet.iInter fun m => measurableSet_le (hz n) ((hd' n m).add (hz m))).inter ((MeasurableSet.iInter fun n => MeasurableSet.iInter fun m => measurableSet_le (hd' n m) ((hz n).add (hz m))).inter (MeasurableSet.iInter fun m : ℕ => MeasurableSet.iUnion fun n : ℕ => measurableSet_lt (hz n) measurable_const))) end section open MeasureTheory MeasurableSpace ProbabilityTheory open scoped ENNReal InnerProductSpace variable {Y : Type*} [MeasurableSpace Y] [CountablyGenerated Y] local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ lemma hilbertTestVector_coe (μ : Measure Y) [IsFiniteMeasure μ] (t : HilbertTestKey Y) : ↑(hilbertTestVector μ t) =ᵐ[μ] hilbertTestFunction t := by induction t with | nil => exact Lp.coeFn_zero _ _ _ | cons p t ih => let c : ℂ := (p.1.1 : ℂ) + p.1.2 * Complex.I let f := indicatorConstLp 2 (hilbertTestAlgebra_measurable p.2.property) (measure_ne_top μ p.2.val) (1 : ℂ) change ↑(c • f + hilbertTestVector μ t) =ᵐ[μ] _ filter_upwards [Lp.coeFn_add (c • f) (hilbertTestVector μ t),Lp.coeFn_smul c f, indicatorConstLp_coeFn (p := 2) (hs := hilbertTestAlgebra_measurable p.2.property) (hμs := measure_ne_top μ p.2.val) (c := (1 : ℂ)),ih] with y hadd hsm hf ht change (c • f + hilbertTestVector μ t) y = c * _ + hilbertTestFunction t y rw [hadd] change (c • f) y + hilbertTestVector μ t y = _ rw [hsm] change c * f y + hilbertTestVector μ t y = _ rw [hf,ht] def hilbertTestKeys : ℕ → HilbertTestKey Y := Classical.choose (exists_surjective_nat (HilbertTestKey Y)) lemma hilbertTestKeys_surjective : Function.Surjective (hilbertTestKeys (Y := Y)) := Classical.choose_spec (exists_surjective_nat (HilbertTestKey Y)) def hilbertTests (μ : Measure Y) [IsFiniteMeasure μ] (n : ℕ) : Lp ℂ 2 μ := hilbertTestVector μ (hilbertTestKeys n) lemma hilbertTests_dense (μ : Measure Y) [IsFiniteMeasure μ] : DenseRange (hilbertTests μ) := by have hr : Set.range (hilbertTests μ) = Set.range (hilbertTestVector μ) := by ext v constructor · rintro ⟨n,rfl⟩ exact ⟨hilbertTestKeys n,rfl⟩ · rintro ⟨t,rfl⟩ obtain ⟨n,rfl⟩ := hilbertTestKeys_surjective t exact ⟨n,rfl⟩ intro v simpa only [hr] using hilbertTestVector_dense μ v lemma hilbertTestVector_dist (μ : Measure Y) [IsFiniteMeasure μ] (s t : HilbertTestKey Y) : dist (hilbertTestVector μ s) (hilbertTestVector μ t) = Real.sqrt (Complex.re (∫ y, inner ℂ (hilbertTestFunction s y - hilbertTestFunction t y) (hilbertTestFunction s y - hilbertTestFunction t y) ∂μ)) := by rw [dist_eq_norm] have he : (∫ y, inner ℂ ((hilbertTestVector μ s - hilbertTestVector μ t) y) ((hilbertTestVector μ s - hilbertTestVector μ t) y) ∂μ) = ∫ y, inner ℂ (hilbertTestFunction s y - hilbertTestFunction t y) (hilbertTestFunction s y - hilbertTestFunction t y) ∂μ := by apply integral_congr_ae filter_upwards [Lp.coeFn_sub (hilbertTestVector μ s) (hilbertTestVector μ t), hilbertTestVector_coe μ s,hilbertTestVector_coe μ t] with y hd hs ht rw [hd] change inner ℂ (hilbertTestVector μ s y - hilbertTestVector μ t y) (hilbertTestVector μ s y - hilbertTestVector μ t y) = _ rw [hs,ht] rw [←he,←L2.inner_def] exact norm_eq_sqrt_re_inner (𝕜 := ℂ) _ variable {U : Type*} [MeasurableSpace U] lemma hilbertTests_dist_measurable (κ : Kernel U Y) [IsMarkovKernel κ] (n m : ℕ) : Measurable (fun u => dist (hilbertTests (κ u) n) (hilbertTests (κ u) m)) := by simp only [hilbertTests,hilbertTestVector_dist] apply Real.continuous_sqrt.measurable.comp apply Complex.continuous_re.measurable.comp apply StronglyMeasurable.measurable apply StronglyMeasurable.integral_kernel exact ((hilbertTestFunction_measurable _).sub (hilbertTestFunction_measurable _)).stronglyMeasurable.inner ((hilbertTestFunction_measurable _).sub (hilbertTestFunction_measurable _)).stronglyMeasurable end section open MeasureTheory ProbabilityTheory Filter Topology open scoped ENNReal variable {U Y : Type*} [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] (κ : Kernel U Y) [IsMarkovKernel κ] local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ def HilbertCode := {p : U × (ℕ → ℝ) // IsDistanceCode (fun n m => dist (hilbertTests (κ p.1) n) (hilbertTests (κ p.1) m)) p.2} instance : MeasurableSpace (HilbertCode κ) := inferInstanceAs (MeasurableSpace (Subtype _)) instance [StandardBorelSpace U] : StandardBorelSpace (HilbertCode κ) := by apply MeasurableSet.standardBorel exact measurableSet_distanceCode _ (hilbertTests_dist_measurable κ) namespace HilbertCode variable {κ} def base (p : HilbertCode κ) : U := p.val.1 def code (p : HilbertCode κ) : ℕ → ℝ := p.val.2 lemma measurable_base : Measurable (base (κ := κ)) := measurable_fst.comp measurable_subtype_coe lemma measurable_code (n : ℕ) : Measurable (fun p : HilbertCode κ => p.code n) := ((measurable_pi_apply n).comp measurable_snd).comp measurable_subtype_coe def decode (p : HilbertCode κ) : Lp ℂ 2 (κ p.base) := p.property.exists_eq.choose lemma decode_dist (p : HilbertCode κ) (n : ℕ) : dist p.decode (hilbertTests (κ p.base) n) = p.code n := p.property.exists_eq.choose_spec n def encode (u : U) (v : Lp ℂ 2 (κ u)) : HilbertCode κ := ⟨(u,fun n => dist v (hilbertTests (κ u) n)),distance_isCode (hilbertTests_dense (κ u)) v⟩ lemma decode_encode (u : U) (v : Lp ℂ 2 (κ u)) : (encode u v).decode = v := by apply distance_code_injective (hilbertTests_dense (κ u)) funext n exact (encode u v).decode_dist n lemma encode_decode (p : HilbertCode κ) : encode p.base p.decode = p := by apply Subtype.ext refine Prod.ext rfl ?_ funext n exact p.decode_dist n def approxIndex (p : HilbertCode κ) (m : ℕ) : ℕ := Nat.find (p.property.2.2.2 m) lemma approxIndex_spec (p : HilbertCode κ) (m : ℕ) : dist (hilbertTests (κ p.base) (approxIndex p m)) p.decode < (1/2 : ℝ)^m := by rw [dist_comm,decode_dist] exact Nat.find_spec (p.property.2.2.2 m) lemma measurable_approxIndex (m : ℕ) : Measurable (fun p : HilbertCode κ => approxIndex p m) := by exact Measurable.find (f := fun n (_ : HilbertCode κ) => n) (fun _ => measurable_const) (fun n => measurableSet_lt (measurable_code (κ := κ) n) measurable_const) (fun p => p.property.2.2.2 m) def approximant (m : ℕ) (p : HilbertCode κ × Y) : ℂ := hilbertTestFunction (hilbertTestKeys (approxIndex p.1 m)) p.2 lemma measurable_approximant (m : ℕ) : Measurable (approximant (κ := κ) m) := by have hm : Measurable (fun p : ℕ × Y => hilbertTestFunction (hilbertTestKeys p.1) p.2) := measurable_from_prod_countable_right fun n => hilbertTestFunction_measurable (hilbertTestKeys n) exact hm.comp (((measurable_approxIndex m).comp measurable_fst).prodMk measurable_snd) def representative (p : HilbertCode κ × Y) : ℂ := limUnder atTop (fun m => approximant m p) lemma measurable_representative : Measurable (representative (κ := κ)) := by exact (StronglyMeasurable.limUnder fun m => (measurable_approximant m).stronglyMeasurable).measurable lemma representative_eq_decode (p : HilbertCode κ) : (fun y => representative (p,y)) =ᵐ[κ p.base] (p.decode : Y → ℂ) := by let v (m : ℕ) : Lp ℂ 2 (κ p.base) := hilbertTests (κ p.base) (approxIndex p m) - p.decode have hvnorm (m) : ‖v m‖ ≤ (1/2 : ℝ)^m := by simpa only [v,←dist_eq_norm] using (approxIndex_spec p m).le have hsum : Summable (fun m => ‖v m‖) := Summable.of_nonneg_of_le (fun m => norm_nonneg _) hvnorm summable_geometric_two have hfinite : ∑' m, eLpNorm (v m : Y → ℂ) 2 (κ p.base) ≠ ⊤ := by have he (m) : eLpNorm (v m : Y → ℂ) 2 (κ p.base) = (‖v m‖₊ : ℝ≥0∞) := by rw [Lp.nnnorm_def,ENNReal.coe_toNNReal (Lp.eLpNorm_ne_top _)] simp only [he] exact ENNReal.tsum_coe_ne_top_iff_summable.mpr (NNReal.summable_coe.mp hsum) have hae := summable_norm_of_tsum_eLpNorm_ne_top (by norm_num : (1 : ℝ≥0∞) ≤ 2) hfinite have heq : ∀ᵐ y ∂κ p.base, ∀ m, v m y = approximant m (p,y) - p.decode y := by rw [ae_all_iff] intro m filter_upwards [Lp.coeFn_sub (hilbertTests (κ p.base) (approxIndex p m)) p.decode, hilbertTestVector_coe (κ p.base) (hilbertTestKeys (approxIndex p m))] with y hd hf change (hilbertTests (κ p.base) (approxIndex p m) - p.decode) y = _ rw [hd] change hilbertTestVector (κ p.base) (hilbertTestKeys (approxIndex p m)) y - p.decode y = _ rw [hf] rfl filter_upwards [hae,heq] with y hy he apply Tendsto.limUnder_eq apply tendsto_iff_norm_sub_tendsto_zero.mpr have ht := hy.tendsto_atTop_zero simpa only [he] using ht end HilbertCode end section open MeasureTheory ProbabilityTheory open scoped ENNReal InnerProductSpace variable {U Y : Type*} [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] {κ : Kernel U Y} [IsMarkovKernel κ] local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ omit [MeasurableSpace.CountablyGenerated Y] in lemma lp_dist_of_representatives {μ : Measure Y} (v w : Lp ℂ 2 μ) (f g : Y → ℂ) (hf : (v : Y → ℂ) =ᵐ[μ] f) (hg : (w : Y → ℂ) =ᵐ[μ] g) : dist v w = Real.sqrt (Complex.re (∫ y, inner ℂ (f y - g y) (f y - g y) ∂μ)) := by rw [dist_eq_norm] have he : (∫ y, inner ℂ ((v-w) y) ((v-w) y) ∂μ) = ∫ y, inner ℂ (f y-g y) (f y-g y) ∂μ := by apply integral_congr_ae filter_upwards [Lp.coeFn_sub v w,hf,hg] with y hd hv hw rw [hd] change inner ℂ (v y-w y) (v y-w y) = _ rw [hv,hw] rw [←he,←L2.inner_def] exact norm_eq_sqrt_re_inner (𝕜 := ℂ) _ namespace HilbertCode def distance (p q : HilbertCode κ) : ℝ := Real.sqrt (Complex.re (∫ y, inner ℂ (representative (p,y) - representative (q,y)) (representative (p,y) - representative (q,y)) ∂κ p.base)) lemma distance_encode (u : U) (v w : Lp ℂ 2 (κ u)) : distance (encode u v) (encode u w) = dist v w := by have hv := representative_eq_decode (encode u v) have hw := representative_eq_decode (encode u w) simp only [decode_encode] at hv hw exact (lp_dist_of_representatives v w _ _ hv.symm hw.symm).symm lemma measurable_distance : Measurable (fun p : HilbertCode κ × HilbertCode κ => distance p.1 p.2) := by let ξ : Kernel (HilbertCode κ × HilbertCode κ) Y := κ.comap (fun p : HilbertCode κ × HilbertCode κ => p.1.base) (measurable_base.comp measurable_fst) have hf : StronglyMeasurable (fun p : (HilbertCode κ × HilbertCode κ) × Y => inner ℂ (representative (p.1.1,p.2)-representative (p.1.2,p.2)) (representative (p.1.1,p.2)-representative (p.1.2,p.2))) := by have h₁ : Measurable (fun p : (HilbertCode κ × HilbertCode κ) × Y => representative (p.1.1,p.2)) := (measurable_representative (κ := κ)).comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd) have h₂ : Measurable (fun p : (HilbertCode κ × HilbertCode κ) × Y => representative (p.1.2,p.2)) := (measurable_representative (κ := κ)).comp ((measurable_snd.comp measurable_fst).prodMk measurable_snd) exact (h₁.sub h₂).stronglyMeasurable.inner (h₁.sub h₂).stronglyMeasurable exact Real.continuous_sqrt.measurable.comp (Complex.continuous_re.measurable.comp (hf.integral_kernel_prod_right (κ := ξ) (f := fun p y => inner ℂ (representative (p.1,y)-representative (p.2,y)) (representative (p.1,y)-representative (p.2,y)))).measurable) end HilbertCode lemma measurable_hilbert_encode {V : Type*} [MeasurableSpace V] (u : V → U) (hu : Measurable u) (s : ∀ v, Lp ℂ 2 (κ (u v))) (f : V × Y → ℂ) (hf : Measurable f) (he : ∀ v, (s v : Y → ℂ) =ᵐ[κ (u v)] (fun y => f (v,y))) : Measurable (fun v => HilbertCode.encode (u v) (s v)) := by apply Measurable.subtype_mk apply hu.prodMk apply Measurable.of_eval intro n have hdist : (fun v => dist (s v) (hilbertTests (κ (u v)) n)) = fun v => Real.sqrt (Complex.re (∫ y, inner ℂ (f (v,y)-hilbertTestFunction (hilbertTestKeys n) y) (f (v,y)-hilbertTestFunction (hilbertTestKeys n) y) ∂κ (u v))) := by funext v exact lp_dist_of_representatives _ _ _ _ (he v) (hilbertTestVector_coe _ _) change Measurable (fun v => dist (s v) (hilbertTests (κ (u v)) n)) rw [hdist] have hh := hf.sub ((hilbertTestFunction_measurable (hilbertTestKeys n)).comp measurable_snd) exact Real.continuous_sqrt.measurable.comp (Complex.continuous_re.measurable.comp ((hh.stronglyMeasurable.inner hh.stronglyMeasurable).integral_kernel_prod_right (κ := κ.comap u hu)).measurable) end section open scoped InnerProductSpace variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] def RayCanonical (t : ℕ → H) (v : H) : Prop := ∃ n, (∀ m < n, inner ℂ (t m) v = 0) ∧ 0 < (inner ℂ (t n) v).re ∧ (inner ℂ (t n) v).im = 0 lemma exists_inner_ne_zero_of_dense (t : ℕ → H) (ht : DenseRange t) {v : H} (hv : v ≠ 0) : ∃ n, inner ℂ (t n) v ≠ 0 := by by_contra! h have he : ∀ w, inner ℂ w v = 0 := by intro w exact ht.induction_on w (isClosed_eq (continuous_id.inner continuous_const) continuous_const) h exact hv ((inner_self_eq_zero (𝕜 := ℂ)).mp (he v)) lemma exists_rayCanonical (t : ℕ → H) (ht : DenseRange t) {v : H} (hv : v ≠ 0) : ∃ a : ℂ, ‖a‖ = 1 ∧ RayCanonical t (a • v) := by have hex := exists_inner_ne_zero_of_dense t ht hv let n := Nat.find hex let z := inner ℂ (t n) v have hz : z ≠ 0 := Nat.find_spec hex have hnz : 0 < ‖z‖ := norm_pos_iff.mpr hz let a : ℂ := (‖z‖ : ℂ) / z have han : ‖a‖ = 1 := by dsimp [a] rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hnz,div_self hnz.ne'] refine ⟨a,han,n,?_,?_,?_⟩ · intro m hm rw [inner_smul_right,not_ne_iff.mp (Nat.find_min hex hm),mul_zero] · have he : inner ℂ (t n) (a • v) = (‖z‖ : ℂ) := by rw [inner_smul_right] exact div_mul_cancel₀ _ hz rw [he,Complex.ofReal_re] exact hnz · have he : inner ℂ (t n) (a • v) = (‖z‖ : ℂ) := by rw [inner_smul_right] exact div_mul_cancel₀ _ hz rw [he,Complex.ofReal_im] lemma rayCanonical_unique {t : ℕ → H} {v w : H} (hv : RayCanonical t v) (hw : RayCanonical t w) {a : ℂ} (ha : ‖a‖ = 1) (he : v = a • w) : v = w := by obtain ⟨n,hn,hpn,hin⟩ := hv obtain ⟨m,hm,hpm,him⟩ := hw have hne : inner ℂ (t n) v ≠ 0 := by intro h; rw [h] at hpn; simp at hpn have hme : inner ℂ (t m) w ≠ 0 := by intro h; rw [h] at hpm; simp at hpm have han : a ≠ 0 := norm_ne_zero_iff.mp (by rw [ha]; norm_num) have hnm : n = m := by rcases lt_trichotomy n m with h | h | h · have hh := hm n h rw [he,inner_smul_right,hh,mul_zero] at hne exact (hne rfl).elim · exact h · have hh := hn m h rw [he,inner_smul_right] at hh exact (mul_ne_zero han hme hh).elim subst m have hev : inner ℂ (t n) v = ((inner ℂ (t n) v).re : ℂ) := by apply Complex.ext <;> simp [hin] have hew : inner ℂ (t n) w = ((inner ℂ (t n) w).re : ℂ) := by apply Complex.ext <;> simp [him] have hnorm : ‖inner ℂ (t n) v‖ = ‖inner ℂ (t n) w‖ := by rw [he,inner_smul_right,norm_mul,ha,one_mul] have hcoeff : inner ℂ (t n) v = inner ℂ (t n) w := by rw [hev,hew] at hnorm ⊢ have hh : (inner ℂ (t n) v).re = (inner ℂ (t n) w).re := by simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_pos hpn,abs_of_pos hpm] using hnorm rw [hh] have ha1 : a = 1 := by apply (mul_right_cancel₀ hme) simpa only [one_mul,←inner_smul_right,←he,one_smul] using hcoeff simpa [ha1] using he end section open MeasureTheory MeasurableSpace Set variable {P T : Type*} [MeasurableSpace P] [CountablySeparated P] [MeasurableSpace T] [StandardBorelSpace T] lemma borel_injective_cover_branches (p : T → P) (hp : Measurable p) (U : ℕ → Set T) (hU : ∀ n, MeasurableSet (U n)) (hcover : (⋃ n, U n) = Set.univ) (hinj : ∀ n, Set.InjOn p (U n)) : ∃ C : ℕ → Set T, (∀ n, MeasurableSet (C n)) ∧ (⋃ n, C n) = Set.univ ∧ Pairwise (fun i j => Disjoint (C i) (C j)) ∧ (∀ n, MeasurableEmbedding (fun x : C n => p x)) := by refine ⟨disjointed U, fun n => MeasurableSet.disjointed hU n, ?_, disjoint_disjointed U, ?_⟩ · rwa [iUnion_disjointed] · intro n let : StandardBorelSpace ↥(disjointed U n) := MeasurableSet.standardBorel (MeasurableSet.disjointed hU n) apply (hp.comp measurable_subtype_coe).measurableEmbedding intro x y hxy apply Subtype.ext exact hinj n (disjointed_subset U n x.property) (disjointed_subset U n y.property) hxy end namespace HilbertCode open MeasureTheory ProbabilityTheory open scoped ENNReal InnerProductSpace variable {U Y : Type*} [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] {κ : Kernel U Y} [IsMarkovKernel κ] def coefficient (p : HilbertCode κ) (n : ℕ) : ℂ := ∫ y, inner ℂ (hilbertTestFunction (hilbertTestKeys n) y) (representative (p,y)) ∂κ p.base lemma coefficient_eq (p : HilbertCode κ) (n : ℕ) : coefficient p n = inner ℂ (hilbertTests (κ p.base) n) p.decode := by rw [L2.inner_def] apply integral_congr_ae filter_upwards [hilbertTestVector_coe (κ p.base) (hilbertTestKeys n),representative_eq_decode p] with y ht hv rw [hv] change inner ℂ _ _ = inner ℂ (hilbertTestVector _ _ y) _ rw [ht] lemma measurable_coefficient (n : ℕ) : Measurable (fun p : HilbertCode κ => coefficient p n) := by have hh : StronglyMeasurable (fun p : HilbertCode κ × Y => inner ℂ (hilbertTestFunction (hilbertTestKeys n) p.2) (representative p)) := ((hilbertTestFunction_measurable _).comp measurable_snd).stronglyMeasurable.inner measurable_representative.stronglyMeasurable exact (hh.integral_kernel_prod_right (κ := κ.comap base measurable_base) (f := fun p y => inner ℂ (hilbertTestFunction (hilbertTestKeys n) y) (representative (p,y)))).measurable def IsCanonical (p : HilbertCode κ) : Prop := RayCanonical (hilbertTests (κ p.base)) p.decode lemma measurableSet_isCanonical : MeasurableSet {p : HilbertCode κ | IsCanonical p} := by simp only [IsCanonical,RayCanonical,←coefficient_eq,Set.ofPred_exists,Set.ofPred_and,Set.ofPred_forall] apply MeasurableSet.iUnion intro n apply MeasurableSet.inter · exact MeasurableSet.iInter fun m => MeasurableSet.iInter fun _ => measurableSet_eq_fun (measurable_coefficient m) measurable_const · exact (measurableSet_lt measurable_const (Complex.continuous_re.measurable.comp (measurable_coefficient n))).inter (measurableSet_eq_fun (Complex.continuous_im.measurable.comp (measurable_coefficient n)) measurable_const) def IsUnimodular (p : HilbertCode κ) : Prop := ∀ᵐ y ∂κ p.base, ‖representative (p,y)‖ = 1 lemma measurableSet_isUnimodular : MeasurableSet {p : HilbertCode κ | IsUnimodular p} := by have ht : MeasurableSet {p : HilbertCode κ × Y | ‖representative p‖ ≠ 1} := (measurableSet_eq_fun measurable_representative.norm measurable_const).compl have hm := Kernel.measurable_kernel_prodMk_left (κ := κ.comap base measurable_base) ht exact measurableSet_eq_fun hm measurable_const lemma norm_decode_eq_one {p : HilbertCode κ} (hp : IsUnimodular p) : ‖p.decode‖ = 1 := by have hn : ∀ᵐ y ∂κ p.base, ‖p.decode y‖ = ‖(1 : ℂ)‖ := by filter_upwards [hp,representative_eq_decode p] with y hy he simpa only [norm_one,he] using hy rw [Lp.norm_def,eLpNorm_congr_norm_ae (Lp.aestronglyMeasurable p.decode) aestronglyMeasurable_const hn,eLpNorm_const (1 : ℂ) (by norm_num) (IsProbabilityMeasure.ne_zero (κ p.base))] simp lemma distance_le_codes (p q : HilbertCode κ) (he : p.base = q.base) (n : ℕ) : distance p q ≤ p.code n + q.code n := by rcases p with ⟨⟨u,a⟩,ha⟩ rcases q with ⟨⟨v,b⟩,hb⟩ change u = v at he subst v let p : HilbertCode κ := ⟨(u,a),ha⟩ let q : HilbertCode κ := ⟨(u,b),hb⟩ change distance p q ≤ p.code n + q.code n have hp : encode u p.decode = p := encode_decode p have hq : encode u q.decode = q := encode_decode q have hd : distance p q = dist p.decode q.decode := by exact (congrArg₂ distance hp hq).symm.trans (distance_encode u (show Lp ℂ 2 (κ u) from p.decode) (show Lp ℂ 2 (κ u) from q.decode)) rw [hd] rw [←decode_dist p n,←decode_dist q n] exact (dist_triangle _ (hilbertTests (κ u) n) _).trans_eq (by rw [dist_comm (hilbertTests (κ u) n) q.decode]; rfl) end HilbertCode section open MeasureTheory ProbabilityTheory Filter open scoped BigOperators ComplexConjugate theorem exact_mean_dichotomy {ι : Type*} [Fintype ι] [DecidableEq ι] (hk : 3 ≤ Fintype.card ι) (a : ι → ℝ) (ha : ∀ i, 0 ≤ a i) (heq : ∀ i, a i = ∏ j ∈ Finset.univ.erase i, a j) : (∀ i, a i = 0) ∨ (∀ i, a i = 1) := by classical by_cases hz : ∃ i, a i = 0 · obtain ⟨i, hi⟩ := hz left intro j by_cases hij : i = j · simpa [← hij] using hi · rw [heq j] exact Finset.prod_eq_zero (by simp [hij]) hi · have hn (i) : a i ≠ 0 := fun hi => hz ⟨i, hi⟩ have hp (i) : 0 < a i := lt_of_le_of_ne (ha i) (hn i).symm have hl (i) : 2 * Real.log (a i) = ∑ j, Real.log (a j) := by have hh := congrArg Real.log (heq i) rw [Real.log_prod (fun j _ => hn j)] at hh have hs := Finset.sum_erase_add (s := Finset.univ) (fun j => Real.log (a j)) (Finset.mem_univ i) linarith have hs := Finset.sum_congr (s₁ := Finset.univ) (s₂ := Finset.univ) rfl (fun i _ => hl i) simp only [← Finset.mul_sum, Finset.sum_const, Finset.card_univ, nsmul_eq_mul] at hs have hc : (3 : ℝ) ≤ Fintype.card ι := by exact_mod_cast hk have hsum : ∑ j, Real.log (a j) = 0 := by nlinarith right intro i have hi : Real.log (a i) = 0 := by simpa [hsum] using hl i rw [← Real.exp_log (hp i), hi, Real.exp_zero] variable {Ω ι : Type*} [MeasurableSpace Ω] [Fintype ι] [DecidableEq ι] {μ : Measure Ω} [IsProbabilityMeasure μ] omit [IsProbabilityMeasure μ] in theorem exact_tuple_mean_equations (f : ι → Ω → ℂ) (hm : ∀ i, AEStronglyMeasurable (f i) μ) (hn : ∀ i, ∀ᵐ ω ∂μ, ‖f i ω‖ = 1) (hind : ∀ i, iIndepFun (fun j : (Finset.univ.erase i : Finset ι) => f j) μ) (c : ℂ) (hc : ‖c‖ = 1) (hprod : ∀ᵐ ω ∂μ, ∏ i, f i ω = c) : ∀ i, ‖∫ ω, f i ω ∂μ‖ = ∏ j ∈ Finset.univ.erase i, ‖∫ ω, f j ω ∂μ‖ := by intro i have hi : (fun ω => ∏ j ∈ Finset.univ.erase i, f j ω) =ᵐ[μ] (fun ω => c * conj (f i ω)) := by filter_upwards [hprod, hn i] with ω hω hnω rw [← Complex.inv_eq_conj hnω, ← hω] rw [← Finset.prod_erase_mul Finset.univ (fun j => f j ω) (Finset.mem_univ i)] have hne : f i ω ≠ 0 := by intro hh; simp [hh] at hnω rw [mul_assoc, mul_inv_cancel₀ hne, mul_one] have hp : ∫ ω, ∏ j ∈ Finset.univ.erase i, f j ω ∂μ = ∏ j ∈ Finset.univ.erase i, ∫ ω, f j ω ∂μ := by have h := (hind i).integral_fun_prod_eq_prod_integral (fun j => hm j) have hh (g : ι → ℂ) : ∏ j : (Finset.univ.erase i : Finset ι), g j = ∏ j ∈ Finset.univ.erase i, g j := Finset.prod_coe_sort _ _ refine (integral_congr_ae (Filter.Eventually.of_forall fun ω => (hh (fun j => f j ω)).symm)).trans ?_ exact h.trans (hh (fun j => ∫ ω, f j ω ∂μ)) have hh : ∏ j ∈ Finset.univ.erase i, ∫ ω, f j ω ∂μ = c * conj (∫ ω, f i ω ∂μ) := by rw [← hp, integral_congr_ae hi, integral_const_mul, integral_conj] have hw := congrArg norm hh simpa only [norm_prod, norm_mul, hc, Complex.norm_conj, one_mul] using hw.symm theorem ae_eq_mean_of_norm_mean_eq_one (f : Ω → ℂ) (hn : ∀ᵐ ω ∂μ, ‖f ω‖ = 1) (hi : ‖∫ ω, f ω ∂μ‖ = 1) : f =ᵐ[μ] (fun _ => ∫ ω, f ω ∂μ) := by rcases ae_eq_const_or_norm_integral_lt_of_norm_le_const (hn.mono fun _ h => h.le) with h | h · simpa only [average_eq_integral, Function.const_def] using h · simp [hi] at h theorem exact_tuple_dichotomy (hk : 3 ≤ Fintype.card ι) (f : ι → Ω → ℂ) (hm : ∀ i, AEStronglyMeasurable (f i) μ) (hn : ∀ i, ∀ᵐ ω ∂μ, ‖f i ω‖ = 1) (hind : ∀ i, iIndepFun (fun j : (Finset.univ.erase i : Finset ι) => f j) μ) (c : ℂ) (hc : ‖c‖ = 1) (hprod : ∀ᵐ ω ∂μ, ∏ i, f i ω = c) : (∀ i, ∫ ω, f i ω ∂μ = 0) ∨ (∀ i, f i =ᵐ[μ] (fun _ => ∫ ω, f i ω ∂μ) ∧ ‖∫ ω, f i ω ∂μ‖ = 1) := by rcases exact_mean_dichotomy hk (fun i => ‖∫ ω, f i ω ∂μ‖) (fun _ => norm_nonneg _) (exact_tuple_mean_equations f hm hn hind c hc hprod) with h | h · exact Or.inl fun i => norm_eq_zero.mp (h i) · exact Or.inr fun i => ⟨ae_eq_mean_of_norm_mean_eq_one (f i) (hn i) (h i), h i⟩ theorem exact_quotient_dichotomy {S : ι → Type*} [∀ i, MeasurableSpace (S i)] (hk : 3 ≤ Fintype.card ι) (X : ∀ i, Ω → S i) (hX : ∀ i, AEMeasurable (X i) μ) (hind : ∀ i, iIndepFun (fun j : (Finset.univ.erase i : Finset ι) => X j) μ) (f g : ∀ i, S i → ℂ) (hf : ∀ i, Measurable (f i)) (hg : ∀ i, Measurable (g i)) (hnf : ∀ i, ∀ᵐ ω ∂μ, ‖f i (X i ω)‖ = 1) (hng : ∀ i, ∀ᵐ ω ∂μ, ‖g i (X i ω)‖ = 1) (c d : ℂ) (hc : ‖c‖ = 1) (hd : ‖d‖ = 1) (hpf : ∀ᵐ ω ∂μ, ∏ i, f i (X i ω) = c) (hpg : ∀ᵐ ω ∂μ, ∏ i, g i (X i ω) = d) : (∀ i, ∫ ω, f i (X i ω) * conj (g i (X i ω)) ∂μ = 0) ∨ (∀ i, ∃ a : ℂ, ‖a‖ = 1 ∧ (fun ω => f i (X i ω)) =ᵐ[μ] (fun ω => a * g i (X i ω))) := by let q (i : ι) (ω : Ω) := f i (X i ω) * conj (g i (X i ω)) have hqm (i) : AEStronglyMeasurable (q i) μ := ((hf i).comp_aemeasurable (hX i)).aestronglyMeasurable.mul (((hg i).comp_aemeasurable (hX i)).aestronglyMeasurable.star) have hqn (i) : ∀ᵐ ω ∂μ, ‖q i ω‖ = 1 := by filter_upwards [hnf i, hng i] with ω hfω hgω simp [q, hfω, hgω] have hqi (i) : iIndepFun (fun j : (Finset.univ.erase i : Finset ι) => q j) μ := by exact (hind i).comp (fun j y => f j y * conj (g j y)) (fun j => (hf j).mul (Complex.continuous_conj.measurable.comp (hg j))) have hqp : ∀ᵐ ω ∂μ, ∏ i, q i ω = c * conj d := by filter_upwards [hpf, hpg] with ω hfω hgω simp only [q, Finset.prod_mul_distrib, ← map_prod, hfω, hgω] have hcd : ‖c * conj d‖ = 1 := by simp [hc, hd] rcases exact_tuple_dichotomy hk q hqm hqn hqi (c * conj d) hcd hqp with h | h · exact Or.inl h · right intro i refine ⟨∫ ω, q i ω ∂μ, (h i).2, ?_⟩ filter_upwards [(h i).1, hng i] with ω hh hn rw [← hh] change f i (X i ω) = (f i (X i ω) * conj (g i (X i ω))) * g i (X i ω) rw [← Complex.inv_eq_conj hn, mul_assoc, inv_mul_cancel₀, mul_one] intro hz simp [hz] at hn end namespace HilbertCode open MeasureTheory ProbabilityTheory Set open scoped InnerProductSpace ComplexConjugate section DecodingInnerProducts variable {U Y I : Type*} [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] {κ : Kernel U Y} [IsMarkovKernel κ] [Fintype I] [DecidableEq I] lemma inner_decode (p q : HilbertCode κ) (he : p.base = q.base) : ∫ y, representative (p,y) * conj (representative (q,y)) ∂κ p.base = inner ℂ (he ▸ q.decode) p.decode := by cases p with | mk p hp => cases q with | mk q hq => rcases p with ⟨u,a⟩ rcases q with ⟨v,b⟩ change u = v at he subst v rw [L2.inner_def] apply integral_congr_ae filter_upwards [representative_eq_decode (⟨(u,a),hp⟩ : HilbertCode κ), representative_eq_decode (⟨(u,b),hq⟩ : HilbertCode κ)] with y hy hz rw [hy,hz] exact (RCLike.inner_apply _ _).symm lemma eq_of_decode_smul {p q : HilbertCode κ} (he : p.base = q.base) (hp : IsCanonical p) (hq : IsCanonical q) {a : ℂ} (ha : ‖a‖ = 1) (hv : (fun y => representative (p,y)) =ᵐ[κ p.base] (fun y => a * representative (q,y))) : p = q := by rcases p with ⟨⟨u,f⟩,hf⟩ rcases q with ⟨⟨v,g⟩,hg⟩ change u = v at he subst v let p : HilbertCode κ := ⟨(u,f),hf⟩ let q : HilbertCode κ := ⟨(u,g),hg⟩ have hs : (p.decode : Lp ℂ 2 (κ u)) = a • (q.decode : Lp ℂ 2 (κ u)) := by apply Lp.ext change (fun y => (p.decode : Lp ℂ 2 (κ u)) y) =ᵐ[κ u] (fun y => (a • (q.decode : Lp ℂ 2 (κ u)) : Lp ℂ 2 (κ u)) y) filter_upwards [hv,representative_eq_decode p,representative_eq_decode q, Lp.coeFn_smul a (q.decode : Lp ℂ 2 (κ u))] with y hy hpy hqy hs exact hpy.symm.trans ((hy.trans (congrArg (a*·) hqy)).trans hs.symm) have hvv : (p.decode : Lp ℂ 2 (κ u)) = (q.decode : Lp ℂ 2 (κ u)) := rayCanonical_unique hp hq ha hs exact (encode_decode p).symm.trans ((congrArg (encode u) hvv).trans (encode_decode q)) lemma distance_ge_one_of_orthogonal {p q : HilbertCode κ} (he : p.base = q.base) (hp : IsUnimodular p) (hq : IsUnimodular q) (hz : ∫ y, representative (p,y) * conj (representative (q,y)) ∂κ p.base = 0) : 1 ≤ distance p q := by rcases p with ⟨⟨u,a⟩,ha⟩ rcases q with ⟨⟨t,b⟩,hb⟩ change u = t at he subst t let p : HilbertCode κ := ⟨(u,a),ha⟩ let q : HilbertCode κ := ⟨(u,b),hb⟩ let v : Lp ℂ 2 (κ u) := p.decode let w : Lp ℂ 2 (κ u) := q.decode have hid : inner ℂ w v = 0 := (inner_decode p q rfl).symm.trans hz have hn1 : ‖v‖ = 1 := norm_decode_eq_one hp have hn2 : ‖w‖ = 1 := norm_decode_eq_one hq have hn := norm_sub_sq (𝕜 := ℂ) w v rw [hid,hn1,hn2] at hn have hd : distance p q = dist v w := (congrArg₂ distance (encode_decode p) (encode_decode q)).symm.trans (distance_encode u v w) change 1 ≤ distance p q rw [hd,dist_comm,dist_eq_norm] norm_num at hn have hh := norm_nonneg (w-v) nlinarith omit [Fintype I] [DecidableEq I] in lemma ae_norm_representative_comp (μ : Measure (I → Y)) (p q : HilbertCode κ) (he : p.base = q.base) (hq : IsUnimodular q) (i : I) (hm : MeasurePreserving (fun y : I → Y => y i) μ (κ p.base)) : ∀ᵐ y ∂μ, ‖representative (q,y i)‖ = 1 := by have h : ∀ᵐ y ∂κ p.base, ‖representative (q,y)‖ = 1 := by rw [he]; exact hq exact hm.quasiMeasurePreserving.ae h omit [DecidableEq I] in lemma ae_norm_prod_representative (μ : Measure (I → Y)) (p : I → HilbertCode κ) (hn : ∀ i, ∀ᵐ y ∂μ, ‖representative (p i,y i)‖ = 1) : ∀ᵐ y ∂μ, ‖∏ i, representative (p i,y i)‖ = 1 := by filter_upwards [ae_all_iff.mpr hn] with y hy simp only [norm_prod,hy,Finset.prod_const_one] lemma code_tuple_quotient (hk : 3 ≤ Fintype.card I) (μ : Measure (I → Y)) [IsProbabilityMeasure μ] (p q : I → HilbertCode κ) (he : ∀ i, (p i).base = (q i).base) (hp : ∀ i, IsUnimodular (p i)) (hq : ∀ i, IsUnimodular (q i)) (hm : ∀ i, MeasurePreserving (fun y : I → Y => y i) μ (κ (p i).base)) (hind : ∀ i, iIndepFun (fun j : (Finset.univ.erase i : Finset I) => fun y : I → Y => y j) μ) (hpe : ‖∫ y, ∏ i, representative (p i,y i) ∂μ‖ = 1) (hqe : ‖∫ y, ∏ i, representative (q i,y i) ∂μ‖ = 1) : (∀ i, ∫ y, representative (p i,y i) * conj (representative (q i,y i)) ∂μ = 0) ∨ (∀ i, ∃ a : ℂ, ‖a‖ = 1 ∧ (fun y : I → Y => representative (p i,y i)) =ᵐ[μ] (fun y => a * representative (q i,y i))) := by have hpm (i) : Measurable (fun y => representative (p i,y)) := measurable_representative.comp measurable_prodMk_left have hqm (i) : Measurable (fun y => representative (q i,y)) := measurable_representative.comp measurable_prodMk_left have hpn (i) := (hm i).quasiMeasurePreserving.ae (hp i) have hqn (i) := ae_norm_representative_comp μ (p i) (q i) (he i) (hq i) i (hm i) have hpProd := ae_norm_prod_representative μ p hpn have hqProd := ae_norm_prod_representative μ q hqn exact exact_quotient_dichotomy (μ := μ) (S := fun _ : I => Y) hk (fun i y => y i) (fun i => (hm i).measurable.aemeasurable) hind (fun i y => representative (p i,y)) (fun i y => representative (q i,y)) hpm hqm hpn hqn _ _ hpe hqe (ae_eq_mean_of_norm_mean_eq_one _ hpProd hpe) (ae_eq_mean_of_norm_mean_eq_one _ hqProd hqe) lemma exact_codes_dichotomy (hk : 3 ≤ Fintype.card I) (μ : Measure (I → Y)) [IsProbabilityMeasure μ] (p q : I → HilbertCode κ) (he : ∀ i, (p i).base = (q i).base) (hp : ∀ i, IsCanonical (p i) ∧ IsUnimodular (p i)) (hq : ∀ i, IsCanonical (q i) ∧ IsUnimodular (q i)) (hm : ∀ i, MeasurePreserving (fun y : I → Y => y i) μ (κ (p i).base)) (hind : ∀ i, iIndepFun (fun j : (Finset.univ.erase i : Finset I) => fun y : I → Y => y j) μ) (hpe : ‖∫ y, ∏ i, representative (p i,y i) ∂μ‖ = 1) (hqe : ‖∫ y, ∏ i, representative (q i,y i) ∂μ‖ = 1) : p = q ∨ ∀ i, 1 ≤ distance (p i) (q i) := by have hpm (i) : Measurable (fun y => representative (p i,y)) := measurable_representative.comp measurable_prodMk_left have hqm (i) : Measurable (fun y => representative (q i,y)) := measurable_representative.comp measurable_prodMk_left rcases code_tuple_quotient hk μ p q he (fun i => (hp i).2) (fun i => (hq i).2) hm hind hpe hqe with hz | hs · right intro i have hz' : ∫ y, representative (p i,y) * conj (representative (q i,y)) ∂κ (p i).base = 0 := by rw [←(hm i).map_eq] have hf : Measurable (fun y => representative (p i,y) * conj (representative (q i,y))) := (hpm i).mul (Complex.continuous_conj.measurable.comp (hqm i)) exact (integral_map (hm i).measurable.aemeasurable hf.aestronglyMeasurable).trans (hz i) exact distance_ge_one_of_orthogonal (he i) (hp i).2 (hq i).2 hz' · left funext i obtain ⟨a,ha,hv⟩ := hs i apply eq_of_decode_smul (he i) (hp i).1 (hq i).1 ha rw [←(hm i).map_eq] apply (ae_map_iff (hm i).measurable.aemeasurable (measurableSet_eq_fun (hpm i) (measurable_const.mul (hqm i)))).mpr exact hv end DecodingInnerProducts variable {U Y P T : Type*} [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] {κ : Kernel U Y} [IsMarkovKernel κ] [MeasurableSpace P] [MeasurableSpace.CountablySeparated P] [MeasurableSpace T] [StandardBorelSpace T] lemma separated_branches (p : T → P) (hp : Measurable p) (v : T → HilbertCode κ) (hv : Measurable v) (hb : ∀ x y, p x = p y → (v x).base = (v y).base) (δ : ℝ) (hδ : 0 < δ) (hsep : ∀ x y, p x = p y → x ≠ y → δ ≤ distance (v x) (v y)) : ∃ C : ℕ → Set T, (∀ n, MeasurableSet (C n)) ∧ (⋃ n, C n) = Set.univ ∧ Pairwise (fun i j => Disjoint (C i) (C j)) ∧ (∀ n, MeasurableEmbedding (fun x : C n => p x)) := by let B : ℕ → Set T := fun n => {x | (v x).code n < δ / 3} apply borel_injective_cover_branches p hp B · intro n exact measurableSet_lt ((measurable_code n).comp hv) measurable_const · apply Set.eq_univ_of_forall intro x obtain ⟨n,hn⟩ := (hilbertTests_dense (κ (v x).base)).exists_dist_lt (v x).decode (by positivity : 0 < δ/3) apply Set.mem_iUnion.mpr refine ⟨n,?_⟩ change (v x).code n < δ/3 rw [←decode_dist] exact hn · intro n x hx y hy hxy by_contra hne have h := hsep x y hxy hne have ht := distance_le_codes (v x) (v y) (hb x y hxy) n change (v x).code n < δ/3 at hx change (v y).code n < δ/3 at hy linarith end HilbertCode section open MeasureTheory ProbabilityTheory Set open scoped BigOperators variable {P U Y I : Type*} [MeasurableSpace P] [StandardBorelSpace P] [MeasurableSpace U] [StandardBorelSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] [Fintype I] [DecidableEq I] (κ : Kernel U Y) [IsMarkovKernel κ] (Λ : Kernel P (I → Y)) [IsMarkovKernel Λ] (u : P → I → U) def exactCodeMean (t : P × (I → HilbertCode κ)) : ℂ := ∫ y, ∏ i, HilbertCode.representative (t.2 i,y i) ∂Λ t.1 omit [StandardBorelSpace P] [StandardBorelSpace U] [DecidableEq I] in lemma measurable_exactCodeMean : Measurable (exactCodeMean κ Λ) := by have hh : StronglyMeasurable (fun t : (P × (I → HilbertCode κ)) × (I → Y) => ∏ i, HilbertCode.representative (t.1.2 i,t.2 i)) := by apply Measurable.stronglyMeasurable apply Finset.measurable_prod intro i _ exact HilbertCode.measurable_representative.comp (((measurable_pi_apply i).comp measurable_fst.snd).prodMk ((measurable_pi_apply i).comp measurable_snd)) exact (hh.integral_kernel_prod_right (κ := Λ.comap Prod.fst measurable_fst) (f := fun (t : P × (I → HilbertCode κ)) (y : I → Y) => ∏ i, HilbertCode.representative (t.2 i,y i))).measurable def exactCodeSet : Set (P × (I → HilbertCode κ)) := {t | (∀ i, (t.2 i).base = u t.1 i) ∧ (∀ i, HilbertCode.IsCanonical (t.2 i) ∧ HilbertCode.IsUnimodular (t.2 i)) ∧ ‖exactCodeMean κ Λ t‖ = 1} omit [DecidableEq I] in omit [StandardBorelSpace P] in lemma measurableSet_exactCodeSet (hu : Measurable u) : MeasurableSet (exactCodeSet κ Λ u) := by have hbase (i : I) : MeasurableSet {t : P × (I → HilbertCode κ) | (t.2 i).base = u t.1 i} := measurableSet_eq_fun ((HilbertCode.measurable_base (κ := κ)).comp ((measurable_pi_apply i).comp measurable_snd)) (((measurable_pi_apply i).comp hu).comp measurable_fst) have hcan (i : I) : MeasurableSet {t : P × (I → HilbertCode κ) | HilbertCode.IsCanonical (t.2 i) ∧ HilbertCode.IsUnimodular (t.2 i)} := (HilbertCode.measurableSet_isCanonical.inter HilbertCode.measurableSet_isUnimodular).preimage ((measurable_pi_apply i).comp measurable_snd) have hb : MeasurableSet {t : P × (I → HilbertCode κ) | ∀ i, (t.2 i).base = u t.1 i} := by convert MeasurableSet.iInter hbase using 1 ext t simp have hc : MeasurableSet {t : P × (I → HilbertCode κ) | ∀ i, HilbertCode.IsCanonical (t.2 i) ∧ HilbertCode.IsUnimodular (t.2 i)} := by convert MeasurableSet.iInter hcan using 1 ext t simp exact hb.inter (hc.inter (measurableSet_eq_fun (measurable_exactCodeMean κ Λ).norm measurable_const)) abbrev ExactCodeTuple := exactCodeSet κ Λ u lemma exactCodeTuple_branches (hu : Measurable u) (hk : 3 ≤ Fintype.card I) (hm : ∀ p i, MeasurePreserving (fun y : I → Y => y i) (Λ p) (κ (u p i))) (hind : ∀ p i, iIndepFun (fun j : (Finset.univ.erase i : Finset I) => fun y : I → Y => y j) (Λ p)) : ∃ C : ℕ → Set (ExactCodeTuple κ Λ u), (∀ n, MeasurableSet (C n)) ∧ (⋃ n, C n) = Set.univ ∧ Pairwise (fun i j => Disjoint (C i) (C j)) ∧ (∀ n, MeasurableEmbedding (fun x : C n => x.1.1.1)) := by let : StandardBorelSpace (ExactCodeTuple κ Λ u) := (measurableSet_exactCodeSet κ Λ u hu).standardBorel obtain ⟨i₀⟩ : Nonempty I := Fintype.card_pos_iff.mp (by omega) refine HilbertCode.separated_branches (fun t : ExactCodeTuple κ Λ u => t.1.1) measurable_subtype_coe.fst (fun t => t.1.2 i₀) ((measurable_pi_apply i₀).comp measurable_subtype_coe.snd) ?_ 1 zero_lt_one ?_ · intro x y hxy exact (x.property.1 i₀).trans ((congrArg (fun p => u p i₀) hxy).trans (y.property.1 i₀).symm) · intro x y hxy hne have hb i : (x.1.2 i).base = (y.1.2 i).base := (x.property.1 i).trans ((congrArg (fun p => u p i) hxy).trans (y.property.1 i).symm) have hmp i : MeasurePreserving (fun z : I → Y => z i) (Λ x.1.1) (κ (x.1.2 i).base) := by rw [x.property.1 i] exact hm x.1.1 i have hymean : ‖exactCodeMean κ Λ (x.1.1,y.1.2)‖ = 1 := by rw [show (x.1.1,y.1.2) = y.1 from Prod.ext hxy rfl] exact y.property.2.2 rcases HilbertCode.exact_codes_dichotomy hk (Λ x.1.1) x.1.2 y.1.2 hb x.property.2.1 y.property.2.1 hmp (hind x.1.1) x.property.2.2 hymean with he | hd · exact (hne (Subtype.ext (Prod.ext hxy he))).elim · exact hd i₀ end section open MeasureTheory Set Function variable {P T Z : Type*} [MeasurableSpace P] [MeasurableSpace T] [MeasurableSpace Z] lemma measurable_branch_sections (f : T → P) (g : T → Z) (hg : Measurable g) (d : P → Z) (hd : Measurable d) (C : ℕ → Set T) (hcover : (⋃ n, C n) = Set.univ) (hemb : ∀ n, MeasurableEmbedding (fun x : C n => f x)) : ∃ (D : ℕ → Set P) (s : ℕ → P → Z), (∀ n, MeasurableSet (D n)) ∧ (∀ n, Measurable (s n)) ∧ (∀ n p, p ∉ D n → s n p = d p) ∧ (∀ n p, p ∈ D n → ∃ t, f t = p ∧ s n p = g t) ∧ (∀ t, ∃ n, f t ∈ D n ∧ s n (f t) = g t) := by let F (n : ℕ) : C n → P := fun x => f x let D (n : ℕ) : Set P := Set.range (F n) let s (n : ℕ) := Function.extend (F n) (fun t => g t) d refine ⟨D,s,fun n => (hemb n).measurableSet_range,?_,?_,?_,?_⟩ · intro n exact (hemb n).measurable_extend (hg.comp measurable_subtype_coe) hd · intro n p hp exact Function.extend_apply' _ _ p hp · intro n p hp obtain ⟨t,rfl⟩ := hp exact ⟨t,rfl,(hemb n).injective.extend_apply _ _ t⟩ · intro t have ht : t ∈ ⋃ n, C n := by rw [hcover]; trivial obtain ⟨n,hn⟩ := Set.mem_iUnion.mp ht exact ⟨n,⟨⟨t,hn⟩,rfl⟩,(hemb n).injective.extend_apply _ _ ⟨t,hn⟩⟩ lemma measurableSet_branched_relation [MeasurableEq Z] (f : T → P) (g : T → Z) (hg : Measurable g) (d : P → Z) (hd : Measurable d) (C : ℕ → Set T) (hcover : (⋃ n, C n) = Set.univ) (hemb : ∀ n, MeasurableEmbedding (fun x : C n => f x)) : MeasurableSet {pz : P × Z | ∃ t, f t = pz.1 ∧ g t = pz.2} := by obtain ⟨D,s,hD,hs,_,hhas,hevery⟩ := measurable_branch_sections f g hg d hd C hcover hemb have he : {pz : P × Z | ∃ t, f t = pz.1 ∧ g t = pz.2} = ⋃ n, {pz : P × Z | pz.1 ∈ D n ∧ s n pz.1 = pz.2} := by ext pz simp only [Set.mem_ofPred_eq,Set.mem_iUnion] constructor · rintro ⟨t,hf,hg⟩ obtain ⟨n,hn,he⟩ := hevery t exact ⟨n,hf ▸ hn,by rw [←hf,he,hg]⟩ · rintro ⟨n,hn,he⟩ obtain ⟨t,hf,hg⟩ := hhas n pz.1 hn exact ⟨t,hf,hg.symm.trans he⟩ rw [he] exact MeasurableSet.iUnion fun n => (hD n).preimage measurable_fst |>.inter (measurableSet_eq_fun ((hs n).comp measurable_fst) measurable_snd) end section open MeasureTheory ProbabilityTheory Set Topology variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℂ H] abbrev RationalSpanKey := List ((ℚ × ℚ) × ℕ) def rationalSpanTest (v : ℕ → H) (t : RationalSpanKey) : H := (t.map (fun p => ((p.1.1 : ℂ) + p.1.2 * Complex.I) • v p.2)).sum lemma rationalSpanTest_append (v : ℕ → H) (s t : RationalSpanKey) : rationalSpanTest v (s ++ t) = rationalSpanTest v s + rationalSpanTest v t := by simp [rationalSpanTest] lemma closure_rationalSpanTest (v : ℕ → H) : closure (Set.range (rationalSpanTest v)) = ((Submodule.span ℂ (Set.range v)).topologicalClosure : Set H) := by let S : AddSubmonoid H := { carrier := Set.range (rationalSpanTest v) zero_mem' := ⟨[],by simp [rationalSpanTest]⟩ add_mem' := by rintro _ _ ⟨s,rfl⟩ ⟨t,rfl⟩ exact ⟨s ++ t,rationalSpanTest_append v s t⟩ } let D := S.topologicalClosure have hd : IsClosed (D : Set H) := AddSubmonoid.isClosed_topologicalClosure S apply Set.Subset.antisymm · apply closure_minimal _ (Submodule.isClosed_topologicalClosure _) rintro _ ⟨t,rfl⟩ apply Submodule.le_topologicalClosure induction t with | nil => exact Submodule.zero_mem _ | cons p t ih => exact Submodule.add_mem _ (Submodule.smul_mem _ _ (Submodule.subset_span ⟨p.2,rfl⟩)) ih · have hsingle (n : ℕ) (c : ℂ) : c • v n ∈ D := by refine rationalComplex_dense.induction_on c (hd.preimage (continuous_id.smul continuous_const)) ?_ intro q apply (show (S : Set H) ⊆ D from subset_closure) exact ⟨[(q,n)],by simp [rationalSpanTest]⟩ apply closure_minimal _ hd intro w hw have h : ∀ c : ℂ, c • w ∈ D := by induction hw using Submodule.span_induction with | mem x hx => obtain ⟨n,rfl⟩ := hx; exact hsingle n | zero => intro c; simp only [smul_zero]; exact D.zero_mem | add x y hx hy ihx ihy => intro c simpa only [smul_add] using D.add_mem (ihx c) (ihy c) | smul a x hx ih => intro c; simpa only [smul_smul] using ih (c*a) simpa using h 1 variable {P U Y : Type*} [MeasurableSpace P] [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] {κ : Kernel U Y} [IsMarkovKernel κ] {u : P → U} namespace HilbertCode def atBase (q : HilbertCode κ) (a : U) (h : q.base = a) : Lp ℂ 2 (κ a) := h ▸ q.decode lemma encode_atBase (q : HilbertCode κ) (a : U) (h : q.base = a) : encode a (atBase q a h) = q := by subst a exact encode_decode q end HilbertCode def MeasurableSection (v : ∀ p, Lp ℂ 2 (κ (u p))) : Prop := Measurable (fun p => HilbertCode.encode (u p) (v p)) lemma MeasurableSection.zero (hu : Measurable u) : MeasurableSection (κ := κ) (u := u) (fun _ => 0) := by exact measurable_hilbert_encode u hu _ (fun _ => 0) measurable_const (fun _ => Lp.coeFn_zero _ _ _) lemma MeasurableSection.add {v w : ∀ p, Lp ℂ 2 (κ (u p))} (hv : MeasurableSection v) (hw : MeasurableSection w) : MeasurableSection (fun p => v p + w p) := by have hu : Measurable u := HilbertCode.measurable_base.comp hv let f : P × Y → ℂ := fun py => HilbertCode.representative (HilbertCode.encode (u py.1) (v py.1),py.2) let g : P × Y → ℂ := fun py => HilbertCode.representative (HilbertCode.encode (u py.1) (w py.1),py.2) have hf : Measurable f := HilbertCode.measurable_representative.comp ((hv.comp measurable_fst).prodMk measurable_snd) have hg : Measurable g := HilbertCode.measurable_representative.comp ((hw.comp measurable_fst).prodMk measurable_snd) refine measurable_hilbert_encode u hu _ (f+g) (hf.add hg) ?_ intro p filter_upwards [Lp.coeFn_add (v p) (w p), HilbertCode.representative_eq_decode (HilbertCode.encode (u p) (v p)), HilbertCode.representative_eq_decode (HilbertCode.encode (u p) (w p))] with y ha hv hw simp only [HilbertCode.decode_encode] at hv hw rw [ha] change (v p) y + (w p) y = _ exact congrArg₂ (fun a b : ℂ => a+b) hv.symm hw.symm lemma MeasurableSection.smul {v : ∀ p, Lp ℂ 2 (κ (u p))} (hv : MeasurableSection v) (c : ℂ) : MeasurableSection (fun p => c • v p) := by have hu : Measurable u := HilbertCode.measurable_base.comp hv let f : P × Y → ℂ := fun py => c * HilbertCode.representative (HilbertCode.encode (u py.1) (v py.1),py.2) have hf : Measurable f := measurable_const.mul (HilbertCode.measurable_representative.comp ((hv.comp measurable_fst).prodMk measurable_snd)) refine measurable_hilbert_encode u hu _ f hf ?_ intro p filter_upwards [Lp.coeFn_smul c (v p), HilbertCode.representative_eq_decode (HilbertCode.encode (u p) (v p))] with y ha hv simp only [HilbertCode.decode_encode] at hv rw [ha] change c * (v p) y = _ exact congrArg (c * ·) hv.symm lemma MeasurableSection.rationalSpanTest (hu : Measurable u) {s : ℕ → ∀ p, Lp ℂ 2 (κ (u p))} (hs : ∀ n, MeasurableSection (s n)) (t : RationalSpanKey) : MeasurableSection (fun p => rationalSpanTest (fun n => s n p) t) := by induction t with | nil => exact MeasurableSection.zero hu | cons p t ih => exact ((hs p.2).smul _).add ih lemma measurableSet_section_mem_span (hu : Measurable u) {s : ℕ → ∀ p, Lp ℂ 2 (κ (u p))} (hs : ∀ n, MeasurableSection (s n)) {v : ∀ p, Lp ℂ 2 (κ (u p))} (hv : MeasurableSection v) : MeasurableSet {p | v p ∈ (Submodule.span ℂ (Set.range (fun n => s n p))).topologicalClosure} := by have he (p) : v p ∈ (Submodule.span ℂ (Set.range (fun n => s n p))).topologicalClosure ↔ ∀ m : ℕ, ∃ t : RationalSpanKey, dist (v p) (rationalSpanTest (fun n => s n p) t) < (1 : ℝ) / (m+1) := by change v p ∈ ((Submodule.span ℂ (Set.range (fun n => s n p))).topologicalClosure : Set _) ↔ _ rw [←closure_rationalSpanTest (fun n => s n p)] exact Metric.mem_closure_range_iff_nat simp_rw [he,Set.ofPred_forall,Set.ofPred_exists] apply MeasurableSet.iInter intro m apply MeasurableSet.iUnion intro t have hd := HilbertCode.measurable_distance.comp (hv.prodMk (MeasurableSection.rationalSpanTest hu hs t)) have hd' : Measurable (fun p => dist (v p) (rationalSpanTest (fun n => s n p) t)) := by convert hd using 1 funext p exact (HilbertCode.distance_encode (u p) (v p) (rationalSpanTest (fun n => s n p) t)).symm exact measurableSet_lt hd' measurable_const end section open MeasureTheory ProbabilityTheory Set variable {P U Y I : Type*} [MeasurableSpace P] [StandardBorelSpace P] [MeasurableSpace U] [StandardBorelSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] [Fintype I] [DecidableEq I] (κ : Kernel U Y) [IsMarkovKernel κ] (Λ : Kernel P (I → Y)) [IsMarkovKernel Λ] (u : P → I → U) lemma exactCode_sections (hu : Measurable u) (hk : 3 ≤ Fintype.card I) (hm : ∀ p i, MeasurePreserving (fun y : I → Y => y i) (Λ p) (κ (u p i))) (hind : ∀ p i, iIndepFun (fun j : (Finset.univ.erase i : Finset I) => fun y : I → Y => y j) (Λ p)) : ∃ (D : ℕ → Set P) (s : ℕ → P → I → HilbertCode κ), (∀ n, MeasurableSet (D n)) ∧ (∀ n, Measurable (s n)) ∧ (∀ n p i, (s n p i).base = u p i) ∧ (∀ n p, p ∈ D n → (p,s n p) ∈ exactCodeSet κ Λ u) ∧ (∀ n p, p ∉ D n → s n p = fun i => HilbertCode.encode (u p i) 0) ∧ (∀ t ∈ exactCodeSet κ Λ u, ∃ n, t.1 ∈ D n ∧ s n t.1 = t.2) := by let T := ExactCodeTuple κ Λ u obtain ⟨C,_,hcover,_,hemb⟩ := exactCodeTuple_branches κ Λ u hu hk hm hind let d : P → I → HilbertCode κ := fun p i => HilbertCode.encode (u p i) 0 have hd : Measurable d := by apply Measurable.of_eval intro i exact MeasurableSection.zero ((measurable_pi_apply i).comp hu) obtain ⟨D,s,hD,hs,hzero,hhas,hevery⟩ := measurable_branch_sections (fun t : T => t.1.1) (fun t : T => t.1.2) measurable_subtype_coe.snd d hd C hcover hemb have hmem (n) (p) (hp : p ∈ D n) : (p,s n p) ∈ exactCodeSet κ Λ u := by obtain ⟨t,ht,hs⟩ := hhas n p hp rw [hs,←ht] exact t.property refine ⟨D,s,hD,hs,?_,hmem,hzero,?_⟩ · intro n p i by_cases hp : p ∈ D n · exact (hmem n p hp).1 i · rw [hzero n p hp] rfl · intro t ht exact hevery ⟨t,ht⟩ lemma measurableSet_exactCodeSlot (hu : Measurable u) (hk : 3 ≤ Fintype.card I) (hm : ∀ p i, MeasurePreserving (fun y : I → Y => y i) (Λ p) (κ (u p i))) (hind : ∀ p i, iIndepFun (fun j : (Finset.univ.erase i : Finset I) => fun y : I → Y => y j) (Λ p)) (i : I) : MeasurableSet {pz : P × HilbertCode κ | ∃ a : I → HilbertCode κ, (pz.1,a) ∈ exactCodeSet κ Λ u ∧ a i = pz.2} := by obtain ⟨C,_,hcover,_,hemb⟩ := exactCodeTuple_branches κ Λ u hu hk hm hind have hd : Measurable (fun p : P => HilbertCode.encode (u p i) (0 : Lp ℂ 2 (κ (u p i)))) := MeasurableSection.zero ((measurable_pi_apply i).comp hu) have h := measurableSet_branched_relation (fun t : ExactCodeTuple κ Λ u => t.1.1) (fun t : ExactCodeTuple κ Λ u => t.1.2 i) ((measurable_pi_apply i).comp measurable_subtype_coe.snd) _ hd C hcover hemb convert h using 1 ext pz constructor · rintro ⟨a,ha,he⟩ exact ⟨⟨(pz.1,a),ha⟩,rfl,he⟩ · rintro ⟨t,ht,he⟩ exact ⟨t.1.2,by rw [←ht]; exact t.property,he⟩ end section open MeasureTheory open scoped ENNReal InnerProductSpace ComplexConjugate local instance instMeasurableSpaceCircle_circleLp : MeasurableSpace Circle := inferInstanceAs (MeasurableSpace (Submonoid.unitSphere ℂ)) local instance instBorelSpaceCircle_circleLp : BorelSpace Circle := inferInstanceAs (BorelSpace (Submonoid.unitSphere ℂ)) variable {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ] lemma circle_memLp (f : X → Circle) (hf : Measurable f) : MemLp (fun x => (f x : ℂ)) 2 μ := MemLp.of_bound (measurable_subtype_coe.comp hf).aestronglyMeasurable 1 (Filter.Eventually.of_forall fun x => (Circle.norm_coe (f x)).le) def circleLp (f : X → Circle) (hf : Measurable f) : Lp ℂ 2 μ := (circle_memLp μ f hf).toLp (fun x => (f x : ℂ)) lemma circleLp_coe (f : X → Circle) (hf : Measurable f) : circleLp μ f hf =ᵐ[μ] (fun x => (f x : ℂ)) := MemLp.coeFn_toLp _ lemma circleLp_congr {f g : X → Circle} (hf : Measurable f) (hg : Measurable g) (he : f =ᵐ[μ] g) : circleLp μ f hf = circleLp μ g hg := by apply Lp.ext filter_upwards [circleLp_coe μ f hf, circleLp_coe μ g hg,he] with x h1 h2 h3 rw [h1,h2,h3] lemma circleLp_norm [IsProbabilityMeasure μ] (f : X → Circle) (hf : Measurable f) : ‖circleLp μ f hf‖ = 1 := by have hn : ∀ᵐ x ∂μ, ‖circleLp μ f hf x‖ = ‖(1 : ℂ)‖ := by filter_upwards [circleLp_coe μ f hf] with x hx rw [hx,Circle.norm_coe,norm_one] rw [Lp.norm_def,eLpNorm_congr_norm_ae (Lp.aestronglyMeasurable (circleLp μ f hf)) aestronglyMeasurable_const hn,eLpNorm_const (1 : ℂ) (by norm_num) (IsProbabilityMeasure.ne_zero μ)] simp lemma circleLp_ne_zero [IsProbabilityMeasure μ] (f : X → Circle) (hf : Measurable f) : circleLp μ f hf ≠ 0 := by intro h have hn := circleLp_norm μ f hf simp only [h,norm_zero,zero_ne_one] at hn lemma circleLp_inner (f g : X → Circle) (hf : Measurable f) (hg : Measurable g) : inner ℂ (circleLp μ f hf) (circleLp μ g hg) = ∫ x, conj (f x : ℂ) * (g x : ℂ) ∂μ := by rw [L2.inner_def] apply integral_congr_ae filter_upwards [circleLp_coe μ f hf,circleLp_coe μ g hg] with x h1 h2 rw [h1,h2] simp only [RCLike.inner_apply,starRingEnd_apply] ring lemma circleLp_phase (c : Circle) (f : X → Circle) (hf : Measurable f) : circleLp μ (fun x => c*f x) (measurable_const.mul hf) = (c : ℂ) • circleLp μ f hf := by apply Lp.ext filter_upwards [circleLp_coe μ (fun x => c*f x) (measurable_const.mul hf), circleLp_coe μ f hf,Lp.coeFn_smul (c : ℂ) (circleLp μ f hf)] with x h1 h2 h3 rw [h1,h3] change (c : ℂ) * (f x : ℂ) = (c : ℂ) * (circleLp μ f hf x) rw [h2] end section open MeasureTheory ProbabilityTheory open scoped ENNReal BigOperators attribute [local instance] Rokhlin.instMeasurableSpaceCircle_circleLp Rokhlin.instBorelSpaceCircle_circleLp variable {P U Y I : Type*} [MeasurableSpace P] [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] [Fintype I] [DecidableEq I] (κ : Kernel U Y) [IsMarkovKernel κ] (Λ : Kernel P (I → Y)) [IsMarkovKernel Λ] (u : P → I → U) omit [DecidableEq I] in lemma normalized_exact_codes (p : P) (hm : ∀ i, MeasurePreserving (fun y : I → Y => y i) (Λ p) (κ (u p i))) (f : I → Y → Circle) (hf : ∀ i, Measurable (f i)) (c : Circle) (he : ∀ᵐ y ∂Λ p, ∏ i, f i (y i) = c) : ∃ (a : I → HilbertCode κ) (d : I → ℂ), (p,a) ∈ exactCodeSet κ Λ u ∧ ∀ i, ‖d i‖ = 1 ∧ a i = HilbertCode.encode (u p i) (d i • circleLp (κ (u p i)) (f i) (hf i)) := by have hc (i) := exists_rayCanonical (hilbertTests (κ (u p i))) (hilbertTests_dense (κ (u p i))) (circleLp_ne_zero (κ (u p i)) (f i) (hf i)) choose d hd hdcan using hc let a : I → HilbertCode κ := fun i => HilbertCode.encode (u p i) (d i • circleLp (κ (u p i)) (f i) (hf i)) have hrepr (i) : (fun y => HilbertCode.representative (a i,y)) =ᵐ[κ (u p i)] (fun y => d i * (f i y : ℂ)) := by filter_upwards [HilbertCode.representative_eq_decode (a i), Lp.coeFn_smul (d i) (circleLp (κ (u p i)) (f i) (hf i)), circleLp_coe (κ (u p i)) (f i) (hf i)] with y h1 h2 h3 change HilbertCode.representative (a i,y) = (HilbertCode.decode (HilbertCode.encode (u p i) (d i • circleLp (κ (u p i)) (f i) (hf i))) : Lp ℂ 2 (κ (u p i))) y at h1 rw [HilbertCode.decode_encode] at h1 exact h1.trans (h2.trans (congrArg (d i * ·) h3)) have heprod : (fun y => ∏ i, HilbertCode.representative (a i,y i)) =ᵐ[Λ p] (fun _ => (∏ i, d i) * (c : ℂ)) := by filter_upwards [ae_all_iff.mpr (fun i => (hm i).quasiMeasurePreserving.ae_eq_comp (hrepr i)),he] with y hy he have hcoe : (∏ i, (f i (y i) : ℂ)) = (c : ℂ) := (map_prod Circle.coeHom (fun i => f i (y i)) Finset.univ).symm.trans (congrArg (fun z : Circle => (z : ℂ)) he) calc (∏ i, HilbertCode.representative (a i,y i)) = ∏ i, d i * (f i (y i) : ℂ) := Finset.prod_congr rfl (fun i _ => hy i) _ = (∏ i, d i) * (∏ i, (f i (y i) : ℂ)) := Finset.prod_mul_distrib _ = (∏ i, d i) * (c : ℂ) := congrArg ((∏ i, d i) * ·) hcoe refine ⟨a,d,⟨fun _ => rfl,?_,?_⟩,fun i => ⟨hd i,rfl⟩⟩ · intro i constructor · change RayCanonical (hilbertTests (κ (u p i))) (HilbertCode.decode (HilbertCode.encode (u p i) (d i • circleLp (κ (u p i)) (f i) (hf i)))) rw [HilbertCode.decode_encode] exact hdcan i · change ∀ᵐ y ∂κ (u p i), ‖HilbertCode.representative (a i,y)‖ = 1 filter_upwards [hrepr i] with y hy rw [hy,norm_mul,hd i,Circle.norm_coe,one_mul] · change ‖∫ y, ∏ i, HilbertCode.representative (a i,y i) ∂Λ p‖ = 1 rw [integral_congr_ae heprod,integral_const] simp only [probReal_univ,one_smul,norm_mul,norm_prod,hd,Finset.prod_const_one, Circle.norm_coe,one_mul] end section open MeasureTheory ProbabilityTheory Set open scoped BigOperators attribute [local instance] Rokhlin.instMeasurableSpaceCircle_circleLp Rokhlin.instBorelSpaceCircle_circleLp def circleOfValue (z : ℂ) : Circle := by classical exact ⟨if ‖z‖ = 1 then z else 1, by split_ifs with h · exact mem_sphere_zero_iff_norm.mpr h · exact mem_sphere_zero_iff_norm.mpr (norm_one : ‖(1 : ℂ)‖ = 1)⟩ lemma circleOfValue_coe {z : ℂ} (h : ‖z‖ = 1) : (circleOfValue z : ℂ) = z := by classical change (if ‖z‖ = 1 then z else (1 : ℂ)) = z exact ite_eq_left h lemma measurable_circleOfValue : Measurable circleOfValue := by classical apply Measurable.subtype_mk exact measurable_id.ite (measurableSet_eq_fun measurable_norm measurable_const) measurable_const variable {P U Y I : Type*} [MeasurableSpace P] [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] [Fintype I] [DecidableEq I] (κ : Kernel U Y) [IsMarkovKernel κ] (Λ : Kernel P (I → Y)) [IsMarkovKernel Λ] (u : P → I → U) namespace HilbertCode variable {κ} def circleRep (q : HilbertCode κ) (y : Y) : Circle := circleOfValue (representative (κ := κ) (q,y)) lemma circleRep_coe (q : HilbertCode κ) (y : Y) (h : ‖representative (κ := κ) (q,y)‖ = 1) : (circleRep (κ := κ) q y : ℂ) = representative (κ := κ) (q,y) := circleOfValue_coe (z := representative (κ := κ) (q,y)) h lemma measurable_circleRep : Measurable (fun qy : HilbertCode κ × Y => circleRep qy.1 qy.2) := measurable_circleOfValue.comp measurable_representative lemma measurable_circleRep_fixed (q : HilbertCode κ) : Measurable (circleRep (κ := κ) q) := by have h : Measurable (fun y : Y => (q,y)) := measurable_const.prodMk measurable_id have hh : Measurable (fun y : Y => representative (κ := κ) (q,y)) := (measurable_representative (κ := κ)).comp h exact measurable_circleOfValue.comp hh lemma circleRep_ae (q : HilbertCode κ) (hq : IsUnimodular q) : (fun y => (circleRep (κ := κ) q y : ℂ)) =ᵐ[κ q.base] (fun y => representative (κ := κ) (q,y)) := hq.mono fun _ h => circleOfValue_coe h lemma circleLp_circleRep (q : HilbertCode κ) (hq : IsUnimodular q) : circleLp (κ q.base) (circleRep (κ := κ) q) (measurable_circleRep_fixed (κ := κ) q) = q.decode := by apply Lp.ext exact (circleLp_coe (κ q.base) (circleRep (κ := κ) q) (measurable_circleRep_fixed (κ := κ) q)).trans ((circleRep_ae (κ := κ) q hq).trans (representative_eq_decode q)) end HilbertCode omit [DecidableEq I] in lemma exactCode_circle_tuple {p : P} {a : I → HilbertCode κ} (hm : ∀ i, MeasurePreserving (fun y : I → Y => y i) (Λ p) (κ (u p i))) (ha : (p,a) ∈ exactCodeSet κ Λ u) : ∃ c : Circle, ∀ᵐ y ∂Λ p, ∏ i, HilbertCode.circleRep (κ := κ) (a i) (y i) = c := by let c : Circle := ⟨exactCodeMean κ Λ (p,a),mem_sphere_zero_iff_norm.mpr ha.2.2⟩ have hn (i) : ∀ᵐ y ∂Λ p, ‖HilbertCode.representative (κ := κ) (a i,y i)‖ = 1 := by have h : ∀ᵐ y ∂κ (u p i), ‖HilbertCode.representative (κ := κ) (a i,y)‖ = 1 := by rw [←ha.1 i] exact (ha.2.1 i).2 exact (hm i).quasiMeasurePreserving.ae h have hnp : ∀ᵐ y ∂Λ p, ‖∏ i, HilbertCode.representative (κ := κ) (a i,y i)‖ = 1 := HilbertCode.ae_norm_prod_representative (κ := κ) (Λ p) a hn have hmean : ‖∫ y, ∏ i, HilbertCode.representative (κ := κ) (a i,y i) ∂Λ p‖ = 1 := ha.2.2 have he : (fun y => ∏ i, HilbertCode.representative (κ := κ) (a i,y i)) =ᵐ[Λ p] (fun _ => exactCodeMean κ Λ (p,a)) := ae_eq_mean_of_norm_mean_eq_one (μ := Λ p) _ hnp hmean refine ⟨c,?_⟩ filter_upwards [he,ae_all_iff.mpr hn] with y hy hn apply Circle.coe_injective change ((∏ i, HilbertCode.circleRep (κ := κ) (a i) (y i) : Circle) : ℂ) = exactCodeMean κ Λ (p,a) change Circle.coeHom (∏ i, HilbertCode.circleRep (κ := κ) (a i) (y i)) = _ rw [map_prod] exact (Finset.prod_congr rfl (fun i _ => HilbertCode.circleRep_coe (κ := κ) (a i) (y i) (hn i))).trans hy end namespace HilbertCode open MeasureTheory ProbabilityTheory open scoped ENNReal InnerProductSpace attribute [local instance] instMeasurableSpaceCircle_circleLp instBorelSpaceCircle_circleLp variable {U Y : Type*} [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] {κ : Kernel U Y} [IsMarkovKernel κ] abbrev UnitCode (κ : Kernel U Y) [IsMarkovKernel κ] := {q : HilbertCode κ // q.IsUnimodular} lemma unitCode_exists_coefficient (q : UnitCode κ) : ∃ n, coefficient q.val n ≠ 0 := by have hne : q.val.decode ≠ 0 := norm_ne_zero_iff.mp (by rw [norm_decode_eq_one q.property]; norm_num) simpa only [coefficient_eq] using exists_inner_ne_zero_of_dense (hilbertTests (κ q.val.base)) (hilbertTests_dense _) hne def firstCoefficient (q : UnitCode κ) : ℕ := Nat.find (unitCode_exists_coefficient q) lemma measurable_firstCoefficient : Measurable (firstCoefficient (κ := κ)) := measurable_find unitCode_exists_coefficient (fun n => (measurableSet_eq_fun ((measurable_coefficient n).comp measurable_subtype_coe) measurable_const).compl) def unitPhase (q : UnitCode κ) : ℂ := (‖coefficient q.val (firstCoefficient q)‖ : ℂ) / coefficient q.val (firstCoefficient q) lemma measurable_unitPhase : Measurable (unitPhase (κ := κ)) := by have hm : Measurable (fun q : UnitCode κ => coefficient q.val (firstCoefficient q)) := Measurable.find (fun n => (measurable_coefficient n).comp measurable_subtype_coe) (fun n => (measurableSet_eq_fun ((measurable_coefficient n).comp measurable_subtype_coe) measurable_const).compl) unitCode_exists_coefficient exact (Complex.continuous_ofReal.measurable.comp hm.norm).div hm lemma unitPhase_norm (q : UnitCode κ) : ‖unitPhase q‖ = 1 := by have hz : coefficient q.val (firstCoefficient q) ≠ 0 := Nat.find_spec (unitCode_exists_coefficient q) simp only [unitPhase,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_norm] exact div_self (norm_ne_zero_iff.mpr hz) lemma unitPhase_canonical (q : UnitCode κ) : RayCanonical (hilbertTests (κ q.val.base)) (unitPhase q • q.val.decode) := by have hz : coefficient q.val (firstCoefficient q) ≠ 0 := Nat.find_spec (unitCode_exists_coefficient q) refine ⟨firstCoefficient q,?_,?_,?_⟩ · intro n hn rw [inner_smul_right,←coefficient_eq] rw [not_ne_iff.mp (Nat.find_min (unitCode_exists_coefficient q) hn),mul_zero] · rw [inner_smul_right,←coefficient_eq] unfold unitPhase rw [div_mul_cancel₀ _ hz,Complex.ofReal_re] exact norm_pos_iff.mpr hz · rw [inner_smul_right,←coefficient_eq] unfold unitPhase rw [div_mul_cancel₀ _ hz,Complex.ofReal_im] def normalizeUnit (q : UnitCode κ) : HilbertCode κ := encode q.val.base (unitPhase q • q.val.decode) lemma normalizeUnit_base (q : UnitCode κ) : (normalizeUnit q).base = q.val.base := rfl lemma normalizeUnit_canonical (q : UnitCode κ) : (normalizeUnit q).IsCanonical := by change RayCanonical _ (decode (encode _ _)) rw [decode_encode] exact unitPhase_canonical q lemma normalizeUnit_representative (q : UnitCode κ) : (fun y => representative (normalizeUnit q,y)) =ᵐ[κ q.val.base] fun y => unitPhase q * representative (q.val,y) := by have hr := representative_eq_decode (normalizeUnit q) change (fun y => representative (normalizeUnit q,y)) =ᵐ[κ q.val.base] fun y => (decode (encode _ (unitPhase q • q.val.decode)) : Lp ℂ 2 (κ q.val.base)) y at hr rw [decode_encode] at hr filter_upwards [hr,Lp.coeFn_smul (unitPhase q) q.val.decode,representative_eq_decode q.val] with y h1 h2 h3 exact h1.trans (h2.trans (congrArg (unitPhase q * ·) h3.symm)) lemma normalizeUnit_unimodular (q : UnitCode κ) : (normalizeUnit q).IsUnimodular := by filter_upwards [normalizeUnit_representative q,q.property] with y h1 h2 rw [h1,norm_mul,unitPhase_norm,h2,mul_one] lemma measurable_normalizeUnit : Measurable (normalizeUnit (κ := κ)) := by apply measurable_hilbert_encode (fun q : UnitCode κ => q.val.base) (measurable_base.comp measurable_subtype_coe) _ (fun qy => unitPhase qy.1 * representative (qy.1.val,qy.2)) · exact (measurable_unitPhase.comp measurable_fst).mul (measurable_representative.comp ((measurable_subtype_coe.comp measurable_fst).prodMk measurable_snd)) · intro q filter_upwards [Lp.coeFn_smul (unitPhase q) q.val.decode,representative_eq_decode q.val] with y h1 h2 exact h1.trans (congrArg (unitPhase q * ·) h2.symm) def transportRaw (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : HilbertCode κ := encode (F q.base) (circleLp (κ (F q.base)) (circleRep (κ := κ) q ∘ E.symm) ((measurable_circleRep_fixed q).comp E.symm.measurable)) lemma transportRaw_base (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : (transportRaw F E q).base = F q.base := rfl lemma transportRaw_representative (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : (fun y => representative (transportRaw F E q,y)) =ᵐ[κ (F q.base)] (fun y => (circleRep (κ := κ) q (E.symm y) : ℂ)) := by have h := representative_eq_decode (transportRaw F E q) change (fun y => representative (transportRaw F E q,y)) =ᵐ[κ (F q.base)] (decode (encode _ _) : Y → ℂ) at h rw [decode_encode] at h exact h.trans (circleLp_coe _ _ _) lemma transportRaw_unimodular (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : (transportRaw F E q).IsUnimodular := by filter_upwards [transportRaw_representative F E q] with y hy rw [hy,Circle.norm_coe] lemma measurable_circleRep_comp (E : Y ≃ᵐ Y) : Measurable (fun qy : HilbertCode κ × Y => circleRep (κ := κ) qy.1 (E.symm qy.2)) := by have hm : Measurable (fun qy : HilbertCode κ × Y => (qy.1,E.symm qy.2)) := measurable_fst.prodMk (E.symm.measurable.comp measurable_snd) have hr : Measurable (fun qy : HilbertCode κ × Y => representative (κ := κ) (qy.1,E.symm qy.2)) := (measurable_representative (κ := κ)).comp hm exact measurable_circleOfValue.comp hr lemma measurable_transportRaw (F : U → U) (hF : Measurable F) (E : Y ≃ᵐ Y) : Measurable (transportRaw (κ := κ) F E) := by unfold transportRaw apply measurable_hilbert_encode (κ := κ) (fun q : HilbertCode κ => F q.base) (hF.comp (measurable_base (κ := κ))) (fun q => circleLp (κ (F q.base)) (circleRep (κ := κ) q ∘ E.symm) ((measurable_circleRep_fixed q).comp E.symm.measurable)) (fun qy : HilbertCode κ × Y => (circleRep (κ := κ) qy.1 (E.symm qy.2) : ℂ)) · exact measurable_subtype_coe.comp (measurable_circleRep_comp (κ := κ) E) · intro q exact circleLp_coe _ _ _ def transport (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : HilbertCode κ := normalizeUnit ⟨transportRaw F E q,transportRaw_unimodular F E q⟩ lemma transport_base (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : (transport F E q).base = F q.base := rfl lemma transport_canonical (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : (transport F E q).IsCanonical := normalizeUnit_canonical _ lemma transport_unimodular (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) : (transport F E q).IsUnimodular := normalizeUnit_unimodular _ lemma measurable_transport (F : U → U) (hF : Measurable F) (E : Y ≃ᵐ Y) : Measurable (transport (κ := κ) F E) := measurable_normalizeUnit.comp (Measurable.subtype_mk (measurable_transportRaw F hF E)) lemma transport_representative (F : U → U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) (hq : q.IsUnimodular) (hE : MeasurePreserving E (κ q.base) (κ (F q.base))) : ∃ c : ℂ, ‖c‖ = 1 ∧ (fun y => representative (transport F E q,y)) =ᵐ[κ (F q.base)] (fun y => c * representative (q,E.symm y)) := by let r : UnitCode κ := ⟨transportRaw F E q,transportRaw_unimodular F E q⟩ refine ⟨unitPhase r,unitPhase_norm r,?_⟩ have hinv := MeasurePreserving.symm E hE have hrep := hinv.quasiMeasurePreserving.ae (circleRep_ae q hq) filter_upwards [normalizeUnit_representative r,transportRaw_representative F E q,hrep] with y h1 h2 h3 exact h1.trans (congrArg (unitPhase r * ·) (h2.trans h3)) lemma transport_inverse (F : U ≃ᵐ U) (E : Y ≃ᵐ Y) (q : HilbertCode κ) (hq : q.IsUnimodular) (hc : q.IsCanonical) (hE : MeasurePreserving E (κ q.base) (κ (F q.base))) : transport F.symm E.symm (transport F E q) = q := by have hinv : MeasurePreserving E.symm (κ (transport F E q).base) (κ (F.symm (transport F E q).base)) := by simpa only [transport_base,F.symm_apply_apply] using MeasurePreserving.symm E hE obtain ⟨c,hc1,hr⟩ := transport_representative F E q hq hE obtain ⟨d,hd1,hs⟩ := transport_representative F.symm E.symm (transport F E q) (transport_unimodular F E q) hinv have hs' : (fun y => representative (transport F.symm E.symm (transport F E q),y)) =ᵐ[κ q.base] (fun y => d * representative (transport F E q,E y)) := by simpa only [transport_base,F.symm_apply_apply,MeasurableEquiv.symm_symm] using hs apply eq_of_decode_smul (by simp only [transport_base,F.symm_apply_apply]) (transport_canonical _ _ _) hc (a := d*c) (by rw [norm_mul,hd1,hc1,mul_one]) simp only [transport_base,F.symm_apply_apply] filter_upwards [hs',hE.quasiMeasurePreserving.ae hr] with y h1 h2 rw [h1,h2,E.symm_apply_apply,mul_assoc] end HilbertCode open MeasureTheory ProbabilityTheory Set open scoped BigOperators variable {P U Y I : Type*} [MeasurableSpace P] [MeasurableSpace U] [MeasurableSpace Y] [MeasurableSpace.CountablyGenerated Y] [Fintype I] (κ : Kernel U Y) [IsMarkovKernel κ] (Λ : Kernel P (I → Y)) [IsMarkovKernel Λ] (u : P → I → U) omit [IsMarkovKernel Λ] in lemma exactCode_transport (F : U → U) (E : Y ≃ᵐ Y) (p p' : P) (hu : ∀ i, u p' i = F (u p i)) (hκ : ∀ i, MeasurePreserving E (κ (u p i)) (κ (F (u p i)))) (hΛ : MeasurePreserving (fun y : I → Y => fun i => E (y i)) (Λ p) (Λ p')) (hm : ∀ i, MeasurePreserving (fun y : I → Y => y i) (Λ p) (κ (u p i))) (q : I → HilbertCode κ) (hq : (p,q) ∈ exactCodeSet κ Λ u) : (p',fun i => HilbertCode.transport F E (q i)) ∈ exactCodeSet κ Λ u := by classical have hkq (i) : MeasurePreserving E (κ (q i).base) (κ (F (q i).base)) := by rw [hq.1 i]; exact hκ i choose c hc hrep using fun i => HilbertCode.transport_representative F E (q i) (hq.2.1 i).2 (hkq i) refine ⟨fun i => by rw [HilbertCode.transport_base,hq.1 i,hu i], fun i => ⟨HilbertCode.transport_canonical _ _ _,HilbertCode.transport_unimodular _ _ _⟩,?_⟩ have hr (i) : ∀ᵐ y ∂Λ p, HilbertCode.representative (HilbertCode.transport F E (q i),E (y i)) = c i * HilbertCode.representative (q i,y i) := by have hh := (hkq i).quasiMeasurePreserving.ae (hrep i) have hh' : ∀ᵐ y ∂κ (u p i), HilbertCode.representative (HilbertCode.transport F E (q i),E y) = c i * HilbertCode.representative (q i,y) := by rw [←hq.1 i] simpa only [E.symm_apply_apply] using hh exact (hm i).quasiMeasurePreserving.ae hh' have he : (fun y : I → Y => ∏ i, HilbertCode.representative (HilbertCode.transport F E (q i),E (y i))) =ᵐ[Λ p] (fun y => (∏ i, c i) * ∏ i, HilbertCode.representative (q i,y i)) := by filter_upwards [ae_all_iff.mpr hr] with y hy rw [←Finset.prod_mul_distrib] exact Finset.prod_congr rfl (fun i _ => hy i) have hmeas : AEStronglyMeasurable (fun y : I → Y => ∏ i, HilbertCode.representative (HilbertCode.transport F E (q i),y i)) (Λ p') := by apply Measurable.aestronglyMeasurable apply Finset.measurable_prod intro i _ exact HilbertCode.measurable_representative.comp (measurable_const.prodMk (measurable_pi_apply i)) change ‖∫ y, ∏ i, HilbertCode.representative (HilbertCode.transport F E (q i),y i) ∂Λ p'‖ = 1 rw [←hΛ.hasLaw.integral_comp hmeas] simp only [Function.comp_def] rw [integral_congr_ae he,integral_const_mul,norm_mul] have hn : ‖∏ i, c i‖ = 1 := by rw [norm_prod]; simp only [hc,Finset.prod_const_one] rw [hn,one_mul] exact hq.2.2 end Rokhlin end end OAI