import OAI.Dynamics.MultipleMixing.Joinings namespace OAI noncomputable section namespace Rokhlin section open MeasureTheory open scoped InnerProductSpace variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν] omit [SFinite μ] in lemma memLp_tensor (g : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : MemLp (fun p : X × Y => g p.1 * h p.2) 2 (μ.prod ν) := by apply (memLp_two_iff_integrable_sq_norm ((g.1.stronglyMeasurable.comp_measurable measurable_fst).mul (h.1.stronglyMeasurable.comp_measurable measurable_snd)).aestronglyMeasurable).mpr change Integrable (fun p : X × Y => ‖g p.1 * h p.2‖ ^ 2) (μ.prod ν) simp only [norm_mul, mul_pow] exact ((Lp.memLp g).integrable_norm_pow (by norm_num : (2 : ℕ) ≠ 0)).mul_prod ((Lp.memLp h).integrable_norm_pow (by norm_num : (2 : ℕ) ≠ 0)) def tensorLp (g : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : Lp ℂ 2 (μ.prod ν) := (memLp_tensor g h).toLp _ omit [SFinite μ] in lemma coe_tensorLp (g : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : (tensorLp g h : X × Y → ℂ) =ᵐ[μ.prod ν] (fun p => g p.1 * h p.2) := MemLp.coeFn_toLp _ lemma tensorLp_inner (g g' : Lp ℂ 2 μ) (h h' : Lp ℂ 2 ν) : inner ℂ (tensorLp g h) (tensorLp g' h') = inner ℂ g g' * inner ℂ h h' := by rw [L2.inner_def, L2.inner_def g g', L2.inner_def h h', ← integral_prod_mul] apply integral_congr_ae filter_upwards [coe_tensorLp g h, coe_tensorLp g' h'] with p hp hp' rw [hp, hp'] simp only [RCLike.inner_apply, map_mul] ring lemma tensorLp_norm (g : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : ‖tensorLp g h‖ = ‖g‖ * ‖h‖ := by apply (sq_eq_sq₀ (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))).mp rw [← inner_self_eq_norm_sq (𝕜 := ℂ), tensorLp_inner] simp only [inner_self_eq_norm_sq_to_K] change ((‖g‖ : ℂ) ^ 2 * (‖h‖ : ℂ) ^ 2).re = _ simp only [← Complex.ofReal_pow, ← Complex.ofReal_mul, Complex.ofReal_re, mul_pow] omit [SFinite μ] in lemma tensorLp_add_left (g g' : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : tensorLp (g + g') h = tensorLp g h + tensorLp g' h := by apply Lp.ext have hf : Measure.QuasiMeasurePreserving Prod.fst (μ.prod ν) μ := Measure.quasiMeasurePreserving_fst filter_upwards [coe_tensorLp (g + g') h, coe_tensorLp g h, coe_tensorLp g' h, Lp.coeFn_add (tensorLp g h) (tensorLp g' h), hf.ae (Lp.coeFn_add g g')] with p ha hb hc hd he simp only [Pi.add_apply] at hd he rw [ha, hd, hb, hc, he, add_mul] omit [SFinite μ] in lemma tensorLp_smul_left (c : ℂ) (g : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : tensorLp (c • g) h = c • tensorLp g h := by apply Lp.ext have hf : Measure.QuasiMeasurePreserving Prod.fst (μ.prod ν) μ := Measure.quasiMeasurePreserving_fst filter_upwards [coe_tensorLp (c • g) h, coe_tensorLp g h, Lp.coeFn_smul c (tensorLp g h), hf.ae (Lp.coeFn_smul c g)] with p ha hb hc hd simp only [Pi.smul_apply] at hc hd rw [ha, hc, hb, hd] simp only [smul_eq_mul, mul_assoc] omit [SFinite μ] in lemma tensorLp_add_right (g : Lp ℂ 2 μ) (h h' : Lp ℂ 2 ν) : tensorLp g (h + h') = tensorLp g h + tensorLp g h' := by apply Lp.ext have hf : Measure.QuasiMeasurePreserving Prod.snd (μ.prod ν) ν := Measure.quasiMeasurePreserving_snd filter_upwards [coe_tensorLp g (h + h'), coe_tensorLp g h, coe_tensorLp g h', Lp.coeFn_add (tensorLp g h) (tensorLp g h'), hf.ae (Lp.coeFn_add h h')] with p ha hb hc hd he simp only [Pi.add_apply] at hd he rw [ha, hd, hb, hc, he, mul_add] omit [SFinite μ] in lemma tensorLp_smul_right (c : ℂ) (g : Lp ℂ 2 μ) (h : Lp ℂ 2 ν) : tensorLp g (c • h) = c • tensorLp g h := by apply Lp.ext have hf : Measure.QuasiMeasurePreserving Prod.snd (μ.prod ν) ν := Measure.quasiMeasurePreserving_snd filter_upwards [coe_tensorLp g (c • h), coe_tensorLp g h, Lp.coeFn_smul c (tensorLp g h), hf.ae (Lp.coeFn_smul c h)] with p ha hb hc hd simp only [Pi.smul_apply] at hc hd rw [ha, hc, hb, hd] simp only [smul_eq_mul] ring end namespace JoiningOperator open MeasureTheory ProbabilityTheory ContinuousLinearMap open scoped InnerProductSpace ENNReal variable {Ω U X Y : Type*} [MeasurableSpace Ω] [MeasurableSpace U] [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure Ω} {η : Measure U} {ν : Measure X} {ρ : Measure Y} {u : Ω → U} {x : Ω → X} {y : Ω → Y} [SFinite η] [SFinite ρ] def contraction (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω, y ω)) μ (η.prod ρ)) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : Lp ℂ 2 ν := operator hx huy (tensorLp g h) lemma contraction_norm_le (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω, y ω)) μ (η.prod ρ)) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : ‖contraction hx huy g h‖ ≤ ‖g‖ * ‖h‖ := by calc _ ≤ ‖operator hx huy‖ * ‖tensorLp g h‖ := (operator hx huy).le_opNorm _ _ ≤ ‖tensorLp g h‖ := by grw [operator_norm_le]; simp _ = _ := tensorLp_norm g h omit [SFinite η] in lemma contraction_inner (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω, y ω)) μ (η.prod ρ)) (f : Lp ℂ 2 ν) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : inner ℂ f (contraction hx huy g h) = ∫ ω, g (u ω) * inner ℂ (f (x ω)) (h (y ω)) ∂μ := by rw [contraction, inner_operator] apply integral_congr_ae filter_upwards [huy.quasiMeasurePreserving.ae (coe_tensorLp g h)] with ω hω rw [hω] simp only [RCLike.inner_apply] ring omit [SFinite η] in lemma contraction_integrand (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω, y ω)) μ (η.prod ρ)) (f : Lp ℂ 2 ν) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : Integrable (fun ω => g (u ω) * inner ℂ (f (x ω)) (h (y ω))) μ := by apply (integrable_inner_comp hx huy f (tensorLp g h)).congr filter_upwards [huy.quasiMeasurePreserving.ae (coe_tensorLp g h)] with ω hω rw [hω] simp only [RCLike.inner_apply] ring variable [IsFiniteMeasure η] [SecondCountableTopology (Lp ℂ 2 ν)] {κ : Kernel U (X × Y)} [IsMarkovKernel κ] omit [SFinite η] [SFinite ρ] [IsMarkovKernel κ] in lemma family_weight_integrable (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : Integrable (fun a => g a • family κ ν ρ a h) η := by apply ((Lp.memLp g).integrable (by norm_num : (1 : ℝ≥0∞) ≤ 2)).smul_bdd ‖h‖ (family_aestronglyMeasurable hm h) filter_upwards with a calc _ ≤ ‖family κ ν ρ a‖ * ‖h‖ := (family κ ν ρ a).le_opNorm h _ ≤ _ := by grw [family_norm_le]; simp omit [SFinite η] in lemma contraction_integral (hu : MeasurePreserving u μ η) (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω, y ω)) μ (η.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω, y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : contraction hx huy g h = ∫ a, g a • family κ ν ρ a h ∂η := by have : IsFiniteMeasure (μ.map u) := hu.map_eq.symm ▸ inferInstance apply ext_inner_left ℂ intro f rw [contraction_inner, ← integral_inner (family_weight_integrable hm g h)] let F : U × (X × Y) → ℂ := fun p => g p.1 * inner ℂ (f p.2.1) (h p.2.2) have hFm : StronglyMeasurable F := (g.1.stronglyMeasurable.comp_measurable (by fun_prop)).mul ((f.1.stronglyMeasurable.comp_measurable (by fun_prop)).inner (h.1.stronglyMeasurable.comp_measurable (by fun_prop))) have hFi : Integrable F (μ.map u ⊗ₘ κ) := by rw [← hκ.map_eq] apply (integrable_map_measure hFm.aestronglyMeasurable hκ.aemeasurable).mpr exact contraction_integrand hx huy f g h calc _ = ∫ p, F p ∂μ.map u ⊗ₘ κ := hκ.integral_comp hFm.aestronglyMeasurable _ = ∫ a, ∫ p, F (a,p) ∂κ a ∂μ.map u := Measure.integral_compProd hFi _ = ∫ a, g a * (∫ p, inner ℂ (f p.1) (h p.2) ∂κ a) ∂η := by rw [hu.map_eq] simp only [F, integral_const_mul] _ = _ := by apply integral_congr_ae filter_upwards [family_inner hm f h] with a ha rw [inner_smul_right, ha] omit [SFinite η] in lemma contraction_projection (hu : MeasurePreserving u μ η) (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω, y ω)) μ (η.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω, y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (P : Lp ℂ 2 ν →L[ℂ] Lp ℂ 2 ν) (Q : Lp ℂ 2 ρ →L[ℂ] Lp ℂ 2 ρ) (hPQ : ∀ᵐ a ∂η, P ∘L family κ ν ρ a = family κ ν ρ a ∘L Q) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : P (contraction hx huy g h) = contraction hx huy g (Q h) := by rw [contraction_integral hu hx huy hκ hm, contraction_integral hu hx huy hκ hm, ← P.integral_comp_comm (family_weight_integrable hm g h)] apply integral_congr_ae filter_upwards [hPQ] with a ha rw [map_smul] exact congrArg (g a • ·) (congrArg (fun L : Lp ℂ 2 ρ →L[ℂ] Lp ℂ 2 ν => L h) ha) end JoiningOperator section open MeasureTheory open scoped InnerProductSpace ENNReal variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure X} {ν : Measure Y} [IsFiniteMeasure μ] [IsFiniteMeasure ν] lemma tensorLp_indicator (A : Set X) (hA : MeasurableSet A) (B : Set Y) (hB : MeasurableSet B) : tensorLp (indicatorConstLp 2 hA (measure_ne_top μ A) (1 : ℂ)) (indicatorConstLp 2 hB (measure_ne_top ν B) (1 : ℂ)) = indicatorConstLp 2 (hA.prod hB) (measure_ne_top (μ.prod ν) _) (1 : ℂ) := by apply Lp.ext have hf : Measure.QuasiMeasurePreserving Prod.fst (μ.prod ν) μ := Measure.quasiMeasurePreserving_fst have hs : Measure.QuasiMeasurePreserving Prod.snd (μ.prod ν) ν := Measure.quasiMeasurePreserving_snd filter_upwards [coe_tensorLp (indicatorConstLp 2 hA (measure_ne_top μ A) (1 : ℂ)) (indicatorConstLp 2 hB (measure_ne_top ν B) (1 : ℂ)), hf.ae (indicatorConstLp_coeFn (p := 2) (hs := hA) (hμs := measure_ne_top μ A) (c := (1 : ℂ))), hs.ae (indicatorConstLp_coeFn (p := 2) (hs := hB) (hμs := measure_ne_top ν B) (c := (1 : ℂ))), indicatorConstLp_coeFn (p := 2) (hs := hA.prod hB) (hμs := measure_ne_top (μ.prod ν) _) (c := (1 : ℂ))] with p ht hp hq hr rw [ht, hp, hq, hr] by_cases ha : p.1 ∈ A <;> by_cases hb : p.2 ∈ B <;> simp [ha, hb] lemma dense_tensorLp : (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 μ × Lp ℂ 2 ν => tensorLp p.1 p.2))).topologicalClosure = ⊤ := by let S := Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 μ × Lp ℂ 2 ν => tensorLp p.1 p.2)) have hS : Sᗮ = ⊥ := by apply eq_bot_iff.mpr intro f hf change f = 0 apply Lp.ext apply (JoiningOperator.ae_eq_zero_of_rectangles ((Lp.memLp f).integrable (by norm_num : (1 : ℝ≥0∞) ≤ 2)) ?_).trans (Lp.coeFn_zero _ _ _).symm intro A B hA hB rw [← L2.inner_indicatorConstLp_one (hA.prod hB) (measure_ne_top _ _), ← tensorLp_indicator A hA B hB] exact S.inner_right_of_mem_orthogonal (Submodule.subset_span ⟨(_, _), rfl⟩) hf rw [← Submodule.orthogonal_orthogonal_eq_closure, hS, Submodule.bot_orthogonal_eq_top] end section open MeasureTheory ContinuousLinearMap open scoped InnerProductSpace variable {X Y H : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure X} {ν : Measure Y} [IsFiniteMeasure μ] [IsFiniteMeasure ν] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] omit [CompleteSpace H] in lemma closure_tensor_image (A : Lp ℂ 2 (μ.prod ν) →L[ℂ] H) : (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 μ × Lp ℂ 2 ν => A (tensorLp p.1 p.2)))).topologicalClosure = (LinearMap.range A.toLinearMap).topologicalClosure := by let S := (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 μ × Lp ℂ 2 ν => A (tensorLp p.1 p.2)))).topologicalClosure apply le_antisymm · apply Submodule.topologicalClosure_minimal · apply Submodule.span_le.mpr rintro z ⟨⟨g,h⟩, rfl⟩ exact (Submodule.le_topologicalClosure _) ⟨tensorLp g h, rfl⟩ · exact Submodule.isClosed_topologicalClosure _ · apply Submodule.topologicalClosure_minimal · rintro z ⟨v,rfl⟩ have hclosed : IsClosed (S.comap A.toLinearMap : Set (Lp ℂ 2 (μ.prod ν))) := (Submodule.isClosed_topologicalClosure _).preimage A.continuous have ht : (⊤ : Submodule ℂ (Lp ℂ 2 (μ.prod ν))) ≤ S.comap A.toLinearMap := by rw [← dense_tensorLp] apply Submodule.topologicalClosure_minimal _ (Submodule.span_le.mpr ?_) hclosed rintro z ⟨⟨g,h⟩,rfl⟩ exact (Submodule.le_topologicalClosure _) (Submodule.subset_span ⟨(g,h),rfl⟩) exact ht (Submodule.mem_top : v ∈ (⊤ : Submodule ℂ (Lp ℂ 2 (μ.prod ν)))) · exact Submodule.isClosed_topologicalClosure _ omit [CompleteSpace H] in lemma restricted_contractions_span (A : Lp ℂ 2 (μ.prod ν) →L[ℂ] H) (D : Submodule ℂ H) [D.HasOrthogonalProjection] (P : Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ) (Q : Lp ℂ 2 ν →L[ℂ] Lp ℂ 2 ν) (hA : D ≤ (LinearMap.range A.toLinearMap).topologicalClosure) (hPQ : ∀ g h, D.starProjection (A (tensorLp g h)) = A (tensorLp (P g) (Q h))) : (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 μ × Lp ℂ 2 ν => A (tensorLp (P p.1) (Q p.2))))).topologicalClosure = D := by let S := (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 μ × Lp ℂ 2 ν => A (tensorLp (P p.1) (Q p.2))))).topologicalClosure have hDS : ∀ v ∈ (LinearMap.range A.toLinearMap).topologicalClosure, D.starProjection v ∈ S := by intro v hv have hclosed : IsClosed (S.comap D.starProjection.toLinearMap : Set H) := (Submodule.isClosed_topologicalClosure _).preimage D.starProjection.continuous have hh : (LinearMap.range A.toLinearMap).topologicalClosure ≤ S.comap D.starProjection.toLinearMap := by rw [← closure_tensor_image A] apply Submodule.topologicalClosure_minimal _ (Submodule.span_le.mpr ?_) hclosed rintro z ⟨⟨g,h⟩,rfl⟩ change D.starProjection (A (tensorLp g h)) ∈ S rw [hPQ] exact (Submodule.le_topologicalClosure _) (Submodule.subset_span ⟨(g,h),rfl⟩) exact hh hv apply le_antisymm · apply Submodule.topologicalClosure_minimal · apply Submodule.span_le.mpr rintro z ⟨⟨g,h⟩,rfl⟩ change A (tensorLp (P g) (Q h)) ∈ D rw [← hPQ] exact D.starProjection_apply_mem _ · exact D.isTopCompl_orthogonal.isClosed · intro v hv have hh := hDS v (hA hv) rwa [D.starProjection_eq_self_iff.mpr hv] at hh end namespace JoiningOperator open MeasureTheory ProbabilityTheory ContinuousLinearMap open scoped InnerProductSpace ENNReal variable {Ω U X Y : Type*} [MeasurableSpace Ω] [MeasurableSpace U] [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure Ω} {η : Measure U} {ν : Measure X} {ρ : Measure Y} {u : Ω → U} {x : Ω → X} {y : Ω → Y} lemma contraction_swap [SFinite η] [SFinite ρ] (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω,y ω)) μ (η.prod ρ)) (hyu : MeasurePreserving (fun ω => (y ω,u ω)) μ (ρ.prod η)) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : contraction hx huy g h = contraction hx hyu h g := by apply ext_inner_left ℂ intro f rw [contraction_inner,contraction_inner] apply integral_congr_ae filter_upwards with ω simp only [RCLike.inner_apply] ring lemma contraction_projection_both [IsFiniteMeasure η] [IsFiniteMeasure ρ] (hx : MeasurePreserving x μ ν) (huy : MeasurePreserving (fun ω => (u ω,y ω)) μ (η.prod ρ)) (hyu : MeasurePreserving (fun ω => (y ω,u ω)) μ (ρ.prod η)) (D : Submodule ℂ (Lp ℂ 2 ν)) [D.HasOrthogonalProjection] (P : Submodule ℂ (Lp ℂ 2 η)) [P.HasOrthogonalProjection] (Q : Submodule ℂ (Lp ℂ 2 ρ)) [Q.HasOrthogonalProjection] (hr : ∀ g h, D.starProjection (contraction hx huy g h) = contraction hx huy g (Q.starProjection h)) (hl : ∀ h g, D.starProjection (contraction hx hyu h g) = contraction hx hyu h (P.starProjection g)) (g : Lp ℂ 2 η) (h : Lp ℂ 2 ρ) : D.starProjection (contraction hx huy g h) = contraction hx huy (P.starProjection g) (Q.starProjection h) := by calc _ = D.starProjection (D.starProjection (contraction hx huy g h)) := by rw [D.starProjection_eq_self_iff.mpr (D.starProjection_apply_mem _)] _ = D.starProjection (contraction hx huy g (Q.starProjection h)) := by rw [hr] _ = D.starProjection (contraction hx hyu (Q.starProjection h) g) := by rw [contraction_swap hx huy hyu] _ = contraction hx hyu (Q.starProjection h) (P.starProjection g) := hl _ _ _ = _ := contraction_swap hx hyu huy _ _ lemma active_eq_essentialRange_product [IsProbabilityMeasure μ] [SecondCountableTopology (Lp ℂ 2 ρ)] {κ : Kernel U (X × Y)} [IsMarkovKernel κ] (hu : MeasurePreserving u μ η) (hx : MeasurePreserving x μ ν) (hy : MeasurePreserving y μ ρ) (huy : MeasurePreserving (fun ω => (u ω,y ω)) μ (η.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω,y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) : (LinearMap.range (operator hx huy).toLinearMap).topologicalClosure = essentialRange η (family κ ν ρ) := by have h := active_eq_essentialRange hu.measurable hx hy hκ (by simpa only [hu.map_eq] using hm) calc _ = (LinearMap.range (operator hx ⟨hu.measurable.prodMk hy.measurable,rfl⟩).toLinearMap).topologicalClosure := active_observation_eq hx huy ⟨hu.measurable.prodMk hy.measurable,rfl⟩ rfl _ = _ := by simpa only [hu.map_eq] using h end JoiningOperator namespace ArrayObjects.Object.Rolled open MeasureTheory ProbabilityTheory ContinuousLinearMap open scoped ENNReal section local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ universe u variable {I : Type u} [Fintype I] [DecidableEq I] {X : StandardSpace.{u}} {T : X ≃ᵐ X} {base : InitialLaws I X T} (M : Object I base) lemma commonActive_swapColumns (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e = commonActive M rows.P rows.A rows.B cols.P cols.B cols.A n e := by filter_upwards [rowActive_swap M hM rows.P rows.properP cols n] with e he simp only [commonActive,he] lemma commonActive_swapRows (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e = commonActive M rows.P rows.B rows.A cols.P cols.A cols.B n e := by filter_upwards [columnActive_swap M hM rows cols.P cols.properP n] with e he simp only [commonActive,he] lemma grouped_first_row_edge (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ᵐ s ∂cellLaw M rows.P cols.A n e, (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection ∘L fibreRowEdge M rows.P cols.A cols.P cols.B n e s = fibreRowEdge M rows.P cols.A cols.P cols.B n e s ∘L (commonActive M rows.P rows.A rows.B cols.B cols.P cols.A n e).starProjection := by filter_upwards [grouped_active_intersections M hM rows.swapPA cols.swapPA n, commonActive_swapColumns M hM rows cols.swapPB n] with e he hs dsimp only [TriplePartition.swapPA,TriplePartition.swapPB] at he hs have hp : (commonActive M rows.P rows.A rows.B cols.B cols.A cols.P n e).starProjection = (commonActive M rows.P rows.A rows.B cols.B cols.P cols.A n e).starProjection := by congr 1 filter_upwards [he.1] with s h rw [hp] at h exact h lemma grouped_second_row_edge (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ᵐ t ∂cellLaw M rows.P cols.B n e, (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection ∘L fibreRowEdge M rows.P cols.B cols.P cols.A n e t = fibreRowEdge M rows.P cols.B cols.P cols.A n e t ∘L (commonActive M rows.P rows.A rows.B cols.A cols.P cols.B n e).starProjection := by filter_upwards [grouped_first_row_edge M hM rows cols.swapAB n, commonActive_swapColumns M hM rows cols n] with e he hs have hp : (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection = (commonActive M rows.P rows.A rows.B cols.P cols.B cols.A n e).starProjection := by congr 1 filter_upwards [he] with t h rw [hp] exact h theorem grouped_row_contraction_projection (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ (hr : MeasurePreserving (cells M rows.P cols.P n) (M.ownKernel (n+2) e) (cellLaw M rows.P cols.P n e)) (hst : MeasurePreserving (fun y => (cells M rows.P cols.A n y,cells M rows.P cols.B n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.P cols.A n e).prod (cellLaw M rows.P cols.B n e))) (g : Lp ℂ 2 (cellLaw M rows.P cols.A n e)) (h : Lp ℂ 2 (cellLaw M rows.P cols.B n e)), (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection (JoiningOperator.contraction hr hst g h) = JoiningOperator.contraction hr hst ((commonActive M rows.P rows.A rows.B cols.A cols.P cols.B n e).starProjection g) ((commonActive M rows.P rows.A rows.B cols.B cols.P cols.A n e).starProjection h) := by filter_upwards [grouped_first_row_edge M hM rows cols n,grouped_second_row_edge M hM rows cols n, cells_fibre_marginal M hM rows.P cols.A rows.properP n, cells_fibre_marginal M hM rows.P cols.B rows.properP n, rowPair_fixed_conditional M rows.P cols.A cols.P cols.B n, rowPair_fixed_conditional M rows.P cols.B cols.P cols.A n, rowPair_fixed_marginals M hM rows.P cols.A cols.P cols.B rows.properP cols.disjointPA.symm cols.disjointAB (by simpa [Finset.union_comm] using cols.properPA) cols.properAB n, rowPair_fixed_marginals M hM rows.P cols.B cols.P cols.A rows.properP cols.disjointPB.symm cols.disjointAB.symm (by simpa [Finset.union_comm] using cols.properPB) (by simpa [Finset.union_comm] using cols.properAB) n] with e hq hp hs ht hκ hξ hmκ hmξ intro hr hst g h rw [hs.map_eq] at hmκ rw [ht.map_eq] at hmξ simp only [fibreRowEdge_eq] at hq hp let hts : MeasurePreserving (fun y => (cells M rows.P cols.B n y,cells M rows.P cols.A n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.P cols.B n e).prod (cellLaw M rows.P cols.A n e)) := Measure.measurePreserving_swap.comp hst apply JoiningOperator.contraction_projection_both hr hst hts _ _ _ · intro g h exact JoiningOperator.contraction_projection hs hr hst hκ hmκ _ _ hq g h · intro h g exact JoiningOperator.contraction_projection ht hr hts hξ hmξ _ _ hp h g lemma grouped_first_column_edge (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ᵐ s ∂cellLaw M rows.A cols.P n e, (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection ∘L fibreColumnEdge M rows.A rows.P rows.B cols.P n e s = fibreColumnEdge M rows.A rows.P rows.B cols.P n e s ∘L (commonActive M rows.B rows.P rows.A cols.P cols.A cols.B n e).starProjection := by filter_upwards [grouped_active_intersections M hM rows.swapPA cols.swapPA n, commonActive_swapRows M hM rows.swapPB cols n] with e he hs dsimp only [TriplePartition.swapPA,TriplePartition.swapPB] at he hs have hp : (commonActive M rows.B rows.A rows.P cols.P cols.A cols.B n e).starProjection = (commonActive M rows.B rows.P rows.A cols.P cols.A cols.B n e).starProjection := by congr 1 filter_upwards [he.2.2.1] with s h rw [hp] at h exact h lemma grouped_second_column_edge (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ᵐ t ∂cellLaw M rows.B cols.P n e, (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection ∘L fibreColumnEdge M rows.B rows.P rows.A cols.P n e t = fibreColumnEdge M rows.B rows.P rows.A cols.P n e t ∘L (commonActive M rows.A rows.P rows.B cols.P cols.A cols.B n e).starProjection := by filter_upwards [grouped_first_column_edge M hM rows.swapAB cols n, commonActive_swapRows M hM rows cols n] with e he hs have hp : (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection = (commonActive M rows.P rows.B rows.A cols.P cols.A cols.B n e).starProjection := by congr 1 filter_upwards [he] with t h rw [hp] exact h theorem grouped_column_contraction_projection (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ (hr : MeasurePreserving (cells M rows.P cols.P n) (M.ownKernel (n+2) e) (cellLaw M rows.P cols.P n e)) (hst : MeasurePreserving (fun y => (cells M rows.A cols.P n y,cells M rows.B cols.P n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.A cols.P n e).prod (cellLaw M rows.B cols.P n e))) (g : Lp ℂ 2 (cellLaw M rows.A cols.P n e)) (h : Lp ℂ 2 (cellLaw M rows.B cols.P n e)), (commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e).starProjection (JoiningOperator.contraction hr hst g h) = JoiningOperator.contraction hr hst ((commonActive M rows.A rows.P rows.B cols.P cols.A cols.B n e).starProjection g) ((commonActive M rows.B rows.P rows.A cols.P cols.A cols.B n e).starProjection h) := by let rr := rows.swapPA let rt := rows.swapPB filter_upwards [grouped_first_column_edge M hM rows cols n,grouped_second_column_edge M hM rows cols n, cells_fibre_marginal M hM rows.A cols.P rows.properA n, cells_fibre_marginal M hM rows.B cols.P rows.properB n, columnPair_fixed_conditional M hM rows.A rows.P rows.B cols.P rr.cardP cols.properP n, columnPair_fixed_conditional M hM rows.B rows.P rows.A cols.P rt.cardP cols.properP n, columnPair_fixed_marginals M hM rows.A rows.P rows.B cols.P rr.cardP cols.properP rows.properP rows.properB rows.disjointPA.symm rows.disjointAB (by simpa [Finset.union_comm] using rows.properPA) rows.properAB n, columnPair_fixed_marginals M hM rows.B rows.P rows.A cols.P rt.cardP cols.properP rows.properP rows.properA rows.disjointPB.symm rows.disjointAB.symm (by simpa [Finset.union_comm] using rows.properPB) (by simpa [Finset.union_comm] using rows.properAB) n] with e hq hp hs ht hκ hξ hmκ hmξ intro hr hst g h rw [hs.map_eq] at hmκ rw [ht.map_eq] at hmξ simp only [fibreColumnEdge_eq] at hq hp let hts : MeasurePreserving (fun y => (cells M rows.B cols.P n y,cells M rows.A cols.P n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.B cols.P n e).prod (cellLaw M rows.A cols.P n e)) := Measure.measurePreserving_swap.comp hst apply JoiningOperator.contraction_projection_both hr hst hts _ _ _ · intro g h exact JoiningOperator.contraction_projection hs hr hst hκ hmκ _ _ hq g h · intro h g exact JoiningOperator.contraction_projection ht hr hts hξ hmξ _ _ hp h g end local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ universe u variable {I : Type u} [Fintype I] [DecidableEq I] {X : StandardSpace.{u}} {T : X ≃ᵐ X} {base : InitialLaws I X T} (M : Object I base) lemma rowActive_eq_range (hM : M.IsSaturated) (A : Finset I) (hA : A ≠ Finset.univ) (cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ (hr : MeasurePreserving (cells M A cols.P n) (M.ownKernel (n+2) e) (cellLaw M A cols.P n e)) (hst : MeasurePreserving (fun y => (cells M A cols.A n y,cells M A cols.B n y)) (M.ownKernel (n+2) e) ((cellLaw M A cols.A n e).prod (cellLaw M A cols.B n e))), rowActive M A cols.P cols.A cols.B n e = (LinearMap.range (JoiningOperator.operator hr hst).toLinearMap).topologicalClosure := by filter_upwards [cells_fibre_marginal M hM A cols.A hA n, cells_fibre_marginal M hM A cols.B hA n, rowPair_fixed_conditional M A cols.A cols.P cols.B n, rowPair_fixed_marginals M hM A cols.A cols.P cols.B hA cols.disjointPA.symm cols.disjointAB (by simpa [Finset.union_comm] using cols.properPA) cols.properAB n] with e hs ht hκ hm intro hr hst rw [hs.map_eq] at hm unfold rowActive rw [fibreRowEdge_eq] exact (JoiningOperator.active_eq_essentialRange_product hs hr ht hst hκ hm).symm lemma columnActive_eq_range (hM : M.IsSaturated) (rows : TriplePartition I) (R : Finset I) (hR : R ≠ Finset.univ) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ (hr : MeasurePreserving (cells M rows.P R n) (M.ownKernel (n+2) e) (cellLaw M rows.P R n e)) (hst : MeasurePreserving (fun y => (cells M rows.A R n y,cells M rows.B R n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.A R n e).prod (cellLaw M rows.B R n e))), columnActive M rows.P rows.A rows.B R n e = (LinearMap.range (JoiningOperator.operator hr hst).toLinearMap).topologicalClosure := by filter_upwards [cells_fibre_marginal M hM rows.A R rows.properA n, cells_fibre_marginal M hM rows.B R rows.properB n, columnPair_fixed_conditional M hM rows.A rows.P rows.B R rows.swapPA.cardP hR n, columnPair_fixed_marginals M hM rows.A rows.P rows.B R rows.swapPA.cardP hR rows.properP rows.properB rows.disjointPA.symm rows.disjointAB (by simpa [Finset.union_comm] using rows.properPA) rows.properAB n] with e hs ht hκ hm intro hr hst rw [hs.map_eq] at hm unfold columnActive rw [fibreColumnEdge_eq] exact (JoiningOperator.active_eq_essentialRange_product hs hr ht hst hκ hm).symm theorem grouped_row_contractions_span (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ (hr : MeasurePreserving (cells M rows.P cols.P n) (M.ownKernel (n+2) e) (cellLaw M rows.P cols.P n e)) (hst : MeasurePreserving (fun y => (cells M rows.P cols.A n y,cells M rows.P cols.B n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.P cols.A n e).prod (cellLaw M rows.P cols.B n e))), (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 (cellLaw M rows.P cols.A n e) × Lp ℂ 2 (cellLaw M rows.P cols.B n e) => JoiningOperator.contraction hr hst ((commonActive M rows.P rows.A rows.B cols.A cols.P cols.B n e).starProjection p.1) ((commonActive M rows.P rows.A rows.B cols.B cols.P cols.A n e).starProjection p.2)))).topologicalClosure = commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e := by filter_upwards [rowActive_eq_range M hM rows.P rows.properP cols n, grouped_row_contraction_projection M hM rows cols n] with e he hp intro hr hst apply restricted_contractions_span (JoiningOperator.operator hr hst) · rw [←he hr hst] exact inf_le_left · exact hp hr hst theorem grouped_column_contractions_span (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ (hr : MeasurePreserving (cells M rows.P cols.P n) (M.ownKernel (n+2) e) (cellLaw M rows.P cols.P n e)) (hst : MeasurePreserving (fun y => (cells M rows.A cols.P n y,cells M rows.B cols.P n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.A cols.P n e).prod (cellLaw M rows.B cols.P n e))), (Submodule.span ℂ (Set.range (fun p : Lp ℂ 2 (cellLaw M rows.A cols.P n e) × Lp ℂ 2 (cellLaw M rows.B cols.P n e) => JoiningOperator.contraction hr hst ((commonActive M rows.A rows.P rows.B cols.P cols.A cols.B n e).starProjection p.1) ((commonActive M rows.B rows.P rows.A cols.P cols.A cols.B n e).starProjection p.2)))).topologicalClosure = commonActive M rows.P rows.A rows.B cols.P cols.A cols.B n e := by filter_upwards [columnActive_eq_range M hM rows cols.P cols.properP n, grouped_column_contraction_projection M hM rows cols n] with e he hp intro hr hst apply restricted_contractions_span (JoiningOperator.operator hr hst) · rw [←he hr hst] exact inf_le_right · exact hp hr hst end ArrayObjects.Object.Rolled namespace JoiningOperator open MeasureTheory ProbabilityTheory ContinuousLinearMap open scoped InnerProductSpace ENNReal section ContractionCoefficients variable {Ω U X Y : Type*} [MeasurableSpace Ω] [MeasurableSpace U] [MeasurableSpace X] [MeasurableSpace Y] [StandardBorelSpace X] [StandardBorelSpace Y] [Nonempty X] [Nonempty Y] {μ : Measure Ω} [IsProbabilityMeasure μ] {η : Measure U} {ν : Measure X} {ρ : Measure Y} [IsProbabilityMeasure η] [IsProbabilityMeasure ν] [IsProbabilityMeasure ρ] {u : Ω → U} {x : Ω → X} {y : Ω → Y} {κ : Kernel U (X × Y)} [IsMarkovKernel κ] omit [IsProbabilityMeasure η] [IsProbabilityMeasure ν] [IsProbabilityMeasure ρ] in lemma contraction_coefficient [SFinite ρ] (hu : MeasurePreserving u μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω,y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (g : Lp ℂ 2 ν) (h : Lp ℂ 2 ρ) : contraction hu hxy g h =ᵐ[η] (fun a => inner ℂ (star g) (family κ ν ρ a h)) := by let F : X × Y → ℂ := fun p => g p.1 * h p.2 have hFm : StronglyMeasurable F := (g.1.stronglyMeasurable.comp_measurable measurable_fst).mul (h.1.stronglyMeasurable.comp_measurable measurable_snd) have hten : (fun ω => tensorLp g h (x ω,y ω)) =ᵐ[μ] (fun ω => F (x ω,y ω)) := hxy.quasiMeasurePreserving.ae (coe_tensorLp g h) have hFi : Integrable (fun ω => F (x ω,y ω)) μ := by apply (((Lp.memLp (tensorLp g h)).comp_measurePreserving hxy).integrable (by norm_num : (1 : ℝ≥0∞) ≤ 2)).congr exact hten have hcond := (operator_ae_eq_condExp hu hxy (tensorLp g h)).trans ((condExp_congr_ae hten).trans (condExp_of_kernel hu.measurable hκ F hFm hFi)) have hi : contraction hu hxy g h =ᵐ[η] (fun a => ∫ p, F p ∂κ a) := by have ht := (ae_map_iff hu.aemeasurable (measurableSet_eq_fun (Lp.stronglyMeasurable (contraction hu hxy g h)).measurable hFm.integral_kernel.measurable)).mpr hcond change ∀ᵐ a ∂η, contraction hu hxy g h a = ∫ p, F p ∂κ a simpa only [hu.map_eq] using ht apply hi.trans filter_upwards [hm, family_inner hm (star g) h] with a ha hai rw [hai] apply integral_congr_ae filter_upwards [ha.1.quasiMeasurePreserving.ae (Lp.coeFn_star g)] with p hp rw [hp] simp [F, RCLike.inner_apply, mul_comm] end ContractionCoefficients variable {Ω Z X Y A B C : Type*} [MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] {μ : Measure Ω} {ν : Measure X} {ρ : Measure Y} {η : Measure Z} {α : Measure A} {β : Measure B} {τ : Measure C} {x : Ω → X} {y : Ω → Y} {z : Ω → Z} {a : X → A} {b : Y → B} {c : Z → C} [SFinite ν] [SFinite ρ] lemma tensor_pull [SFinite β] (ha : MeasurePreserving a ν α) (hb : MeasurePreserving b ρ β) (g : Lp ℂ 2 α) (h : Lp ℂ 2 β) : tensorLp (pull ha g) (pull hb h) = pull (ha.prod hb) (tensorLp g h) := by apply Lp.ext filter_upwards [coe_tensorLp (pull ha g) (pull hb h), (ha.prod hb).quasiMeasurePreserving.ae (coe_tensorLp g h), Lp.coeFn_compMeasurePreserving (tensorLp g h) (ha.prod hb), Measure.quasiMeasurePreserving_fst.ae (Lp.coeFn_compMeasurePreserving g ha), Measure.quasiMeasurePreserving_snd.ae (Lp.coeFn_compMeasurePreserving h hb)] with p hp ht hpull hga hhb exact hp.trans ((congrArg₂ (· * ·) hga hhb).trans (ht.symm.trans hpull.symm)) lemma contraction_pull [SFinite β] (hz : MeasurePreserving z μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (ha : MeasurePreserving a ν α) (hb : MeasurePreserving b ρ β) (g : Lp ℂ 2 α) (h : Lp ℂ 2 β) : contraction hz hxy (pull ha g) (pull hb h) = operator hz ((ha.prod hb).comp hxy) (tensorLp g h) := by unfold contraction operator rw [tensor_pull ha hb, pull_comp hxy (ha.prod hb)] rfl lemma contraction_mem_screen [IsProbabilityMeasure μ] [StandardBorelSpace A] [Nonempty A] [StandardBorelSpace B] [Nonempty B] (hz : MeasurePreserving z μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (ha : MeasurePreserving a ν α) (hb : MeasurePreserving b ρ β) (hc : MeasurePreserving c η τ) {κ : Kernel C (A × B)} [IsMarkovKernel κ] (hκ : HasCondDistrib (fun ω => (a (x ω),b (y ω))) z (κ.comap c hc.measurable) μ) (g : Lp ℂ 2 ν) (h : Lp ℂ 2 ρ) (hg : g ∈ lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 ν) (hh : h ∈ lpMeas ℂ ℂ (MeasurableSpace.comap b inferInstance) 2 ρ) : contraction hz hxy g h ∈ lpMeas ℂ ℂ (MeasurableSpace.comap c inferInstance) 2 η := by have : SFinite β := hb.map_eq ▸ inferInstanceAs (SFinite (ρ.map b)) rw [←pull_range ha] at hg rw [←pull_range hb] at hh obtain ⟨g,rfl⟩ := hg obtain ⟨h,rfl⟩ := hh change contraction hz hxy (pull ha g) (pull hb h) ∈ _ rw [contraction_pull hz hxy ha hb g h, screened_operator hz ((ha.prod hb).comp hxy) hc hκ] rw [←pull_range hc] exact ⟨operator (hc.comp hz) ((ha.prod hb).comp hxy) (tensorLp g h),rfl⟩ end JoiningOperator section open MeasureTheory ProbabilityTheory variable {G : Type*} [Group G] [MeasurableSpace G] [MeasurableMul₂ G] lemma measure_eq_graph_of_ratio (μ : Measure (G × G)) (ρ : Measure G) (a : G) (hρ : μ.map Prod.snd = ρ) (hr : ∀ᵐ p ∂μ, p.1 * p.2⁻¹ = a) : μ = ρ.map (fun b => (a*b,b)) := by have hg : Measurable (fun b : G => (a*b,b)) := (measurable_const.mul measurable_id).prodMk measurable_id rw [←hρ, Measure.map_map hg measurable_snd] calc μ = μ.map id := (Measure.map_id).symm _ = μ.map ((fun b => (a*b,b)) ∘ Prod.snd) := by apply Measure.map_congr filter_upwards [hr] with p hp apply Prod.ext · change p.1 = a*p.2 rw [←hp, mul_assoc, inv_mul_cancel, mul_one] · rfl def ratioKernel (ρ : Measure G) : Kernel G (G × G) := (Kernel.id.prod (Kernel.const G ρ)).map (fun p : G × G => (p.1*p.2,p.2)) lemma ratioKernel_apply (ρ : Measure G) [SFinite ρ] (a : G) : ratioKernel ρ a = ρ.map (fun b => (a*b,b)) := by have hm : Measurable (fun p : G × G => (p.1*p.2,p.2)) := (measurable_fst.mul measurable_snd).prodMk measurable_snd rw [ratioKernel, Kernel.map_apply _ hm] simp only [Kernel.prod_apply, Kernel.id_apply, Kernel.const_apply] rw [Measure.dirac_prod] rw [Measure.map_map hm (measurable_prodMk_left (x := a))] rfl instance ratioKernel_markov (ρ : Measure G) [IsProbabilityMeasure ρ] : IsMarkovKernel (ratioKernel ρ) := by unfold ratioKernel exact Kernel.IsMarkovKernel.map _ (measurable_fst.mul measurable_snd |>.prodMk measurable_snd) variable {Ω Z : Type*} [MeasurableSpace Ω] [MeasurableSpace Z] {μ : Measure Ω} {z : Ω → Z} {v : Ω → G × G} lemma hasCondDistrib_ratio {κ : Kernel Z (G × G)} [IsMarkovKernel κ] (hκ : HasCondDistrib v z κ μ) (ρ : Measure G) [IsProbabilityMeasure ρ] (c : Z → G) (hc : Measurable c) (hm : ∀ᵐ t ∂μ.map z, (κ t).map Prod.snd = ρ) (hr : ∀ᵐ t ∂μ.map z, ∀ᵐ p ∂κ t, p.1*p.2⁻¹ = c t) : HasCondDistrib v z ((ratioKernel ρ).comap c hc) μ := by refine ⟨hκ.aemeasurable, hκ.map_eq.trans (Measure.compProd_congr ?_)⟩ filter_upwards [hm,hr] with t ht htr change κ t = ratioKernel ρ (c t) rw [ratioKernel_apply] exact measure_eq_graph_of_ratio (κ t) ρ (c t) ht htr end namespace JoiningOperator open MeasureTheory ProbabilityTheory ContinuousLinearMap open scoped InnerProductSpace ENNReal section FactorFields variable {Ω Z X Y A B C : Type*} [MeasurableSpace Ω] [mZ : MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C] {μ : Measure Ω} {ν : Measure X} {ρ : Measure Y} {η : Measure Z} {α : Measure A} {β : Measure B} {τ : Measure C} {x : Ω → X} {y : Ω → Y} {z : Ω → Z} {a : X → A} {b : Y → B} {c : Z → C} lemma mem_field_of_eq_comp (hc : Measurable c) (f : Lp ℂ 2 η) (F : C → ℂ) (hF : Measurable F) (he : f =ᵐ[η] (fun v => F (c v))) : f ∈ lpMeas ℂ ℂ (MeasurableSpace.comap c inferInstance) 2 η := by let : Fact (MeasurableSpace.comap c inferInstance ≤ mZ) := ⟨hc.comap_le⟩ apply mem_lpMeas_iff_aestronglyMeasurable.mpr have hm : @StronglyMeasurable Z ℂ _ (MeasurableSpace.comap c inferInstance) (fun v => F (c v)) := (hF.comp (Measurable.of_comap_le le_rfl)).stronglyMeasurable exact hm.aestronglyMeasurable.congr he.symm lemma common_space_le_screen [IsProbabilityMeasure μ] [IsFiniteMeasure ν] [IsFiniteMeasure ρ] [StandardBorelSpace A] [Nonempty A] [StandardBorelSpace B] [Nonempty B] (hz : MeasurePreserving z μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (ha : MeasurePreserving a ν α) (hb : MeasurePreserving b ρ β) (hc : MeasurePreserving c η τ) {κ : Kernel C (A × B)} [IsMarkovKernel κ] (hκ : HasCondDistrib (fun ω => (a (x ω),b (y ω))) z (κ.comap c hc.measurable) μ) (D : Submodule ℂ (Lp ℂ 2 η)) [D.HasOrthogonalProjection] (P : Lp ℂ 2 ν →L[ℂ] Lp ℂ 2 ν) (Q : Lp ℂ 2 ρ →L[ℂ] Lp ℂ 2 ρ) (hD : D ≤ (LinearMap.range (operator hz hxy).toLinearMap).topologicalClosure) (hPQ : ∀ g h, D.starProjection (contraction hz hxy g h) = contraction hz hxy (P g) (Q h)) (hP : ∀ g, P g ∈ lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 ν) (hQ : ∀ h, Q h ∈ lpMeas ℂ ℂ (MeasurableSpace.comap b inferInstance) 2 ρ) : D ≤ lpMeas ℂ ℂ (MeasurableSpace.comap c inferInstance) 2 η := by rw [←restricted_contractions_span (operator hz hxy) D P Q hD hPQ] apply Submodule.topologicalClosure_minimal · apply Submodule.span_le.mpr rintro _ ⟨⟨g,h⟩,rfl⟩ exact contraction_mem_screen hz hxy ha hb hc hκ (P g) (Q h) (hP g) (hQ h) · exact isClosed_aestronglyMeasurable hc.measurable.comap_le lemma common_space_le_ratio {G : Type*} [gG : Group G] [mG : MeasurableSpace G] [hG : MeasurableMul₂ G] [StandardBorelSpace G] [Nonempty G] [IsProbabilityMeasure μ] [IsFiniteMeasure ν] [IsFiniteMeasure ρ] {a : X → G} {b : Y → G} {c : Z → G} (hz : MeasurePreserving z μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (ha : Measurable a) (hb : Measurable b) (hc : Measurable c) {κ : Kernel Z (G × G)} [IsMarkovKernel κ] (hκ : HasCondDistrib (fun ω => (a (x ω),b (y ω))) z κ μ) (σ : Measure G) [IsProbabilityMeasure σ] (hm : ∀ᵐ t ∂η, (κ t).map Prod.snd = σ) (hr : ∀ᵐ t ∂η, ∀ᵐ p ∂κ t, p.1*p.2⁻¹ = c t) (D : Submodule ℂ (Lp ℂ 2 η)) [D.HasOrthogonalProjection] (P : Lp ℂ 2 ν →L[ℂ] Lp ℂ 2 ν) (Q : Lp ℂ 2 ρ →L[ℂ] Lp ℂ 2 ρ) (hD : D ≤ (LinearMap.range (operator hz hxy).toLinearMap).topologicalClosure) (hPQ : ∀ g h, D.starProjection (contraction hz hxy g h) = contraction hz hxy (P g) (Q h)) (hP : ∀ g, P g ∈ lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 ν) (hQ : ∀ h, Q h ∈ lpMeas ℂ ℂ (MeasurableSpace.comap b inferInstance) 2 ρ) : D ≤ lpMeas ℂ ℂ (MeasurableSpace.comap c inferInstance) 2 η := by have hscreen := hasCondDistrib_ratio hκ σ c hc (by simpa only [hz.map_eq] using hm) (by simpa only [hz.map_eq] using hr) exact common_space_le_screen hz hxy ⟨ha,rfl⟩ ⟨hb,rfl⟩ ⟨hc,rfl⟩ hscreen D P Q hD hPQ hP hQ end FactorFields variable {Ω U X Y C : Type*} [MeasurableSpace Ω] [MeasurableSpace U] [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace C] [StandardBorelSpace X] [StandardBorelSpace Y] [Nonempty X] [Nonempty Y] {μ : Measure Ω} [IsProbabilityMeasure μ] {η : Measure U} {ν : Measure X} {ρ : Measure Y} {u : Ω → U} {x : Ω → X} {y : Ω → Y} {κ : Kernel U (X × Y)} [IsMarkovKernel κ] local instance : MeasurableSpace (Lp ℂ 2 ν) := borel _ local instance : BorelSpace (Lp ℂ 2 ν) := ⟨rfl⟩ lemma contraction_mem_operatorField [SFinite ρ] (hu : MeasurePreserving u μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω,y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (V : Submodule ℂ (Lp ℂ 2 ρ)) (W : Submodule ℂ (Lp ℂ 2 ν)) [V.HasOrthogonalProjection] [W.HasOrthogonalProjection] (hA : ∀ᵐ a ∂η, W.starProjection ∘L family κ ν ρ a = family κ ν ρ a ∘L V.starProjection) (c : U → C) (hc : Measurable c) (F : C → V → W) (hF : ∀ v, Measurable (fun q => (F q v : Lp ℂ 2 ν))) (hf : ∀ᵐ a ∂η, ∀ v, compress (family κ ν ρ a) V W v = F (c a) v) (g : Lp ℂ 2 ν) (h : V) : contraction hu hxy g h ∈ lpMeas ℂ ℂ (MeasurableSpace.comap c inferInstance) 2 η := by apply mem_field_of_eq_comp hc _ (fun q => inner ℂ (star g) (F q h : Lp ℂ 2 ν)) ((continuous_const.inner (𝕜 := ℂ) continuous_id).measurable.comp (hF h)) apply (contraction_coefficient hu hxy hκ hm g h).trans filter_upwards [hA,hf] with a ha hf rw [←compress_apply _ V W ha h, hf] lemma common_space_le_operatorField [IsFiniteMeasure ν] [IsFiniteMeasure ρ] (hu : MeasurePreserving u μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω,y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (V : Submodule ℂ (Lp ℂ 2 ρ)) (W : Submodule ℂ (Lp ℂ 2 ν)) [V.HasOrthogonalProjection] [W.HasOrthogonalProjection] (hA : ∀ᵐ a ∂η, W.starProjection ∘L family κ ν ρ a = family κ ν ρ a ∘L V.starProjection) (c : U → C) (hc : Measurable c) (F : C → V → W) (hF : ∀ v, Measurable (fun q => (F q v : Lp ℂ 2 ν))) (hf : ∀ᵐ a ∂η, ∀ v, compress (family κ ν ρ a) V W v = F (c a) v) (D : Submodule ℂ (Lp ℂ 2 η)) [D.HasOrthogonalProjection] (P : Lp ℂ 2 ν →L[ℂ] Lp ℂ 2 ν) (Q : Lp ℂ 2 ρ →L[ℂ] Lp ℂ 2 ρ) (hD : D ≤ (LinearMap.range (operator hu hxy).toLinearMap).topologicalClosure) (hPQ : ∀ g h, D.starProjection (contraction hu hxy g h) = contraction hu hxy (P g) (Q h)) (hQ : ∀ h, Q h ∈ V) : D ≤ lpMeas ℂ ℂ (MeasurableSpace.comap c inferInstance) 2 η := by rw [←restricted_contractions_span (operator hu hxy) D P Q hD hPQ] apply Submodule.topologicalClosure_minimal · apply Submodule.span_le.mpr rintro _ ⟨⟨g,h⟩,rfl⟩ exact contraction_mem_operatorField hu hxy hκ hm V W hA c hc F hF hf (P g) ⟨Q h,hQ h⟩ · exact isClosed_aestronglyMeasurable hc.comap_le end JoiningOperator section open MeasureTheory ContinuousLinearMap variable {X E H : Type*} [MeasurableSpace X] [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] [SecondCountableTopology E] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [SecondCountableTopology H] {μ : Measure X} [IsProbabilityMeasure μ] {F : X → E →L[ℂ] H} omit [SecondCountableTopology H] in lemma MeasurablePolar.fixed_factor (R : MeasurablePolar μ F) (hs : HasFixedInjectiveSquares μ F) : ∃ B : E →L[ℂ] E, ∀ᵐ x ∂μ, ∀ v, F x v = R.unitary x (B v) := by obtain ⟨P,Q,_,_,hP,_⟩ := hs obtain ⟨x₀,hx₀⟩ := hP.exists refine ⟨operatorAbs (F x₀),?_⟩ filter_upwards [hP,R.formula] with x hx hf intro v have he : operatorAbs (F x) = operatorAbs (F x₀) := by unfold operatorAbs rw [hx,hx₀] rw [←hf.2.2] change R.unitary x (operatorAbs (F x) v) = _ rw [he] end section open MeasureTheory ProbabilityTheory ContinuousLinearMap open scoped ENNReal namespace JoiningOperator local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ variable {Ω U X Y : Type*} [MeasurableSpace Ω] [MeasurableSpace U] [MeasurableSpace X] [MeasurableSpace Y] [StandardBorelSpace X] [StandardBorelSpace Y] [Nonempty X] [Nonempty Y] {μ : Measure Ω} [IsProbabilityMeasure μ] {η : Measure U} [IsProbabilityMeasure η] {ν : Measure X} {ρ : Measure Y} [IsFiniteMeasure ν] [IsFiniteMeasure ρ] {u : Ω → U} {x : Ω → X} {y : Ω → Y} {κ : Kernel U (X × Y)} [IsMarkovKernel κ] local instance : MeasurableSpace (Lp ℂ 2 ν) := borel _ local instance : BorelSpace (Lp ℂ 2 ν) := ⟨rfl⟩ lemma common_space_le_polarField (hu : MeasurePreserving u μ η) (hxy : MeasurePreserving (fun ω => (x ω,y ω)) μ (ν.prod ρ)) (hκ : HasCondDistrib (fun ω => (x ω,y ω)) u κ μ) (hm : ∀ᵐ a ∂η, MeasurePreserving Prod.fst (κ a) ν ∧ MeasurePreserving Prod.snd (κ a) ρ) (V : Submodule ℂ (Lp ℂ 2 ρ)) (W : Submodule ℂ (Lp ℂ 2 ν)) [V.HasOrthogonalProjection] [W.HasOrthogonalProjection] [CompleteSpace V] [CompleteSpace W] (hA : ∀ᵐ a ∂η, W.starProjection ∘L family κ ν ρ a = family κ ν ρ a ∘L V.starProjection) (R : MeasurablePolar η (fun a => compress (family κ ν ρ a) V W)) (hs : HasFixedInjectiveSquares η (fun a => compress (family κ ν ρ a) V W)) (D : Submodule ℂ (Lp ℂ 2 η)) [D.HasOrthogonalProjection] (P : Lp ℂ 2 ν →L[ℂ] Lp ℂ 2 ν) (Q : Lp ℂ 2 ρ →L[ℂ] Lp ℂ 2 ρ) (hD : D ≤ (LinearMap.range (operator hu hxy).toLinearMap).topologicalClosure) (hPQ : ∀ g h, D.starProjection (contraction hu hxy g h) = contraction hu hxy (P g) (Q h)) (hQ : ∀ h, Q h ∈ V) : D ≤ lpMeas ℂ ℂ (MeasurableSpace.comap R.unitary inferInstance) 2 η := by obtain ⟨B,hB⟩ := R.fixed_factor hs apply common_space_le_operatorField hu hxy hκ hm V W hA R.unitary R.measurable (fun q v => q (B v)) _ hB D P Q hD hPQ hQ intro v exact (continuous_subtype_val.comp (StrongUnitary.continuous_apply (B v))).measurable end JoiningOperator namespace ArrayObjects.Object.Rolled local instance : Fact ((2 : ℝ≥0∞) ≠ ∞) := ⟨by norm_num⟩ universe u variable {I : Type u} [Fintype I] [DecidableEq I] {X : StandardSpace.{u}} {T : X ≃ᵐ X} {base : InitialLaws I X T} (M : Object I base) theorem grouped_polar_a_field (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ R : GroupedPolarData M rows cols n e, commonActive M rows.A rows.P rows.B cols.P cols.A cols.B n e ≤ lpMeas ℂ ℂ (MeasurableSpace.comap R.a.unitary inferInstance) 2 (cellLaw M rows.A cols.P n e) := by filter_upwards [grouped_rectangle_spaces M hM rows cols n, grouped_restricted_fixed_squares M hM rows cols n, grouped_active_intersections M hM rows cols n, grouped_row_contraction_projection M hM rows.swapPA cols n, rowActive_eq_range M hM rows.A rows.properA cols n, cells_fibre_marginal M hM rows.A cols.P rows.properA n, cells_fibre_marginal M hM rows.A cols.A rows.properA n, cells_fibre_marginal M hM rows.A cols.B rows.properA n, cells_indep_columns_fibre M rows.A rows.A cols.A cols.B cols.disjointAB cols.properAB n, rowPair_fixed_conditional M rows.A cols.P cols.A cols.B n, rowPair_fixed_marginals M hM rows.A cols.P cols.A cols.B rows.properA cols.disjointPA cols.disjointPB cols.properPA cols.properPB n] with e hspaces hs hA hproj hrange hx hc hd hcd hκ hm intro R rw [hx.map_eq] at hm let hpair : MeasurePreserving (fun y => (cells M rows.A cols.A n y,cells M rows.A cols.B n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.A cols.A n e).prod (cellLaw M rows.A cols.B n e)) := ⟨hc.measurable.prodMk hd.measurable, by rw [←hc.map_eq,←hd.map_eq] exact hcd.map_prod_eq_prod_map_map hc.measurable.aemeasurable hd.measurable.aemeasurable⟩ let V := rectangleSpace₂ (cellLaw M rows.A cols.P n e) (cellLaw M rows.P cols.B n e) (fibreRowEdge M rows.A cols.P cols.A cols.B n e) (fibreColumnEdge M rows.P rows.A rows.B cols.B n e) let W := rectangleSpace₁ (cellLaw M rows.A cols.P n e) (cellLaw M rows.P cols.A n e) (fibreRowEdge M rows.A cols.P cols.A cols.B n e) (fibreColumnEdge M rows.P rows.A rows.B cols.A n e) have hv : V = commonActive M rows.A rows.P rows.B cols.B cols.P cols.A n e := hspaces.2.1 have hw : W = commonActive M rows.A rows.P rows.B cols.A cols.P cols.B n e := hspaces.1 have ha : ∀ᵐ a ∂cellLaw M rows.A cols.P n e, W.starProjection ∘L JoiningOperator.family ((rowPairKernel M rows.A cols.P cols.A cols.B n).sectR e) (cellLaw M rows.A cols.A n e) (cellLaw M rows.A cols.B n e) a = JoiningOperator.family ((rowPairKernel M rows.A cols.P cols.A cols.B n).sectR e) (cellLaw M rows.A cols.A n e) (cellLaw M rows.A cols.B n e) a ∘L V.starProjection := by simpa only [hv,hw,fibreRowEdge_eq] using hA.1 apply JoiningOperator.common_space_le_polarField hx hpair hκ hm V W ha R.a hs.1 _ (commonActive M rows.A rows.P rows.B cols.A cols.P cols.B n e).starProjection (commonActive M rows.A rows.P rows.B cols.B cols.P cols.A n e).starProjection · rw [←hrange hx hpair] exact inf_le_left · exact hproj hx hpair · intro h rw [hv] exact Submodule.starProjection_apply_mem _ h theorem grouped_polar_b_field (hM : M.IsSaturated) (rows cols : TriplePartition I) (n : ℕ) : ∀ᵐ e ∂(M.law (n+2)).map (amap M.par (n+2)), ∀ R : GroupedPolarData M rows cols n e, commonActive M rows.B rows.P rows.A cols.P cols.A cols.B n e ≤ lpMeas ℂ ℂ (MeasurableSpace.comap R.b.unitary inferInstance) 2 (cellLaw M rows.B cols.P n e) := by filter_upwards [grouped_rectangle_spaces M hM rows cols n, grouped_restricted_fixed_squares M hM rows cols n, grouped_active_intersections M hM rows cols n, grouped_row_contraction_projection M hM rows.swapAB.swapPA cols n, rowActive_eq_range M hM rows.B rows.properB cols n, cells_fibre_marginal M hM rows.B cols.P rows.properB n, cells_fibre_marginal M hM rows.B cols.A rows.properB n, cells_fibre_marginal M hM rows.B cols.B rows.properB n, cells_indep_columns_fibre M rows.B rows.B cols.A cols.B cols.disjointAB cols.properAB n, rowPair_fixed_conditional M rows.B cols.P cols.A cols.B n, rowPair_fixed_marginals M hM rows.B cols.P cols.A cols.B rows.properB cols.disjointPA cols.disjointPB cols.properPA cols.properPB n] with e hspaces hs hA hproj hrange hx hc hd hcd hκ hm intro R rw [hx.map_eq] at hm let hpair : MeasurePreserving (fun y => (cells M rows.B cols.A n y,cells M rows.B cols.B n y)) (M.ownKernel (n+2) e) ((cellLaw M rows.B cols.A n e).prod (cellLaw M rows.B cols.B n e)) := ⟨hc.measurable.prodMk hd.measurable, by rw [←hc.map_eq,←hd.map_eq] exact hcd.map_prod_eq_prod_map_map hc.measurable.aemeasurable hd.measurable.aemeasurable⟩ let V := rectangleSpace₄ (cellLaw M rows.B cols.P n e) (cellLaw M rows.P cols.B n e) (fibreRowEdge M rows.B cols.P cols.A cols.B n e) (fibreColumnEdge M rows.P rows.A rows.B cols.B n e) let W := rectangleSpace₃ (cellLaw M rows.B cols.P n e) (cellLaw M rows.P cols.A n e) (fibreRowEdge M rows.B cols.P cols.A cols.B n e) (fibreColumnEdge M rows.P rows.A rows.B cols.A n e) have hv : V = commonActive M rows.B rows.P rows.A cols.B cols.P cols.A n e := hspaces.2.2.2 have hw : W = commonActive M rows.B rows.P rows.A cols.A cols.P cols.B n e := hspaces.2.2.1 have ha : ∀ᵐ a ∂cellLaw M rows.B cols.P n e, W.starProjection ∘L JoiningOperator.family ((rowPairKernel M rows.B cols.P cols.A cols.B n).sectR e) (cellLaw M rows.B cols.A n e) (cellLaw M rows.B cols.B n e) a = JoiningOperator.family ((rowPairKernel M rows.B cols.P cols.A cols.B n).sectR e) (cellLaw M rows.B cols.A n e) (cellLaw M rows.B cols.B n e) a ∘L V.starProjection := by simpa only [hv,hw,fibreRowEdge_eq] using hA.2.1 apply JoiningOperator.common_space_le_polarField hx hpair hκ hm V W ha R.b hs.2.1 _ (commonActive M rows.B rows.P rows.A cols.A cols.P cols.B n e).starProjection (commonActive M rows.B rows.P rows.A cols.B cols.P cols.A n e).starProjection · rw [←hrange hx hpair] exact inf_le_left · exact hproj hx hpair · intro h rw [hv] exact Submodule.starProjection_apply_mem _ h end ArrayObjects.Object.Rolled end section open MeasureTheory open scoped ENNReal variable {Ω A B : Type*} [MeasurableSpace Ω] [MeasurableSpace A] [MeasurableSpace B] {μ : Measure Ω} {a a' : Ω → A} {b : Ω → B} lemma lpMeas_comap_congr (he : a =ᵐ[μ] a') : lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 μ = lpMeas ℂ ℂ (MeasurableSpace.comap a' inferInstance) 2 μ := by suffices h : ∀ (a a' : Ω → A), a =ᵐ[μ] a' → lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 μ ≤ lpMeas ℂ ℂ (MeasurableSpace.comap a' inferInstance) 2 μ from le_antisymm (h a a' he) (h a' a he.symm) intro a a' hae f hf have hm := mem_lpMeas_iff_aestronglyMeasurable.mp hf obtain ⟨g,hg,hgcomp⟩ := hm.stronglyMeasurable_mk.exists_eq_measurable_comp apply mem_lpMeas_iff_aestronglyMeasurable.mpr have hstrong : StronglyMeasurable[MeasurableSpace.comap a' inferInstance] (g ∘ a') := hg.comp_measurable (Measurable.of_comap_le le_rfl) apply hstrong.aestronglyMeasurable.congr filter_upwards [hm.ae_eq_mk,hae] with ω hfω haω exact (congrArg g haω).symm.trans ((congrFun hgcomp ω).symm.trans hfω.symm) lemma lpMeas_comap_mono (hab : MeasurableSpace.comap a inferInstance ≤ MeasurableSpace.comap b inferInstance) : lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 μ ≤ lpMeas ℂ ℂ (MeasurableSpace.comap b inferInstance) 2 μ := by intro f hf apply mem_lpMeas_iff_aestronglyMeasurable.mpr exact (mem_lpMeas_iff_aestronglyMeasurable.mp hf).mono hab end section open MeasureTheory lemma comap_eq_of_measurable_leftInverse {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] (a : X → Y) (f : Y → Z) (g : Z → Y) (hf : Measurable f) (hg : Measurable g) (hgf : Function.LeftInverse g f) : MeasurableSpace.comap (f ∘ a) inferInstance = MeasurableSpace.comap a inferInstance := by apply le_antisymm · rw [←MeasurableSpace.comap_comp] exact MeasurableSpace.comap_mono hf.comap_le · have he : a = g ∘ (f ∘ a) := funext fun x => (hgf (a x)).symm conv_lhs => rw [he,←MeasurableSpace.comap_comp] exact MeasurableSpace.comap_mono hg.comap_le namespace StrongUnitary variable {X H₁ H₂ H₃ H₄ : Type*} [MeasurableSpace X] [NormedAddCommGroup H₁] [NormedSpace ℂ H₁] [SecondCountableTopology H₁] [NormedAddCommGroup H₂] [NormedSpace ℂ H₂] [SecondCountableTopology H₂] [NormedAddCommGroup H₃] [NormedSpace ℂ H₃] [SecondCountableTopology H₃] [NormedAddCommGroup H₄] [NormedSpace ℂ H₄] [SecondCountableTopology H₄] omit [SecondCountableTopology H₁] in lemma comap_normalizeA (a₀ : H₂ ≃ₗᵢ[ℂ] H₁) (d₀ : H₄ ≃ₗᵢ[ℂ] H₂) (a : X → H₂ ≃ₗᵢ[ℂ] H₁) : MeasurableSpace.comap (normalizeA a₀ d₀ ∘ a) inferInstance = MeasurableSpace.comap a inferInstance := by let g (u : H₄ ≃ₗᵢ[ℂ] H₄) := d₀.symm.trans (u.trans (d₀.trans a₀)) have hg : Continuous g := by apply continuous_iff.mpr intro v exact (d₀.trans a₀).continuous.comp (continuous_apply (d₀.symm v)) apply comap_eq_of_measurable_leftInverse a _ g (continuous_normalizeA a₀ d₀).measurable hg.measurable intro u ext v simp [g,normalizeA] omit [SecondCountableTopology H₃] in lemma comap_normalizeB (b₀ : H₄ ≃ₗᵢ[ℂ] H₃) (b : X → H₄ ≃ₗᵢ[ℂ] H₃) : MeasurableSpace.comap (normalizeB b₀ ∘ b) inferInstance = MeasurableSpace.comap b inferInstance := by let g (u : H₄ ≃ₗᵢ[ℂ] H₄) := u.trans b₀ have hg : Continuous g := by apply continuous_iff.mpr intro v exact b₀.continuous.comp (continuous_apply v) apply comap_eq_of_measurable_leftInverse b _ g (continuous_normalizeB b₀).measurable hg.measurable intro u ext v simp [g,normalizeB] end StrongUnitary end namespace StrongUnitary open MeasureTheory section MeasurableEigenvalues variable {H I X : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [SecondCountableTopology H] [MeasurableSpace X] lemma measurable_of_eigenvalues (b : HilbertBasis I ℂ H) (U : X → H ≃ₗᵢ[ℂ] H) (χ : I → X → ℂ) (hχ : ∀ i, Measurable (χ i)) (he : ∀ i x, U x (b i) = χ i x • b i) : Measurable U := by borelize H let : Countable I := hilbertBasis_countable b apply measurable_iff.mpr intro v have hes (x : X) : (∑' i, (b.repr v i * χ i x) • b i) = U x v := by have hs := (U x : H →L[ℂ] H).hasSum (b.hasSum_repr v) change HasSum (fun i => U x (b.repr v i • b i)) (U x v) at hs simpa only [map_smul, he, smul_smul] using hs.tsum_eq have hm : Measurable (fun x => ∑' i, (b.repr v i * χ i x) • b i) := Measurable.tsum (fun i => (measurable_const.mul (hχ i)).smul measurable_const) simpa only [hes] using hm.stronglyMeasurable end MeasurableEigenvalues section variable {H I X : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [SecondCountableTopology H] local instance : MeasurableSpace Circle := inferInstanceAs (MeasurableSpace (Submonoid.unitSphere ℂ)) local instance : BorelSpace Circle := inferInstanceAs (BorelSpace (Submonoid.unitSphere ℂ)) lemma measurable_iff_eigenCoords [MeasurableSpace X] (b : HilbertBasis I ℂ H) (U : X → H ≃ₗᵢ[ℂ] H) (γ : X → I → Circle) (he : ∀ i x, U x (b i) = (γ x i : ℂ) • b i) : Measurable U ↔ Measurable γ := by borelize H constructor · intro hU apply Measurable.of_eval intro i apply Measurable.subtype_mk have hm : Measurable (fun x => inner ℂ (b i) (U x (b i))) := (measurable_const (a := b i)).inner (𝕜 := ℂ) ((continuous_apply (b i)).measurable.comp hU) have heq (x : X) : (γ x i : ℂ) = inner ℂ (b i) (U x (b i)) := by rw [he, inner_smul_right] simp [inner_self_eq_norm_sq_to_K, b.orthonormal.1 i] convert hm using 1 ext x exact heq x · intro hγ apply measurable_of_eigenvalues b U (fun i x => (γ x i : ℂ)) _ he intro i exact measurable_subtype_coe.comp ((measurable_pi_apply i).comp hγ) lemma comap_eigenCoords (b : HilbertBasis I ℂ H) (U : X → H ≃ₗᵢ[ℂ] H) (γ : X → I → Circle) (he : ∀ i x, U x (b i) = (γ x i : ℂ) • b i) : MeasurableSpace.comap U inferInstance = MeasurableSpace.comap γ inferInstance := by apply le_antisymm · let : MeasurableSpace X := MeasurableSpace.comap γ inferInstance have hm : Measurable γ := Measurable.of_comap_le le_rfl exact ((measurable_iff_eigenCoords b U γ he).mpr hm).comap_le · let : MeasurableSpace X := MeasurableSpace.comap U inferInstance have hm : Measurable U := Measurable.of_comap_le le_rfl exact ((measurable_iff_eigenCoords b U γ he).mp hm).comap_le end variable {H J X : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [SecondCountableTopology H] local instance : MeasurableSpace Circle := inferInstanceAs (MeasurableSpace (Submonoid.unitSphere ℂ)) local instance : BorelSpace Circle := inferInstanceAs (BorelSpace (Submonoid.unitSphere ℂ)) def subgroupVersion (S : Subgroup (H ≃ₗᵢ[ℂ] H)) (g : H ≃ₗᵢ[ℂ] H) : S := by classical exact if h : g ∈ S then ⟨g,h⟩ else 1 omit [CompleteSpace H] in lemma measurable_subgroupVersion (S : Subgroup (H ≃ₗᵢ[ℂ] H)) (hS : IsClosed (S : Set (H ≃ₗᵢ[ℂ] H))) : Measurable (subgroupVersion S) := by classical exact Measurable.dite measurable_id measurable_const hS.measurableSet omit [CompleteSpace H] [SecondCountableTopology H] in lemma subgroupVersion_coe (S : Subgroup (H ≃ₗᵢ[ℂ] H)) {g : H ≃ₗᵢ[ℂ] H} (hg : g ∈ S) : (subgroupVersion S g : H ≃ₗᵢ[ℂ] H) = g := by simp [subgroupVersion,hg] def spectralCoords (S : Subgroup (H ≃ₗᵢ[ℂ] H)) (γ : J → S →* Circle) (g : H ≃ₗᵢ[ℂ] H) : J → Circle := fun j => γ j (subgroupVersion S g) omit [CompleteSpace H] in lemma measurable_spectralCoords (S : Subgroup (H ≃ₗᵢ[ℂ] H)) (hS : IsClosed (S : Set (H ≃ₗᵢ[ℂ] H))) (γ : J → S →* Circle) (hγ : ∀ j, Continuous (γ j)) : Measurable (spectralCoords S γ) := Measurable.of_eval fun j => (hγ j).measurable.comp (measurable_subgroupVersion S hS) omit [CompleteSpace H] [SecondCountableTopology H] in lemma spectralCoords_mul_inv (S : Subgroup (H ≃ₗᵢ[ℂ] H)) (γ : J → S →* Circle) {g h : H ≃ₗᵢ[ℂ] H} (hg : g ∈ S) (hh : h ∈ S) : spectralCoords S γ (g*h⁻¹) = spectralCoords S γ g * (spectralCoords S γ h)⁻¹ := by have hs : subgroupVersion S (g*h⁻¹) = subgroupVersion S g * (subgroupVersion S h)⁻¹ := by apply Subtype.ext simp only [Subgroup.coe_mul,Subgroup.coe_inv,subgroupVersion_coe S hg, subgroupVersion_coe S hh,subgroupVersion_coe S (S.mul_mem hg (S.inv_mem hh))] funext j simp only [spectralCoords,hs,map_mul,map_inv,Pi.mul_apply,Pi.inv_apply] lemma lpMeas_spectralCoords [MeasurableSpace X] (μ : Measure X) (S : Subgroup (H ≃ₗᵢ[ℂ] H)) (b : HilbertBasis J ℂ H) (γ : J → S →* Circle) (heig : ∀ j (g : S), (g : H ≃ₗᵢ[ℂ] H) (b j) = (γ j g : ℂ) • b j) (U : X → H ≃ₗᵢ[ℂ] H) (hU : ∀ᵐ x ∂μ, U x ∈ S) : lpMeas ℂ ℂ (MeasurableSpace.comap U inferInstance) 2 μ = lpMeas ℂ ℂ (MeasurableSpace.comap (spectralCoords S γ ∘ U) inferInstance) 2 μ := by let V (x : X) : H ≃ₗᵢ[ℂ] H := subgroupVersion S (U x) have huv : U =ᵐ[μ] V := hU.mono fun x hx => (subgroupVersion_coe S hx).symm rw [lpMeas_comap_congr huv] rw [comap_eigenCoords b V (spectralCoords S γ ∘ U) (fun j x => heig j (subgroupVersion S (U x)))] rfl end StrongUnitary section open MeasureTheory ProbabilityTheory lemma exists_const_of_ae_prod_eq {X Y V : Type*} [MeasurableSpace X] [MeasurableSpace Y] {μ : Measure X} {ν : Measure Y} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] {f : X → V} {g : Y → V} (h : ∀ᵐ p ∂μ.prod ν, f p.1 = g p.2) : ∃ v, f =ᵐ[μ] (fun _ => v) ∧ g =ᵐ[ν] (fun _ => v) := by have hh : ∀ᵐ x ∂μ, ∀ᵐ y ∂ν, f x = g y := Measure.ae_ae_of_ae_prod h obtain ⟨x₀,hx₀⟩ := hh.exists refine ⟨f x₀, ?_, hx₀.mono (fun _ hy => hy.symm)⟩ filter_upwards [hh] with x hx obtain ⟨y,hy,h0y⟩ := (hx.and hx₀).exists exact hy.trans h0y.symm lemma measurable_conull_subset {Z : Type*} [MeasurableSpace Z] {σ : Measure Z} {p : Z → Prop} (h : ∀ᵐ z ∂σ, p z) : ∃ s : Set Z, MeasurableSet s ∧ (∀ᵐ z ∂σ, z ∈ s) ∧ ∀ z ∈ s, p z := by obtain ⟨t,ht,htm,ht0⟩ := exists_measurable_superset_of_null (ae_iff.mp h) refine ⟨tᶜ, htm.compl, ?_, ?_⟩ · apply mem_ae_iff.mpr change σ (tᶜ)ᶜ = 0 simpa only [compl_compl] using ht0 · intro z hz by_contra hp exact hz (ht hp) lemma exists_measurable_common_unitary {Z X Y H : Type*} [MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] [NormedAddCommGroup H] [NormedSpace ℂ H] [SecondCountableTopology H] [CompleteSpace H] {σ : Measure Z} (κ : Kernel Z X) (ξ : Kernel Z Y) [IsMarkovKernel κ] [IsMarkovKernel ξ] (f : X → H ≃ₗᵢ[ℂ] H) (g : Y → H ≃ₗᵢ[ℂ] H) (hf : Measurable f) (heq : ∀ᵐ z ∂σ, ∀ᵐ p ∂(κ z).prod (ξ z), f p.1 = g p.2) : ∃ U : Z → H ≃ₗᵢ[ℂ] H, Measurable U ∧ ∀ᵐ z ∂σ, f =ᵐ[κ z] (fun _ => U z) ∧ g =ᵐ[ξ z] (fun _ => U z) := by classical have he : ∀ᵐ z ∂σ, ∃ U, f =ᵐ[κ z] (fun _ => U) ∧ g =ᵐ[ξ z] (fun _ => U) := heq.mono fun _ hz => exists_const_of_ae_prod_eq hz obtain ⟨s,hsm,hs,hsprop⟩ := measurable_conull_subset he choose U₀ hU₀ using fun z : s => hsprop z z.property let U : Z → H ≃ₗᵢ[ℂ] H := fun z => if hz : z ∈ s then U₀ ⟨z,hz⟩ else 1 refine ⟨U, ?_, ?_⟩ · apply StrongUnitary.measurable_iff.mpr intro v have hev : (fun z => U z v) = s.piecewise (fun z => ∫ x, f x v ∂κ z) (fun _ => v) := by funext z by_cases hz : z ∈ s · simp only [U, dite_eq_left hz, Set.piecewise_eq_of_mem s _ _ hz] calc U₀ ⟨z,hz⟩ v = ∫ _x, U₀ ⟨z,hz⟩ v ∂κ z := by simp _ = ∫ x, f x v ∂κ z := integral_congr_ae ((hU₀ ⟨z,hz⟩).1.fun_comp (fun A => A v)).symm · simp [U, hz] rw [hev] exact ((StrongUnitary.measurable_iff.mp hf v).integral_kernel).piecewise hsm stronglyMeasurable_const · filter_upwards [hs] with z hz simpa only [U,dite_eq_left hz] using hU₀ ⟨z,hz⟩ namespace StrongUnitary lemma rectangle_ratio_eq {G : Type*} [Group G] {a b c d : G} (hbd : Commute b d) (h : a * d = c * b) : a * b⁻¹ = c * d⁻¹ := by apply mul_inv_eq_iff_eq_mul.mpr calc a = c * b * d⁻¹ := by rw [←h, mul_assoc, mul_inv_cancel, mul_one] _ = (c * d⁻¹) * b := by rw [mul_assoc, hbd.inv_right.eq, ←mul_assoc] variable {Z H : Type*} [MeasurableSpace Z] [NormedAddCommGroup H] [NormedSpace ℂ H] [CompleteSpace H] [SecondCountableTopology H] lemma measurable_corner_ratio (σ : Measure Z) (κ ξ : Kernel Z ((H ≃ₗᵢ[ℂ] H) × (H ≃ₗᵢ[ℂ] H))) [IsMarkovKernel κ] [IsMarkovKernel ξ] (ρ τ : ProbabilityMeasure (H ≃ₗᵢ[ℂ] H)) (G : Subgroup (H ≃ₗᵢ[ℂ] H)) (hρ : (ρ : Measure _).support = (G : Set (H ≃ₗᵢ[ℂ] H))) (hτ : (τ : Measure _).support = (G : Set (H ≃ₗᵢ[ℂ] H))) (hcomm : ∀ u ∈ G, ∀ v ∈ G, u * v = v * u) (hm : ∀ᵐ z ∂σ, (κ z).map Prod.fst = ρ ∧ (κ z).map Prod.snd = ρ ∧ (ξ z).map Prod.fst = τ ∧ (ξ z).map Prod.snd = τ) (hr : ∀ᵐ z ∂σ, ∀ᵐ p ∂(κ z).prod (ξ z), p.1.1*p.2.2 = p.2.1*p.1.2) : ∃ U : Z → H ≃ₗᵢ[ℂ] H, Measurable U ∧ ∀ᵐ z ∂σ, U z ∈ G ∧ (∀ᵐ p ∂κ z, p.1 * p.2⁻¹ = U z) ∧ (∀ᵐ p ∂ξ z, p.1 * p.2⁻¹ = U z) := by have hρm : ∀ᵐ u ∂(ρ : Measure _), u ∈ G := by change (G : Set (H ≃ₗᵢ[ℂ] H)) ∈ ae (ρ : Measure _) rw [←hρ] exact Measure.support_mem_ae have hτm : ∀ᵐ u ∂(τ : Measure _), u ∈ G := by change (G : Set (H ≃ₗᵢ[ℂ] H)) ∈ ae (τ : Measure _) rw [←hτ] exact Measure.support_mem_ae have hpairs : ∀ᵐ z ∂σ, (∀ᵐ p ∂κ z, p.1 ∈ G ∧ p.2 ∈ G) ∧ (∀ᵐ p ∂ξ z, p.1 ∈ G ∧ p.2 ∈ G) := by filter_upwards [hm] with z hz constructor · have ha := ae_of_ae_map measurable_fst.aemeasurable (hz.1 ▸ hρm) have hb := ae_of_ae_map measurable_snd.aemeasurable (hz.2.1 ▸ hρm) exact ha.and hb · have hc := ae_of_ae_map measurable_fst.aemeasurable (hz.2.2.1 ▸ hτm) have hd := ae_of_ae_map measurable_snd.aemeasurable (hz.2.2.2 ▸ hτm) exact hc.and hd have he : ∀ᵐ z ∂σ, ∀ᵐ p ∂(κ z).prod (ξ z), p.1.1 * p.1.2⁻¹ = p.2.1 * p.2.2⁻¹ := by filter_upwards [hpairs,hr] with z hz hzr filter_upwards [Measure.quasiMeasurePreserving_fst.ae hz.1, Measure.quasiMeasurePreserving_snd.ae hz.2,hzr] with p hp hq hrect exact rectangle_ratio_eq (hcomm _ hp.2 _ hq.2) hrect obtain ⟨U,hU,heU⟩ := exists_measurable_common_unitary κ ξ (fun p => p.1 * p.2⁻¹) (fun p => p.1 * p.2⁻¹) (measurable_fst.mul measurable_snd.inv) he refine ⟨U,hU,?_⟩ filter_upwards [hpairs,heU] with z hz hzu obtain ⟨p,hp,heq⟩ := (hz.1.and hzu.1).exists change p.1 * p.2⁻¹ = U z at heq exact ⟨heq ▸ G.mul_mem hp.1 (G.inv_mem hp.2), hzu⟩ end StrongUnitary namespace ConditionalFibres lemma ae_of_ae_kernel {Ω Z V : Type*} [MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace V] {μ : Measure Ω} {z : Ω → Z} {v : Ω → V} {κ : Kernel Z V} (h : HasCondDistrib v z κ μ) {p : Z × V → Prop} (hp : MeasurableSet {q | p q}) (hv : ∀ᵐ a ∂μ.map z, ∀ᵐ b ∂κ a, p (a,b)) : ∀ᵐ ω ∂μ, p (z ω,v ω) := by have hh := Measure.ae_compProd_of_ae_ae hp hv rw [←h.map_eq] at hh exact ae_of_ae_map h.aemeasurable hh end ConditionalFibres namespace StrongUnitary local instance : MeasurableSpace Circle := inferInstanceAs (MeasurableSpace (Submonoid.unitSphere ℂ)) local instance : BorelSpace Circle := inferInstanceAs (BorelSpace (Submonoid.unitSphere ℂ)) universe u variable {Ω Z X Y C D : Type*} {H : Type u} [MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace C] [MeasurableSpace D] [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [SecondCountableTopology H] {μ : Measure Ω} {ν : Measure X} {η : Measure Y} {ξC : Measure C} {ξD : Measure D} {z : Ω → Z} {x : Ω → X} {y : Ω → Y} {c : Ω → C} {d : Ω → D} lemma spectral_corner (G : Subgroup (H ≃ₗᵢ[ℂ] H)) (hG : IsClosed (G : Set (H ≃ₗᵢ[ℂ] H))) (ρ τ : ProbabilityMeasure (H ≃ₗᵢ[ℂ] H)) (hρ : (ρ : Measure _).support = (G : Set (H ≃ₗᵢ[ℂ] H))) (hτ : (τ : Measure _).support = (G : Set (H ≃ₗᵢ[ℂ] H))) (hcomm : ∀ g ∈ G, ∀ h ∈ G, g*h = h*g) (hinv : ∀ g ∈ G, (ρ : Measure _).map (g * ·) = ρ) (a : X → H ≃ₗᵢ[ℂ] H) (b : Y → H ≃ₗᵢ[ℂ] H) (f : C → H ≃ₗᵢ[ℂ] H) (g : D → H ≃ₗᵢ[ℂ] H) (ha : MeasurePreserving a ν ρ) (hb : MeasurePreserving b η ρ) (hf : MeasurePreserving f ξC τ) (hg : MeasurePreserving g ξD τ) (κ ξ : Kernel Z ((H ≃ₗᵢ[ℂ] H) × (H ≃ₗᵢ[ℂ] H))) [IsMarkovKernel κ] [IsMarkovKernel ξ] (hκ : HasCondDistrib (fun ω => (a (x ω),b (y ω))) z κ μ) (hξ : HasCondDistrib (fun ω => (f (c ω),g (d ω))) z ξ μ) (hm : ∀ᵐ t ∂μ.map z, (κ t).map Prod.fst = ρ ∧ (κ t).map Prod.snd = ρ ∧ (ξ t).map Prod.fst = τ ∧ (ξ t).map Prod.snd = τ) (hr : ∀ᵐ t ∂μ.map z, ∀ᵐ p ∂(κ t).prod (ξ t), p.1.1*p.2.2 = p.2.1*p.1.2) : ∃ (J : Type u) (_ : Countable J) (A : X → J → Circle) (B : Y → J → Circle) (F : C → J → Circle) (Q : D → J → Circle) (K : Z → J → Circle), Measurable A ∧ Measurable B ∧ Measurable F ∧ Measurable Q ∧ Measurable K ∧ lpMeas ℂ ℂ (MeasurableSpace.comap a inferInstance) 2 ν = lpMeas ℂ ℂ (MeasurableSpace.comap A inferInstance) 2 ν ∧ lpMeas ℂ ℂ (MeasurableSpace.comap b inferInstance) 2 η = lpMeas ℂ ℂ (MeasurableSpace.comap B inferInstance) 2 η ∧ (∀ᵐ ω ∂μ, K (z ω) = A (x ω) * (B (y ω))⁻¹) ∧ (∀ᵐ ω ∂μ, K (z ω) = F (c ω) * (Q (d ω))⁻¹) := by obtain ⟨U,hU,hUr⟩ := measurable_corner_ratio (μ.map z) κ ξ ρ τ G hρ hτ hcomm hm hr obtain ⟨J,hJ,e,γ,hγ,he⟩ := polar_spectral G hG ρ hρ hcomm hinv let sc := spectralCoords G γ have hsc : Measurable sc := measurable_spectralCoords G hG γ hγ have hgρ : ∀ᵐ v ∂(ρ : Measure _), v ∈ G := by change (G : Set (H ≃ₗᵢ[ℂ] H)) ∈ ae (ρ : Measure _) rw [←hρ] exact Measure.support_mem_ae have hgτ : ∀ᵐ v ∂(τ : Measure _), v ∈ G := by change (G : Set (H ≃ₗᵢ[ℂ] H)) ∈ ae (τ : Measure _) rw [←hτ] exact Measure.support_mem_ae have hmκ : ∀ᵐ t ∂μ.map z, ∀ᵐ p ∂κ t, p.1 ∈ G ∧ p.2 ∈ G := by filter_upwards [hm] with t ht exact (ae_of_ae_map measurable_fst.aemeasurable (ht.1 ▸ hgρ)).and (ae_of_ae_map measurable_snd.aemeasurable (ht.2.1 ▸ hgρ)) have hmξ : ∀ᵐ t ∂μ.map z, ∀ᵐ p ∂ξ t, p.1 ∈ G ∧ p.2 ∈ G := by filter_upwards [hm] with t ht exact (ae_of_ae_map measurable_fst.aemeasurable (ht.2.2.1 ▸ hgτ)).and (ae_of_ae_map measurable_snd.aemeasurable (ht.2.2.2 ▸ hgτ)) refine ⟨J,hJ,sc ∘ a,sc ∘ b,sc ∘ f,sc ∘ g,sc ∘ U, hsc.comp ha.measurable,hsc.comp hb.measurable,hsc.comp hf.measurable, hsc.comp hg.measurable,hsc.comp hU,?_,?_,?_,?_⟩ · exact lpMeas_spectralCoords ν G e γ he a (ha.quasiMeasurePreserving.ae hgρ) · exact lpMeas_spectralCoords η G e γ he b (hb.quasiMeasurePreserving.ae hgρ) · have hh : ∀ᵐ t ∂μ.map z, ∀ᵐ p ∂κ t, sc (U t) = sc p.1 * (sc p.2)⁻¹ := by filter_upwards [hUr,hmκ] with t ht hm filter_upwards [ht.2.1,hm] with p hp hmem rw [←hp] exact spectralCoords_mul_inv G γ hmem.1 hmem.2 exact ConditionalFibres.ae_of_ae_kernel hκ (measurableSet_eq_fun ((hsc.comp hU).comp measurable_fst) ((hsc.comp measurable_snd.fst).mul (hsc.comp measurable_snd.snd).inv)) hh · have hh : ∀ᵐ t ∂μ.map z, ∀ᵐ p ∂ξ t, sc (U t) = sc p.1 * (sc p.2)⁻¹ := by filter_upwards [hUr,hmξ] with t ht hm filter_upwards [ht.2.2,hm] with p hp hmem rw [←hp] exact spectralCoords_mul_inv G γ hmem.1 hmem.2 exact ConditionalFibres.ae_of_ae_kernel hξ (measurableSet_eq_fun ((hsc.comp hU).comp measurable_fst) ((hsc.comp measurable_snd.fst).mul (hsc.comp measurable_snd.snd).inv)) hh end StrongUnitary end section open MeasureTheory ProbabilityTheory ContinuousLinearMap local instance : MeasurableSpace Circle := inferInstanceAs (MeasurableSpace (Submonoid.unitSphere ℂ)) local instance : BorelSpace Circle := inferInstanceAs (BorelSpace (Submonoid.unitSphere ℂ)) universe u variable {Ω Z X Y C D H₁ H₂ H₃ : Type*} {H₄ : Type u} [MeasurableSpace Ω] [MeasurableSpace Z] [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace C] [MeasurableSpace D] [NormedAddCommGroup H₁] [InnerProductSpace ℂ H₁] [CompleteSpace H₁] [SecondCountableTopology H₁] [NormedAddCommGroup H₂] [InnerProductSpace ℂ H₂] [CompleteSpace H₂] [SecondCountableTopology H₂] [NormedAddCommGroup H₃] [InnerProductSpace ℂ H₃] [CompleteSpace H₃] [SecondCountableTopology H₃] [NormedAddCommGroup H₄] [InnerProductSpace ℂ H₄] [CompleteSpace H₄] [SecondCountableTopology H₄] {μ : Measure Ω} [IsProbabilityMeasure μ] {μX : Measure X} {μY : Measure Y} {μC : Measure C} {μD : Measure D} [IsProbabilityMeasure μX] [IsProbabilityMeasure μY] [IsProbabilityMeasure μC] [IsProbabilityMeasure μD] {x : Ω → X} {y : Ω → Y} {c : Ω → C} {d : Ω → D} {F : X → H₂ →L[ℂ] H₁} {G : Y → H₄ →L[ℂ] H₃} {K : C → H₃ →L[ℂ] H₁} {N : D → H₄ →L[ℂ] H₂} omit [SecondCountableTopology H₁] [IsProbabilityMeasure μY] [IsProbabilityMeasure μD] in lemma MeasurablePolarRectangle.torus_coordinates (R : MeasurablePolarRectangle μ μX μY μC μD F G K N x y c d) (hR : R.HasHaarNormalization) {z : Ω → Z} (κ : Kernel Z (X × Y)) (ξ : Kernel Z (C × D)) [IsMarkovKernel κ] [IsMarkovKernel ξ] (hκ : HasCondDistrib (fun ω => (x ω,y ω)) z κ μ) (hξ : HasCondDistrib (fun ω => (c ω,d ω)) z ξ μ) (hpair : HasCondDistrib (fun ω => ((x ω,y ω),(c ω,d ω))) z (κ ×ₖ ξ) μ) (hκm : ∀ᵐ v ∂μ.map z, (κ v).map Prod.fst = μX ∧ (κ v).map Prod.snd = μY) (hξm : ∀ᵐ v ∂μ.map z, (ξ v).map Prod.fst = μC ∧ (ξ v).map Prod.snd = μD) : ∃ (J : Type u) (_ : Countable J) (A : X → J → Circle) (B : Y → J → Circle) (P : C → J → Circle) (Q : D → J → Circle) (H : Z → J → Circle), Measurable A ∧ Measurable B ∧ Measurable P ∧ Measurable Q ∧ Measurable H ∧ lpMeas ℂ ℂ (MeasurableSpace.comap R.a.unitary inferInstance) 2 μX = lpMeas ℂ ℂ (MeasurableSpace.comap A inferInstance) 2 μX ∧ lpMeas ℂ ℂ (MeasurableSpace.comap R.b.unitary inferInstance) 2 μY = lpMeas ℂ ℂ (MeasurableSpace.comap B inferInstance) 2 μY ∧ (∀ᵐ ω ∂μ, H (z ω) = A (x ω) * (B (y ω))⁻¹) ∧ (∀ᵐ ω ∂μ, H (z ω) = P (c ω) * (Q (d ω))⁻¹) := by obtain ⟨a₀,b₀,d₀,S,hS,hAB,hCD,hρ,hτ,hcomm,hinv⟩ := hR let na := StrongUnitary.normalizeA a₀ d₀ let nb := StrongUnitary.normalizeB b₀ let nc := StrongUnitary.normalizeC a₀ b₀ d₀ let nd := StrongUnitary.normalizeD d₀ have hna : Measurable na := (StrongUnitary.continuous_normalizeA a₀ d₀).measurable have hnb : Measurable nb := (StrongUnitary.continuous_normalizeB b₀).measurable have hnc : Measurable nc := (StrongUnitary.continuous_normalizeC a₀ b₀ d₀).measurable have hnd : Measurable nd := (StrongUnitary.continuous_normalizeD d₀).measurable let aa := na ∘ R.a.unitary let bb := nb ∘ R.b.unitary let cc := nc ∘ R.p.unitary let dd := nd ∘ R.q.unitary have ha : Measurable aa := hna.comp R.a.measurable have hb : Measurable bb := hnb.comp R.b.measurable have hc : Measurable cc := hnc.comp R.p.measurable have hd : Measurable dd := hnd.comp R.q.measurable let ρ : ProbabilityMeasure (H₄ ≃ₗᵢ[ℂ] H₄) := ⟨μX.map aa,(fun mapMeasurable => (Measure.isProbabilityMeasure_map_iff mapMeasurable).mpr inferInstance) ha.aemeasurable⟩ let τ : ProbabilityMeasure (H₄ ≃ₗᵢ[ℂ] H₄) := ⟨μC.map cc,(fun mapMeasurable => (Measure.isProbabilityMeasure_map_iff mapMeasurable).mpr inferInstance) hc.aemeasurable⟩ have hρeq : (μX.map R.a.unitary).map na = (ρ : Measure _) := Measure.map_map hna R.a.measurable have hτeq : (μC.map R.p.unitary).map nc = (τ : Measure _) := Measure.map_map hnc R.p.measurable change (μX.map R.a.unitary).map na = (μY.map R.b.unitary).map nb at hAB change (μC.map R.p.unitary).map nc = (μD.map R.q.unitary).map nd at hCD have hab : μY.map bb = (ρ : Measure _) := (Measure.map_map hnb R.b.measurable).symm.trans (hAB.symm.trans hρeq) have hcd : μD.map dd = (τ : Measure _) := (Measure.map_map hnd R.q.measurable).symm.trans (hCD.symm.trans hτeq) let α := κ.map (Prod.map aa bb) let β := ξ.map (Prod.map cc dd) let : IsMarkovKernel α := Kernel.IsMarkovKernel.map κ (ha.prodMap hb) let : IsMarkovKernel β := Kernel.IsMarkovKernel.map ξ (hc.prodMap hd) have hα : HasCondDistrib (fun ω => (aa (x ω),bb (y ω))) z α μ := hκ.comp_left (ha.prodMap hb) have hβ : HasCondDistrib (fun ω => (cc (c ω),dd (d ω))) z β μ := hξ.comp_left (hc.prodMap hd) have hm : ∀ᵐ v ∂μ.map z, (α v).map Prod.fst = ρ ∧ (α v).map Prod.snd = ρ ∧ (β v).map Prod.fst = τ ∧ (β v).map Prod.snd = τ := by filter_upwards [hκm,hξm] with v hv hw dsimp only [α,β] rw [kernel_map_pair_fst _ _ ha hb,kernel_map_pair_snd _ _ ha hb, kernel_map_pair_fst _ _ hc hd,kernel_map_pair_snd _ _ hc hd,hv.1,hv.2,hw.1,hw.2,hab,hcd] exact ⟨rfl,rfl,rfl,rfl⟩ have hr : ∀ᵐ v ∂μ.map z, ∀ᵐ w ∂(α v).prod (β v), w.1.1*w.2.2 = w.2.1*w.1.2 := by have hh := hpair.comp_left ((ha.prodMap hb).prodMap (hc.prodMap hd)) have he : (κ ×ₖ ξ).map (Prod.map (Prod.map aa bb) (Prod.map cc dd)) = α ×ₖ β := (Kernel.map_prod_map κ ξ (ha.prodMap hb) (hc.prodMap hd)).symm rw [he] at hh have hr := ConditionalFibres.ae_kernel hh (isClosed_eq (continuous_fst.fst.mul continuous_snd.snd) (continuous_snd.fst.mul continuous_fst.snd)).measurableSet (R.rectangle.mono (fun ω hω => StrongUnitary.normalized_rectangle a₀ (R.a.unitary (x ω)) b₀ (R.b.unitary (y ω)) (R.p.unitary (c ω)) d₀ (R.q.unitary (d ω)) hω)) simpa only [Kernel.prod_apply, Pi.mul_apply] using hr have hρs : (ρ : Measure _).support = (S : Set (H₄ ≃ₗᵢ[ℂ] H₄)) := by rw [←hρeq]; exact hρ have hτs : (τ : Measure _).support = (S : Set (H₄ ≃ₗᵢ[ℂ] H₄)) := by rw [←hτeq]; exact hτ have hρi : ∀ g ∈ S, (ρ : Measure _).map (g*·) = ρ := by rw [←hρeq]; exact hinv obtain ⟨J,hJ,A,B,P,Q,H,hA,hB,hP,hQ,hH,hfa,hfb,hra,hrb⟩ := StrongUnitary.spectral_corner S hS ρ τ hρs hτs hcomm hρi aa bb cc dd ⟨ha,rfl⟩ ⟨hb,hab⟩ ⟨hc,rfl⟩ ⟨hd,hcd⟩ α β hα hβ hm hr refine ⟨J,hJ,A,B,P,Q,H,hA,hB,hP,hQ,hH,?_,?_,hra,hrb⟩ · rw [←StrongUnitary.comap_normalizeA a₀ d₀ R.a.unitary] exact hfa · rw [←StrongUnitary.comap_normalizeB b₀ R.b.unitary] exact hfb end section open MeasureTheory variable {X G : Type*} [TopologicalSpace X] [CompactSpace X] [Group G] def characterAlgebra (χ : G →* C(X,ℂ)) (hs : ∀ g, star (χ g) = χ g⁻¹) : StarSubalgebra ℂ C(X,ℂ) where toSubalgebra := Algebra.adjoin ℂ (Set.range χ) star_mem' := by intro f hf induction hf using Algebra.adjoin_induction with | mem f hf => obtain ⟨g,rfl⟩ := hf rw [hs] exact Algebra.subset_adjoin ⟨g⁻¹,rfl⟩ | algebraMap c => exact (Algebra.adjoin ℂ (Set.range χ)).algebraMap_mem (star c) | add f g hf hg hfs hgs => rw [star_add] exact (Algebra.adjoin ℂ (Set.range χ)).add_mem hfs hgs | mul f g hf hg hfs hgs => rw [star_mul] exact (Algebra.adjoin ℂ (Set.range χ)).mul_mem hgs hfs lemma characterAlgebra_span (χ : G →* C(X,ℂ)) (hs : ∀ g, star (χ g) = χ g⁻¹) : (characterAlgebra χ hs).toSubmodule = Submodule.span ℂ (Set.range χ) := by change (Algebra.adjoin ℂ (Set.range χ)).toSubmodule = _ rw [Algebra.adjoin_eq_span] rw [←MonoidHom.coe_mrange χ, (MonoidHom.mrange χ).closure_eq] lemma dense_character_span (χ : G →* C(X,ℂ)) (hs : ∀ g, star (χ g) = χ g⁻¹) (hsep : ∀ x y : X, x ≠ y → ∃ g, χ g x ≠ χ g y) : (Submodule.span ℂ (Set.range χ)).topologicalClosure = ⊤ := by have hA : (characterAlgebra χ hs).SeparatesPoints := by intro x y hxy obtain ⟨g,hg⟩ := hsep x y hxy refine ⟨(χ g : X → ℂ), ⟨χ g, Algebra.subset_adjoin ⟨g,rfl⟩, rfl⟩, hg⟩ have h := ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints (characterAlgebra χ hs) hA have hh := congrArg (fun A : StarSubalgebra ℂ C(X,ℂ) => A.toSubalgebra.toSubmodule) h change (characterAlgebra χ hs).toSubmodule.topologicalClosure = ⊤ at hh rwa [characterAlgebra_span] at hh end section open MeasureTheory open scoped ENNReal variable {I : Type*} def torusWordCircle : FreeGroup I →* C((I → Circle), Circle) := FreeGroup.lift (fun i => ⟨fun x => x i, continuous_apply i⟩) def circleMapComplex {X : Type*} [TopologicalSpace X] : C(X,Circle) →* C(X,ℂ) where toFun f := ⟨fun x => (f x : ℂ), continuous_subtype_val.comp f.continuous⟩ map_one' := by ext x; rfl map_mul' f g := by ext x; rfl def torusWord : FreeGroup I →* C((I → Circle),ℂ) := circleMapComplex.comp torusWordCircle lemma torusWord_star (w : FreeGroup I) : star (torusWord w) = torusWord w⁻¹ := by ext x change star ((torusWordCircle w x : Circle) : ℂ) = (torusWordCircle w⁻¹ x : ℂ) rw [map_inv] exact (Circle.coe_inv_eq_conj (torusWordCircle w x)).symm @[simp] lemma torusWord_of (i : I) (x : I → Circle) : torusWord (FreeGroup.of i) x = (x i : ℂ) := by change (torusWordCircle (FreeGroup.of i) x : ℂ) = _ simp only [torusWordCircle, FreeGroup.lift_apply_of] rfl lemma torusWord_dense : (Submodule.span ℂ (Set.range (torusWord (I := I)))).topologicalClosure = ⊤ := by apply dense_character_span _ torusWord_star intro x y hxy obtain ⟨i,hi⟩ := Function.ne_iff.mp hxy refine ⟨FreeGroup.of i, ?_⟩ simpa only [torusWord_of] using (fun h => hi (Subtype.ext h)) variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedAddCommGroup F] [NormedSpace ℂ F] lemma dense_span_image (A : E →L[ℂ] F) (hA : DenseRange A) (s : Set E) (hs : (Submodule.span ℂ s).topologicalClosure = ⊤) : (Submodule.span ℂ (A '' s)).topologicalClosure = ⊤ := by let P := (Submodule.span ℂ (A '' s)).topologicalClosure have hspan : Submodule.span ℂ s ≤ P.comap A.toLinearMap := by apply Submodule.span_le.mpr intro x hx exact Submodule.le_topologicalClosure _ (Submodule.subset_span ⟨x,hx,rfl⟩) have htop : ⊤ ≤ P.comap A.toLinearMap := by rw [←hs] exact Submodule.topologicalClosure_minimal _ hspan ((Submodule.isClosed_topologicalClosure _).preimage A.continuous) apply top_unique intro x hx exact (Submodule.isClosed_topologicalClosure _).closure_subset_iff.mpr (Set.range_subset_iff.mpr (fun v => htop (show v ∈ (⊤ : Submodule ℂ E) from trivial))) (by rw [hA.closure_range]; trivial : x ∈ closure (Set.range A)) lemma dense_span_continuous_toLp {X J : Type*} [TopologicalSpace X] [CompactSpace X] [NormalSpace X] [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ] [μ.WeaklyRegular] (f : J → C(X,ℂ)) (hf : (Submodule.span ℂ (Set.range f)).topologicalClosure = ⊤) : (Submodule.span ℂ (Set.range (fun j => ContinuousMap.toLp 2 μ ℂ (f j)))).topologicalClosure = ⊤ := by have hD : DenseRange (ContinuousMap.toLp 2 μ ℂ : C(X,ℂ) →L[ℂ] Lp ℂ 2 μ) := ContinuousMap.toLp_denseRange ℂ μ ℂ (by norm_num : (2 : ENNReal) ≠ ∞) have h := dense_span_image (ContinuousMap.toLp 2 μ ℂ : C(X,ℂ) →L[ℂ] Lp ℂ 2 μ) hD (Set.range f) hf simpa only [←Set.range_comp, Function.comp_def] using h local instance : MeasurableSpace Circle := inferInstanceAs (MeasurableSpace (Submonoid.unitSphere ℂ)) local instance : BorelSpace Circle := inferInstanceAs (BorelSpace (Submonoid.unitSphere ℂ)) lemma torusWord_L2_dense [Countable I] (μ : Measure (I → Circle)) [IsFiniteMeasure μ] : (Submodule.span ℂ (Set.range (fun w => ContinuousMap.toLp 2 μ ℂ (torusWord (I := I) w)))).topologicalClosure = ⊤ := by exact dense_span_continuous_toLp μ (torusWord (I := I)) torusWord_dense end variable {I : Type*} lemma torusWordCircle_of (i : I) (x : I → Circle) : torusWordCircle (FreeGroup.of i) x = x i := by simp only [torusWordCircle, FreeGroup.lift_apply_of] rfl lemma torusWordCircle_mul (w : FreeGroup I) (x y : I → Circle) : torusWordCircle w (x*y) = torusWordCircle w x * torusWordCircle w y := by induction w using FreeGroup.induction_on with | one => simp only [map_one, ContinuousMap.one_apply, one_mul] | of i => simp only [torusWordCircle_of, Pi.mul_apply] | inv_of i hi => simp only [map_inv, ContinuousMap.inv_apply, hi, mul_inv_rev] exact mul_comm _ _ | mul a b ha hb => simp only [map_mul, ContinuousMap.mul_apply, ha, hb] ac_rfl lemma torusWordCircle_one (w : FreeGroup I) : torusWordCircle w (1 : I → Circle) = 1 := by have h := torusWordCircle_mul w (1 : I → Circle) 1 simp only [one_mul] at h exact mul_left_cancel (by simpa only [mul_one] using h.symm : torusWordCircle w 1 * torusWordCircle w 1 = torusWordCircle w 1 * 1) lemma torusWordCircle_inv (w : FreeGroup I) (x : I → Circle) : torusWordCircle w x⁻¹ = (torusWordCircle w x)⁻¹ := by symm apply inv_eq_of_mul_eq_one_left rw [←torusWordCircle_mul, inv_mul_cancel, torusWordCircle_one] def torusCharacter (w : FreeGroup I) : (I → Circle) →* Circle where toFun := torusWordCircle w map_one' := torusWordCircle_one w map_mul' := torusWordCircle_mul w lemma torusWord_ratio (w : FreeGroup I) (x y : I → Circle) : torusWord w (x*y⁻¹) = torusWord w x * star (torusWord w y) := by change (torusCharacter w (x*y⁻¹) : ℂ) = (torusCharacter w x : ℂ) * star (torusCharacter w y : ℂ) rw [map_mul, map_inv, Circle.coe_mul, Circle.coe_inv_eq_conj] rfl lemma torusWord_norm (w : FreeGroup I) (x : I → Circle) : ‖torusWord w x‖ = 1 := by change ‖(torusWordCircle w x : ℂ)‖ = 1 exact Circle.norm_coe _ end Rokhlin end end OAI