import OAI.Dynamics.StandardMap.EntropyEndpoint import OAI.Dynamics.StandardMap.Components.FineChartGraph namespace OAI section section namespace StandardMapEntropy open MeasureTheory Set Filter open scoped Topology theorem curved_holonomy_local_bound {S : Set ℝ} {G : ℝ → ℝ → ℝ} {u H : ℝ → ℝ} {κ ℓ K : ℝ} (hκ : 0≤κ) (hℓ : 0≤ℓ) (hK : 0≤K) (hcone : κ*ℓ<1) (hG : ∀ a∈S,∀ s t, |G a s-G a t|≤κ*|s-t|) (hu : ∀ s t,|u s-u t|≤ℓ*|s-t|) (hholo : ∀ x a,a∈S → ∀ᶠ b in 𝓝[S] a,|G b x-G a x|≤K*|b-a|) (hmatch : ∀ a∈S,H a=G a (u (H a))) : ∀ a∈S,∀ᶠ b in 𝓝[S] a,|H b-H a|≤(K/(1-κ*ℓ))*|b-a| := by intro a ha filter_upwards [hholo (u (H a)) a ha,self_mem_nhdsWithin] with b hb hbS by_cases he : H b=H a · rw [he,sub_self,abs_zero] exact mul_nonneg (div_nonneg hK (sub_pos.mpr hcone).le) (abs_nonneg _) have hm : |H b-H a|≤κ*ℓ*|H b-H a|+K*|b-a| := by calc |H b-H a| = |G b (u (H b))-G a (u (H a))| := by rw [←hmatch b hbS,←hmatch a ha] _ ≤ |G b (u (H b))-G b (u (H a))|+|G b (u (H a))-G a (u (H a))| := abs_sub_le _ _ _ _ ≤ κ*(ℓ*|H b-H a|)+K*|b-a| := add_le_add ((hG b hbS _ _).trans (by simpa only [mul_comm] using mul_le_mul (hu _ _) le_rfl hκ (mul_nonneg hℓ (abs_nonneg _)))) hb _ = κ*ℓ*|H b-H a|+K*|b-a| := by ring rw [div_mul_eq_mul_div] apply (le_div_iff₀ (sub_pos.mpr hcone)).mpr nlinarith theorem curved_inverse_holonomy_local_bound {S T : Set ℝ} {G : ℝ → ℝ → ℝ} {u J : ℝ → ℝ} {κ ℓ K : ℝ} (hκ : 0≤κ) (hℓ : 0≤ℓ) (hK : 0≤K) (hG : ∀ a∈S,∀ s t, |G a s-G a t|≤κ*|s-t|) (hu : ∀ s t,|u s-u t|≤ℓ*|s-t|) (hholo : ∀ x a,a∈S → ∀ᶠ b in 𝓝[S] a,|b-a|≤K*|G b x-G a x|) (hJ : ContinuousOn J T) (hJS : MapsTo J T S) (hmatch : ∀ t∈T,t=G (J t) (u t)) : ∀ a∈T,∀ᶠ b in 𝓝[T] a,|J b-J a|≤(K*(1+κ*ℓ))*|b-a| := by intro a ha have ht : Tendsto J (𝓝[T] a) (𝓝[S] (J a)) := tendsto_nhdsWithin_iff.mpr ⟨hJ a ha,Eventually.mono self_mem_nhdsWithin hJS⟩ filter_upwards [ht.eventually (hholo (u a) (J a) (hJS ha)),self_mem_nhdsWithin] with b hb hbT have hm : |G (J b) (u a)-G (J a) (u a)|≤(1+κ*ℓ)*|b-a| := by calc |G (J b) (u a)-G (J a) (u a)| = |G (J b) (u a)-a| := by rw [←hmatch a ha] _ ≤ |G (J b) (u a)-G (J b) (u b)|+|G (J b) (u b)-a| := abs_sub_le _ _ _ _ ≤ κ*(ℓ*|a-b|)+|b-a| := add_le_add ((hG (J b) (hJS hbT) _ _).trans (by simpa only [mul_comm] using mul_le_mul (hu _ _) le_rfl hκ (mul_nonneg hℓ (abs_nonneg _)))) (by rw [←hmatch b hbT]) _ = (1+κ*ℓ)*|b-a| := by rw [abs_sub_comm a b]; ring exact hb.trans ((mul_le_mul_of_nonneg_left hm hK).trans_eq (by ring)) theorem curved_holonomy_null_image {S : Set ℝ} {G : ℝ → ℝ → ℝ} {u H : ℝ → ℝ} {κ ℓ K : ℝ} (hκ : 0≤κ) (hℓ : 0≤ℓ) (hK : 0≤K) (hcone : κ*ℓ<1) (hG : ∀ a∈S,∀ s t, |G a s-G a t|≤κ*|s-t|) (hu : ∀ s t,|u s-u t|≤ℓ*|s-t|) (hholo : ∀ x a,a∈S → ∀ᶠ b in 𝓝[S] a,|G b x-G a x|≤K*|b-a|) (hmatch : ∀ a∈S,H a=G a (u (H a))) {Z : Set ℝ} (hZS : Z⊆S) (hZ : volume Z=0) : volume (H '' Z)=0 := null_image_of_pointwise_local_bound (div_nonneg hK (sub_pos.mpr hcone).le) (curved_holonomy_local_bound hκ hℓ hK hcone hG hu hholo hmatch) hZS hZ end StandardMapEntropy end section namespace StandardMapEntropy open MeasureTheory Set Filter Topology open scoped Topology ENNReal def AffineGraphControl {X : Type*} (G : X → ℝ → ℝ) (I : Set ℝ) (K : ℝ) : Prop := ∀ c∈I,∀ d∈I,∃ H : ℝ → ℝ, ContinuousOn H (range (fun a => G a c)) ∧ (∀ a,H (G a c)=G a d) ∧ ∀ z∈range (fun a => G a c),∀ᶠ z' in 𝓝[range (fun a => G a c)] z,|H z'-H z|≤K*|z'-z| lemma AffineGraphControl.fibres {X : Type*} {G : X → ℝ → ℝ} {I : Set ℝ} {K : ℝ} (h : AffineGraphControl G I K) {c d : ℝ} (hc : c∈I) (hd : d∈I) (a b : X) : G a c=G b c ↔ G a d=G b d := by obtain ⟨H,_,hH,_⟩ := h c hc d hd obtain ⟨J,_,hJ,_⟩ := h d hd c hc constructor · intro he; rw [←hH a,←hH b,he] · intro he; rw [←hJ a,←hJ b,he] theorem compact_curved_inverse_control {X : Type*} [TopologicalSpace X] [CompactSpace X] {G : X → ℝ → ℝ} (hGc : Continuous (fun p : X×ℝ => G p.1 p.2)) {I : Set ℝ} {K κ ℓ : ℝ} (hK : 0≤K) (hκ : 0≤κ) (hℓ : 0≤ℓ) (hcone : κ*ℓ<1) (hG : ∀ a s t,|G a s-G a t|≤κ*|s-t|) (hh : AffineGraphControl G I K) (u : ℝ → ℝ) (hu : ∀ s t,|u s-u t|≤ℓ*|s-t|) (g : X → ℝ) (hg : Continuous g) (hmatch : ∀ a,g a=G a (u (g a))) (hwithin : ∀ a,u (g a)∈I) {c : ℝ} (hc : c∈I) : ∃ J : ℝ → ℝ,ContinuousOn J (range g) ∧ (∀ a,J (g a)=G a c) ∧ ∀ t∈range g,∀ᶠ t' in 𝓝[range g] t,|J t'-J t|≤(K*(1+κ*ℓ))*|t'-t| := by have fibre (a b : X) (d : ℝ) (hd : d∈I) : g a=g b ↔ G a d=G b d := by constructor · intro he apply (hh.fibres (hwithin a) hd a b).mp rw [←hmatch a,he,←he,he,←hmatch b] · intro he have hsame := (hh.fibres (hwithin b) hd a b).mpr he have hdist : |g a-g b|≤κ*(ℓ*|g a-g b|) := by calc |g a-g b| = |G a (u (g a))-G a (u (g b))| := by rw [hsame,←hmatch a,←hmatch b] _ ≤ κ*(ℓ*|g a-g b|) := (hG a _ _).trans (mul_le_mul_of_nonneg_left (hu _ _) hκ) apply sub_eq_zero.mp apply abs_eq_zero.mp exact le_antisymm (by nlinarith [mul_nonneg (sub_pos.mpr hcone).le (abs_nonneg (g a-g b))]) (abs_nonneg _) obtain ⟨J,hJ,hJval⟩ := compact_fibre_descent hg (hGc.comp (continuous_id.prodMk continuous_const)) (fun a b he => (fibre a b c hc).mp he) change ∀ a,J (g a)=G a c at hJval refine ⟨J,hJ,hJval,?_⟩ rintro t ⟨a,rfl⟩ let d := u (g a) have hd : d∈I := hwithin a obtain ⟨H,hH,hHval,hHbound⟩ := hh d hd c hc obtain ⟨V,hV,hVval⟩ := compact_fibre_descent hg (show Continuous (fun b => G b d) from hGc.comp (continuous_id.prodMk continuous_const)) (fun a b he => (fibre a b d hd).mp he) change ∀ b,V (g b)=G b d at hVval have hmaps : MapsTo V (range g) (range (fun b => G b d)) := by rintro _ ⟨b,rfl⟩; exact ⟨b,(hVval b).symm⟩ have hv : Tendsto V (𝓝[range g] (g a)) (𝓝[range (fun b => G b d)] (G a d)) := by rw [←hVval a] exact tendsto_nhdsWithin_iff.mpr ⟨hV (g a) ⟨a,rfl⟩,Eventually.mono self_mem_nhdsWithin hmaps⟩ filter_upwards [hv.eventually (hHbound (G a d) ⟨a,rfl⟩),self_mem_nhdsWithin] with t ht htS obtain ⟨b,rfl⟩ := htS rw [hVval b,hHval b,hHval a] at ht rw [hJval b,hJval a] apply ht.trans have hm : |G b d-G a d|≤(1+κ*ℓ)*|g b-g a| := by have hea : G a d=g a := (hmatch a).symm calc |G b d-G a d| = |G b (u (g a))-g a| := by rw [hea] _ ≤ |G b (u (g a))-G b (u (g b))|+|G b (u (g b))-g a| := abs_sub_le _ _ _ _ ≤ κ*(ℓ*|g a-g b|)+|g b-g a| := add_le_add ((hG b _ _).trans (by simpa only [mul_comm] using mul_le_mul (hu _ _) le_rfl hκ (mul_nonneg hℓ (abs_nonneg _)))) (by rw [←hmatch b]) _ = (1+κ*ℓ)*|g b-g a| := by rw [abs_sub_comm (g a) (g b)]; ring exact (mul_le_mul_of_nonneg_left hm hK).trans_eq (by ring) end StandardMapEntropy namespace StandardMapEntropy open Set Filter Topology open scoped Topology lemma AffineGraphControl.comp {X Y : Type*} {G : Y → ℝ → ℝ} {I : Set ℝ} {K : ℝ} (h : AffineGraphControl G I K) (f : X → Y) : AffineGraphControl (fun a s => G (f a) s) I K := by intro c hc d hd obtain ⟨H,hH,hval,hb⟩ := h c hc d hd have hsub : range (fun a : X => G (f a) c)⊆range (fun a : Y => G a c) := by rintro _ ⟨a,rfl⟩; exact ⟨f a,rfl⟩ exact ⟨H,hH.mono hsub,fun a => hval (f a),fun z hz => (hb z (hsub hz)).filter_mono (nhdsWithin_mono z hsub)⟩ end StandardMapEntropy end section namespace StandardMapEntropy open MeasureTheory Set Filter Topology open scoped Topology ENNReal lemma AffineGraphControl.scale {X : Type*} {G : X → ℝ → ℝ} {I J : Set ℝ} {K α β : ℝ} (h : AffineGraphControl G I K) (hβ : β≠0) (hJ : ∀ t∈J,t/α∈I) : AffineGraphControl (fun a s => β*G a (s/α)) J K := by intro c hc d hd obtain ⟨H,hH,hHval,hHbound⟩ := h (c/α) (hJ c hc) (d/α) (hJ d hd) let S := range (fun a => β*G a (c/α)) have hmaps : MapsTo (fun t : ℝ => t/β) S (range (fun a => G a (c/α))) := by rintro _ ⟨a,rfl⟩ exact ⟨a,(mul_div_cancel_left₀ _ hβ).symm⟩ have hcont : ContinuousOn (fun t : ℝ => H (t/β)) S := hH.comp (continuous_id.div_const β).continuousOn hmaps refine ⟨(fun t => β*H (t/β)),continuousOn_const.mul hcont,?_,?_⟩ · intro a dsimp only rw [mul_div_cancel_left₀ _ hβ,hHval] · intro t ht have hv : Tendsto (fun t : ℝ => t/β) (𝓝[S] t) (𝓝[range (fun a => G a (c/α))] (t/β)) := tendsto_nhdsWithin_iff.mpr ⟨(continuous_id.div_const β).continuousAt.continuousWithinAt, Eventually.mono self_mem_nhdsWithin hmaps⟩ filter_upwards [hv.eventually (hHbound (t/β) (hmaps ht))] with t' hh rw [←mul_sub,abs_mul] apply (mul_le_mul_of_nonneg_left hh (abs_nonneg β)).trans_eq rw [←sub_div,abs_div] field_simp end StandardMapEntropy end section namespace StandardMapEntropy open MeasureTheory Set Filter Topology open scoped Topology ENNReal open NonlinearStable namespace ReversibleRectangleBlock variable {k χ : ℝ} (B : ReversibleRectangleBlock k χ) noncomputable def holonomyConstant : ℝ := 2*Real.exp (8*B.δ/(1-(Real.exp (-χ+B.ε)+B.δ))) lemma holonomyConstant_pos : 0 intro h <;> constructor <;> linarith [h.1,h.2] lemma mem_unstableInterval {s : ℝ} : s∈B.unstableInterval ↔ |s|≤|B.α| * B.r'/2 := by change -|B.α| * B.r'/2 ≤ s ∧ s ≤ |B.α| * B.r'/2 ↔ _ rw [abs_le] constructor <;> intro h <;> constructor <;> linarith [h.1,h.2] lemma reverseCoordinates_chart (p : RealPlane) : B.reverseCoordinates (B.chart p)=(p.2/B.α,p.1/B.β) := by have he := B.coordinates_swap (B.chart p) rw [B.coordinates_chart] at he have he1 := congrArg Prod.fst he have he2 := congrArg Prod.snd he change p.1=B.β*(B.reverseCoordinates (B.chart p)).2 at he1 change p.2=B.α*(B.reverseCoordinates (B.chart p)).1 at he2 apply Prod.ext · apply (eq_div_iff B.α_ne).mpr nlinarith [he2] · apply (eq_div_iff B.β_ne).mpr nlinarith [he1] lemma reverse_chart_coordinates (v : ℂ) : tangentReversal B.centre+ fineFrame k χ B.ε B.δ (complexProjection (tangentReversal B.centre)) (B.reverseCoordinates v)=tangentReversal v := by rw [reverseCoordinates,fineFrame_inverse k χ B.ε B.δ_pos _ B.reverse_regular] abel end ReversibleRectangleBlock namespace ReversibleGraphFamilies variable {k χ : ℝ} {B : ReversibleRectangleBlock k χ} (F : ReversibleGraphFamilies B) lemma stable_control (hk : 0≤k) : AffineGraphControl F.G B.stableInterval B.holonomyConstant := by let _ : CompactSpace B.carrier := isCompact_iff_compactSpace.mp B.compact intro c hc d hd obtain ⟨H,J,hH,hJ,hval,_,_,hHb,_⟩ := actual_graph_holonomy_bounds k hk χ B.ε B.δ B.ε_pos.le B.δ_pos B.δ_small B.contraction B.slow B.centre B.centre_regular (fun a : B.carrier => (a : ℂ)) (fun a => B.regular a a.property) F.G F.stableParameter F.continuous_G B.r_pos B.r_small B.R_pos (fun a => B.radius a a.property) B.size (fun a => (B.frames a a.property).trans (by linarith [B.η_small])) (fun a => B.coframes a a.property) B.stableInterval (fun a s _ => F.stable_bound a s) (fun a s hs => F.stable_representation a s (B.mem_stableInterval.mp hs)) hc hd exact ⟨H,hH,fun a => (hval a).1,hHb⟩ lemma reverse_representation (a : B.carrier) {s : ℝ} (hs : |s|≤B.r'/2) : tangentReversal B.centre+fineFrame k χ B.ε B.δ (complexProjection (tangentReversal B.centre)) (s,F.U a (B.α*s)/B.β)= fineStableCurve k χ B.ε B.δ B.δ_pos B.contraction (tangentReversal a) (B.reversed a a.property) (F.unstableParameter a (B.α*s)) := by have has : |B.α*s|≤|B.α| * B.r'/2 := by rw [abs_mul]; nlinarith [abs_nonneg B.α] have he := F.unstable_representation a (B.α*s) has have hc := congrArg B.reverseCoordinates he rw [B.reverseCoordinates_chart] at hc have heα : B.α*s/B.α=s := mul_div_cancel_left₀ s B.α_ne rw [heα] at hc have hz := congrArg (fun p => tangentReversal B.centre+ fineFrame k χ B.ε B.δ (complexProjection (tangentReversal B.centre)) p) hc rw [B.reverse_chart_coordinates] at hz simpa only [fineUnstableCurve,tangentReversal_involutive] using hz lemma unstable_control (hk : 0≤k) : AffineGraphControl F.U B.unstableInterval B.holonomyConstant := by let _ : CompactSpace B.carrier := isCompact_iff_compactSpace.mp B.compact let GR (a : B.carrier) (s : ℝ) := F.U a (B.α*s)/B.β let σ (a : B.carrier) (s : ℝ) := F.unstableParameter a (B.α*s) let I : Set ℝ := Icc (-B.r'/2) (B.r'/2) have hGR : Continuous (fun p : B.carrier×ℝ => GR p.1 p.2) := (F.continuous_U.comp (continuous_fst.prodMk (continuous_const.mul continuous_snd))).div_const B.β have hcontrol : AffineGraphControl GR I B.holonomyConstant := by intro c hc d hd obtain ⟨H,J,hH,hJ,hval,_,_,hHb,_⟩ := actual_graph_holonomy_bounds k hk χ B.ε B.δ B.ε_pos.le B.δ_pos B.δ_small B.contraction B.slow (tangentReversal B.centre) B.reverse_regular (fun a : B.carrier => tangentReversal a) (fun a => B.reversed a a.property) GR σ hGR B.r'_pos B.r'_small B.R'_pos (fun a => B.reverse_radius a a.property) B.reverse_size (fun a => (B.reverse_frames a a.property).trans (by linarith [B.η_small])) (fun a => B.reverse_coframes a a.property) I (fun a s _ => F.unstable_bound a (B.α*s)) (fun a s hs => F.reverse_representation a (abs_le.mpr ⟨by linarith [hs.1],hs.2⟩)) hc hd exact ⟨H,hH,fun a => (hval a).1,hHb⟩ have hscaled := hcontrol.scale (α := B.α) B.β_ne (J := B.unstableInterval) (by intro s hs have hs' := B.mem_unstableInterval.mp hs have hab : |s/B.α|≤B.r'/2 := by rw [abs_div] exact (div_le_iff₀ (abs_pos.mpr B.α_ne)).mpr (by nlinarith) exact ⟨by linarith [(abs_le.mp hab).1],(abs_le.mp hab).2⟩) have he : (fun a s => B.β*GR a (s/B.α))=F.U := by funext a s dsimp only [GR] rw [mul_div_cancel₀ _ B.β_ne,mul_div_cancel₀ _ B.α_ne] rw [he] at hscaled exact hscaled end ReversibleGraphFamilies end StandardMapEntropy end section namespace StandardMapEntropy open MeasureTheory MeasureTheory.Measure Set Filter Topology open scoped Topology ENNReal attribute [local instance] Measure.Subtype.measureSpace namespace ReversibleRectangleBlock variable {k χ : ℝ} (B : ReversibleRectangleBlock k χ) noncomputable def coordinateCarrier : Set RealPlane := B.coordinates '' B.carrier lemma compact_coordinateCarrier : IsCompact B.coordinateCarrier := B.compact.image B.continuous_coordinates noncomputable def label (p : B.coordinateCarrier) : B.carrier := ⟨B.chart p,by obtain ⟨v,hv,he⟩ := p.property change B.coordinates v=(p : RealPlane) at he rw [←he,B.chart_coordinates] exact hv⟩ lemma continuous_label : Continuous B.label := (B.continuous_chart.comp continuous_subtype_val).subtype_mk _ @[simp] lemma coordinates_label (p : B.coordinateCarrier) : B.coordinates (B.label p)=(p : RealPlane) := B.coordinates_chart p lemma chart_image_carrier : B.chart '' B.coordinateCarrier=B.carrier := by ext v constructor · rintro ⟨p,⟨v',hv',hp⟩,rfl⟩ rw [←hp,B.chart_coordinates] exact hv' · intro hv exact ⟨B.coordinates v,⟨v,hv,rfl⟩,B.chart_coordinates v⟩ lemma volume_chart_image {S : Set RealPlane} (hS : MeasurableSet S) : volume (B.chart '' S)=ENNReal.ofReal (fineVolumeFactor k χ B.ε B.δ (complexProjection B.centre))*volume S := by have he : B.chart '' S=(fun z : ℂ => B.centre+z) '' ((fineSize k χ B.ε B.δ (complexProjection B.centre) • lyapunovFrame k χ (complexProjection B.centre)) '' (pairToComplex '' S)) := by rw [Set.image_image,Set.image_image] rfl rw [he,Set.image_add_left,measure_preimage_add,Measure.addHaar_image_continuousLinearMap,volume_pairToComplex_image hS] rfl lemma positive_coordinateCarrier : 0 ?_⟩ rw [map_apply hf hN] change Measure.comap Subtype.val volume (f ⁻¹' N)=0 rw [comap_subtype_coe_apply hK] let S : Set (ℝ×ℝ) := Subtype.val '' (f ⁻¹' N) have hS : MeasurableSet S := (MeasurableEmbedding.subtype_coe hK).measurableSet_image.mpr (hN.preimage hf) change volume S=0 rw [Measure.volume_eq_prod,Measure.prod_apply hS] apply lintegral_eq_zero_of_ae_eq_zero apply Filter.Eventually.of_forall intro c by_cases hc : c∈I · obtain ⟨J,hJ,hJnull⟩ := hinv c hc apply measure_mono_null (t := J '' (N∩range f)) _ (hJnull _ inter_subset_right (measure_mono_null inter_subset_left hNz)) rintro t ⟨a,ha,he⟩ have h1 : (a : ℝ×ℝ).1=c := congrArg Prod.fst he have h2 : (a : ℝ×ℝ).2=t := congrArg Prod.snd he exact ⟨f a,⟨ha,⟨a,rfl⟩⟩,(hJ a h1).trans h2⟩ · apply measure_mono_null (t := ∅) _ (measure_empty) rintro t ⟨a,ha,he⟩ have he' : (a : ℝ×ℝ).1=c := congrArg Prod.fst he exact (hc (he' ▸ hwithin a)).elim theorem label_nonsingular_of_horizontal_inverse {K : Set (ℝ×ℝ)} (hK : MeasurableSet K) (f : K → ℝ) (hf : Measurable f) {I : Set ℝ} (hwithin : ∀ a : K,(a : ℝ×ℝ).2∈I) (hinv : ∀ c∈I,∃ J : ℝ → ℝ, (∀ a : K,(a : ℝ×ℝ).2=c → J (f a)=(a : ℝ×ℝ).1) ∧ ∀ Z : Set ℝ,Z⊆range f → volume Z=0 → volume (J '' Z)=0) : QuasiMeasurePreserving f (volume : Measure K) volume := by refine ⟨hf,AbsolutelyContinuous.mk fun N hN hNz => ?_⟩ rw [map_apply hf hN] change Measure.comap Subtype.val volume (f ⁻¹' N)=0 rw [comap_subtype_coe_apply hK] let S : Set (ℝ×ℝ) := Subtype.val '' (f ⁻¹' N) have hS : MeasurableSet S := (MeasurableEmbedding.subtype_coe hK).measurableSet_image.mpr (hN.preimage hf) change volume S=0 rw [Measure.volume_eq_prod,Measure.prod_apply_symm hS] apply lintegral_eq_zero_of_ae_eq_zero apply Filter.Eventually.of_forall intro c by_cases hc : c∈I · obtain ⟨J,hJ,hJnull⟩ := hinv c hc apply measure_mono_null (t := J '' (N∩range f)) _ (hJnull _ inter_subset_right (measure_mono_null inter_subset_left hNz)) rintro t ⟨a,ha,he⟩ have h1 : (a : ℝ×ℝ).2=c := congrArg Prod.snd he have h2 : (a : ℝ×ℝ).1=t := congrArg Prod.fst he exact ⟨f a,⟨ha,⟨a,rfl⟩⟩,(hJ a h1).trans h2⟩ · apply measure_mono_null (t := ∅) _ (measure_empty) rintro t ⟨a,ha,he⟩ have he' : (a : ℝ×ℝ).2=c := congrArg Prod.snd he exact (hc (he' ▸ hwithin a)).elim end StandardMapEntropy end section open MeasureTheory MeasureTheory.Measure Set Filter open scoped ENNReal Topology theorem ae_constant_of_product_rows_columns {A B Y : Type*} [MeasurableSpace A] [MeasurableSpace B] {α : Measure A} {β : Measure B} [IsProbabilityMeasure α] [IsProbabilityMeasure β] {P : A×B → Prop} (hP : ∀ᵐ p ∂α.prod β,P p) {F : A×B → Y} (hrow : ∀ a b b',P (a,b) → P (a,b') → F (a,b)=F (a,b')) (hcol : ∀ a a' b,P (a,b) → P (a',b) → F (a,b)=F (a',b)) : ∃ c : Y, ∀ᵐ p ∂α.prod β,F p=c := by have hp : ∀ᵐ a ∂α,∀ᵐ b ∂β,P (a,b) := ae_ae_of_ae_prod hP obtain ⟨a₀,ha₀⟩ := hp.exists obtain ⟨b₀,hb₀⟩ := ha₀.exists refine ⟨F (a₀,b₀),?_⟩ have hrowc : ∀ᵐ a ∂α,∀ b,P (a,b) → F (a,b)=F (a₀,b₀) := by filter_upwards [hp] with a ha obtain ⟨b',hab',ha₀b'⟩ := (ha.and ha₀).exists intro b hab exact (hrow a b b' hab hab').trans ((hcol a a₀ b' hab' ha₀b').trans (hrow a₀ b' b₀ ha₀b' hb₀)) have hr : ∀ᵐ p ∂α.prod β,∀ b,P (p.1,b) → F (p.1,b)=F (a₀,b₀) := (measurePreserving_fst (μ := α) (ν := β)).quasiMeasurePreserving.ae hrowc filter_upwards [hP,hr] with p hp' hr' exact hr' p.2 hp' theorem positive_atom_of_hopf_product {A B X Y : Type*} [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace X] {α : Measure A} {β : Measure B} {μ : Measure X} [IsProbabilityMeasure α] [IsProbabilityMeasure β] {π : A×B → X} (hπ : QuasiMeasurePreserving π (α.prod β) μ) {P : X → Prop} (hP : ∀ᵐ x ∂μ,P x) {F : X → Y} (hrow : ∀ a b b',P (π (a,b)) → P (π (a,b')) → F (π (a,b))=F (π (a,b'))) (hcol : ∀ a a' b,P (π (a,b)) → P (π (a',b)) → F (π (a,b))=F (π (a',b))) : ∃ c : Y,0<μ {x | F x=c} := by obtain ⟨c,hc⟩ := ae_constant_of_product_rows_columns (F := fun p => F (π p)) (hπ.ae hP) hrow hcol refine ⟨c,?_⟩ by_contra hn have hn' : μ {x | F x=c}=0 := le_antisymm (not_lt.mp hn) (bot_le) have he : ∀ᵐ x ∂μ,F x≠c := by rw [ae_iff] simpa only [not_not] using hn' obtain ⟨p,hp,hp'⟩ := (hc.and (hπ.ae he)).exists exact hp' hp end end end OAI