ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.AbsolutelyContinuous.ae_eq
theorem AbsolutelyContinuous.ae_eq (h : μ ≪ ν) {f g : α → δ} (h' : f =ᵐ[ν] g) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {δ : Type u_4} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ.AbsolutelyContinuous ν) {f g : α → δ} (h' : f =ᵐ[ν] g)
Goal: f =ᵐ[μ] g
-- MeasureTheory.AEDisjoint.of_absolutelyContinuous
protected theorem _root_.MeasureTheory.AEDisjoint.of_absolutelyContinuous
    (h : AEDisjoint μ s t) {ν : Measure α} (h' : ν ≪ μ) :
    AEDisjoint ν s t 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t) {ν : MeasureTheory.Measure α} (h' : ν.AbsolutelyContinuous μ)
Goal: MeasureTheory.AEDisjoint ν s t
-- MeasureTheory.AEDisjoint.of_le
protected theorem _root_.MeasureTheory.AEDisjoint.of_le
    (h : AEDisjoint μ s t) {ν : Measure α} (h' : ν ≤ μ) :
    AEDisjoint ν s t 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t) {ν : MeasureTheory.Measure α} (h' : ν ≤ μ)
Goal: MeasureTheory.AEDisjoint ν s t
-- MeasureTheory.Measure.QuasiMeasurePreserving.absolutelyContinuous
structure QuasiMeasurePreserving {m0 : MeasurableSpace α} (f : α → β)
  (μa : Measure α := by volume_tac)
  (μb : Measure β := by volume_tac) : Prop where
  protected measurable : Measurable f
  protected absolutelyContinuous : μa.map f ≪ μb
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {m0 : MeasurableSpace α} {f : α → β} {μa : autoParam (MeasureTheory.Measure α) _auto✝} {μb : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: (MeasureTheory.Measure.map f μa).AbsolutelyContinuous μb
-- MeasureTheory.Measure.QuasiMeasurePreserving.measurable
structure QuasiMeasurePreserving {m0 : MeasurableSpace α} (f : α → β)
  (μa : Measure α := by volume_tac)
  (μb : Measure β := by volume_tac) : Prop where
  protected measurable : Measurable f
  protected absolutelyContinuous : μa.map f ≪ μb
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {m0 : MeasurableSpace α} {f : α → β} {μa : autoParam (MeasureTheory.Measure α) _auto✝} {μb : autoParam (MeasureTheory.Measure β) _auto✝¹} (self : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: Measurable f
-- MeasureTheory.Measure.QuasiMeasurePreserving.id
protected theorem id {_m0 : MeasurableSpace α} (μ : Measure α) : QuasiMeasurePreserving id μ μ 
Arguments: {α : Type u_1} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving id μ μ
-- Measurable.quasiMeasurePreserving
protected theorem _root_.Measurable.quasiMeasurePreserving
    {_m0 : MeasurableSpace α} (hf : Measurable f) (μ : Measure α) :
    QuasiMeasurePreserving f μ (μ.map f) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {f : α → β} {_m0 : MeasurableSpace α} (hf : Measurable f) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f μ (MeasureTheory.Measure.map f μ)
-- MeasureTheory.Measure.QuasiMeasurePreserving.mono_left
theorem mono_left (h : QuasiMeasurePreserving f μa μb) (ha : μa' ≪ μa) :
    QuasiMeasurePreserving f μa' μb 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa μa' : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) (ha : μa'.AbsolutelyContinuous μa)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f μa' μb
-- MeasureTheory.Measure.QuasiMeasurePreserving.mono_right
theorem mono_right (h : QuasiMeasurePreserving f μa μb) (ha : μb ≪ μb') :
    QuasiMeasurePreserving f μa μb' 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb μb' : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) (ha : μb.AbsolutelyContinuous μb')
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f μa μb'
-- MeasureTheory.Measure.QuasiMeasurePreserving.mono
@[mono]
theorem mono (ha : μa' ≪ μa) (hb : μb ≪ μb') (h : QuasiMeasurePreserving f μa μb) :
    QuasiMeasurePreserving f μa' μb' 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa μa' : MeasureTheory.Measure α} {μb μb' : MeasureTheory.Measure β} {f : α → β} (ha : μa'.AbsolutelyContinuous μa) (hb : μb.AbsolutelyContinuous μb') (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f μa' μb'
-- MeasureTheory.Measure.QuasiMeasurePreserving.comp
protected theorem comp {g : β → γ} {f : α → β} (hg : QuasiMeasurePreserving g μb μc)
    (hf : QuasiMeasurePreserving f μa μb) : QuasiMeasurePreserving (g ∘ f) μa μc 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {g : β → γ} {f : α → β} (hg : MeasureTheory.Measure.QuasiMeasurePreserving g μb μc) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving (g ∘ f) μa μc
-- MeasureTheory.Measure.QuasiMeasurePreserving.iterate
protected theorem iterate {f : α → α} (hf : QuasiMeasurePreserving f μa μa) :
    ∀ n, QuasiMeasurePreserving f^[n] μa μa
  
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μa : MeasureTheory.Measure α} {f : α → α} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μa μa) (n : ℕ)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f^[n] μa μa
-- MeasureTheory.Measure.QuasiMeasurePreserving.aemeasurable
protected theorem aemeasurable (hf : QuasiMeasurePreserving f μa μb) : AEMeasurable f μa 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: AEMeasurable f μa
-- MeasureTheory.Measure.QuasiMeasurePreserving.ae_map_le
theorem ae_map_le (h : QuasiMeasurePreserving f μa μb) : ae (μa.map f) ≤ ae μb 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: MeasureTheory.ae (MeasureTheory.Measure.map f μa) ≤ MeasureTheory.ae μb
-- MeasureTheory.Measure.QuasiMeasurePreserving.tendsto_ae
theorem tendsto_ae (h : QuasiMeasurePreserving f μa μb) : Tendsto f (ae μa) (ae μb) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb)
Goal: Filter.Tendsto f (MeasureTheory.ae μa) (MeasureTheory.ae μb)
-- MeasureTheory.Measure.QuasiMeasurePreserving.ae
theorem ae (h : QuasiMeasurePreserving f μa μb) {p : β → Prop} (hg : ∀ᵐ x ∂μb, p x) :
    ∀ᵐ x ∂μa, p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) {p : β → Prop} (hg : ∀ᵐ (x : β) ∂μb, p x)
Goal: ∀ᵐ (x : α) ∂μa, p (f x)
-- MeasureTheory.Measure.QuasiMeasurePreserving.ae_eq
theorem ae_eq (h : QuasiMeasurePreserving f μa μb) {g₁ g₂ : β → δ} (hg : g₁ =ᵐ[μb] g₂) :
    g₁ ∘ f =ᵐ[μa] g₂ ∘ f 
Arguments: {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) {g₁ g₂ : β → δ} (hg : g₁ =ᵐ[μb] g₂)
Goal: g₁ ∘ f =ᵐ[μa] g₂ ∘ f
-- MeasureTheory.Measure.QuasiMeasurePreserving.preimage_null
theorem preimage_null (h : QuasiMeasurePreserving f μa μb) {s : Set β} (hs : μb s = 0) :
    μa (f ⁻¹' s) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) {s : Set β} (hs : μb s = 0)
Goal: μa (f ⁻¹' s) = 0
-- MeasureTheory.Measure.QuasiMeasurePreserving.preimage_mono_ae
theorem preimage_mono_ae {s t : Set β} (hf : QuasiMeasurePreserving f μa μb) (h : s ≤ᵐ[μb] t) :
    f ⁻¹' s ≤ᵐ[μa] f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} {s t : Set β} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) (h : s ≤ᵐ[μb] t)
Goal: f ⁻¹' s ≤ᵐ[μa] f ⁻¹' t
-- MeasureTheory.Measure.QuasiMeasurePreserving.preimage_ae_eq
theorem preimage_ae_eq {s t : Set β} (hf : QuasiMeasurePreserving f μa μb) (h : s =ᵐ[μb] t) :
    f ⁻¹' s =ᵐ[μa] f ⁻¹' t 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {f : α → β} {s t : Set β} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μa μb) (h : s =ᵐ[μb] t)
Goal: f ⁻¹' s =ᵐ[μa] f ⁻¹' t
-- MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq
theorem preimage_iterate_ae_eq {s : Set α} {f : α → α} (hf : QuasiMeasurePreserving f μ μ) (k : ℕ)
    (hs : f ⁻¹' s =ᵐ[μ] s) : f^[k] ⁻¹' s =ᵐ[μ] s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → α} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) (k : ℕ) (hs : f ⁻¹' s =ᵐ[μ] s)
Goal: f^[k] ⁻¹' s =ᵐ[μ] s
-- MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq
theorem image_zpow_ae_eq {s : Set α} {e : α ≃ α} (he : QuasiMeasurePreserving e μ μ)
    (he' : QuasiMeasurePreserving e.symm μ μ) (k : ℤ) (hs : e '' s =ᵐ[μ] s) :
    (⇑(e ^ k)) '' s =ᵐ[μ] s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {e : α ≃ α} (he : MeasureTheory.Measure.QuasiMeasurePreserving (⇑e) μ μ) (he' : MeasureTheory.Measure.QuasiMeasurePreserving (⇑e.symm) μ μ) (k : ℤ) (hs : ⇑e '' s =ᵐ[μ] s)
Goal: ⇑(e ^ k) '' s =ᵐ[μ] s
-- MeasureTheory.Measure.QuasiMeasurePreserving.limsup_preimage_iterate_ae_eq
theorem limsup_preimage_iterate_ae_eq {f : α → α} (hf : QuasiMeasurePreserving f μ μ)
    (hs : f ⁻¹' s =ᵐ[μ] s) : limsup (α := Set α) (fun n => (preimage f)^[n] s) atTop =ᵐ[μ] s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → α} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) (hs : f ⁻¹' s =ᵐ[μ] s)
Goal: Filter.limsup (fun n => (Set.preimage f)^[n] s) Filter.atTop =ᵐ[μ] s
-- MeasureTheory.Measure.QuasiMeasurePreserving.liminf_preimage_iterate_ae_eq
theorem liminf_preimage_iterate_ae_eq {f : α → α} (hf : QuasiMeasurePreserving f μ μ)
    (hs : f ⁻¹' s =ᵐ[μ] s) : liminf (α := Set α) (fun n => (preimage f)^[n] s) atTop =ᵐ[μ] s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → α} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) (hs : f ⁻¹' s =ᵐ[μ] s)
Goal: Filter.liminf (fun n => (Set.preimage f)^[n] s) Filter.atTop =ᵐ[μ] s
-- MeasureTheory.Measure.QuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae
theorem exists_preimage_eq_of_preimage_ae {f : α → α} (h : QuasiMeasurePreserving f μ μ)
    (hs : MeasurableSet s) (hs' : f ⁻¹' s =ᵐ[μ] s) :
    ∃ t : Set α, MeasurableSet t ∧ t =ᵐ[μ] s ∧ f ⁻¹' t = t 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → α} (h : MeasureTheory.Measure.QuasiMeasurePreserving f μ μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s =ᵐ[μ] s)
Goal: ∃ t, MeasurableSet t ∧ t =ᵐ[μ] s ∧ f ⁻¹' t = t
-- MeasureTheory.Measure.QuasiMeasurePreserving.smul_ae_eq_of_ae_eq
@[to_additive]
theorem smul_ae_eq_of_ae_eq {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace α]
    {s t : Set α} {μ : Measure α} (g : G)
    (h_qmp : QuasiMeasurePreserving (g⁻¹ • · : α → α) μ μ)
    (h_ae_eq : s =ᵐ[μ] t) : (g • s : Set α) =ᵐ[μ] (g • t : Set α) 
Arguments: {G : Type u_8} {α : Type u_9} [Group G] [MulAction G α] [MeasurableSpace α] {s t : Set α} {μ : MeasureTheory.Measure α} (g : G) (h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g⁻¹ • x) μ μ) (h_ae_eq : s =ᵐ[μ] t)
Goal: g • s =ᵐ[μ] g • t
-- MeasureTheory.Measure.QuasiMeasurePreserving.vadd_ae_eq_of_ae_eq
@[to_additive]
theorem smul_ae_eq_of_ae_eq {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace α]
    {s t : Set α} {μ : Measure α} (g : G)
    (h_qmp : QuasiMeasurePreserving (g⁻¹ • · : α → α) μ μ)
    (h_ae_eq : s =ᵐ[μ] t) : (g • s : Set α) =ᵐ[μ] (g • t : Set α) 
Arguments: {G : Type u_8} {α : Type u_9} [AddGroup G] [AddAction G α] [MeasurableSpace α] {s t : Set α} {μ : MeasureTheory.Measure α} (g : G) (h_qmp : MeasureTheory.Measure.QuasiMeasurePreserving (fun x => -g +ᵥ x) μ μ) (h_ae_eq : s =ᵐ[μ] t)
Goal: g +ᵥ s =ᵐ[μ] g +ᵥ t
-- MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one
@[to_additive]
theorem pairwise_aedisjoint_of_aedisjoint_forall_ne_one {G α : Type*} [Group G] [MulAction G α]
    [MeasurableSpace α] {μ : Measure α} {s : Set α}
    (h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
    (h_qmp : ∀ g : G, QuasiMeasurePreserving (g • ·) μ μ) :
    Pairwise (AEDisjoint μ on fun g : G => g • s) 
Arguments: {G : Type u_8} {α : Type u_9} [Group G] [MulAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (h_ae_disjoint : ∀ (g : G), g ≠ 1 → MeasureTheory.AEDisjoint μ (g • s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g • x) μ μ)
Goal: Pairwise (MeasureTheory.AEDisjoint μ on fun g => g • s)
-- MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_zero
@[to_additive]
theorem pairwise_aedisjoint_of_aedisjoint_forall_ne_one {G α : Type*} [Group G] [MulAction G α]
    [MeasurableSpace α] {μ : Measure α} {s : Set α}
    (h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
    (h_qmp : ∀ g : G, QuasiMeasurePreserving (g • ·) μ μ) :
    Pairwise (AEDisjoint μ on fun g : G => g • s) 
Arguments: {G : Type u_8} {α : Type u_9} [AddGroup G] [AddAction G α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (h_ae_disjoint : ∀ (g : G), g ≠ 0 → MeasureTheory.AEDisjoint μ (g +ᵥ s) s) (h_qmp : ∀ (g : G), MeasureTheory.Measure.QuasiMeasurePreserving (fun x => g +ᵥ x) μ μ)
Goal: Pairwise (MeasureTheory.AEDisjoint μ on fun g => g +ᵥ s)
-- MeasureTheory.Measure.mem_cofinite
theorem mem_cofinite : s ∈ μ.cofinite ↔ μ sᶜ < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: s ∈ μ.cofinite ↔ μ sᶜ < ⊤
-- MeasureTheory.Measure.compl_mem_cofinite
theorem compl_mem_cofinite : sᶜ ∈ μ.cofinite ↔ μ s < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: sᶜ ∈ μ.cofinite ↔ μ s < ⊤
-- MeasureTheory.Measure.eventually_cofinite
theorem eventually_cofinite {p : α → Prop} : (∀ᶠ x in μ.cofinite, p x) ↔ μ { x | ¬p x } < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop}
Goal: (∀ᶠ (x : α) in μ.cofinite, p x) ↔ μ {x | ¬p x} < ⊤
-- AEMeasurable.nullMeasurable
protected theorem _root_.AEMeasurable.nullMeasurable {f : α → β} (h : AEMeasurable f μ) :
    NullMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (h : AEMeasurable f μ)
Goal: MeasureTheory.NullMeasurable f μ
-- AEMeasurable.nullMeasurableSet_preimage
lemma _root_.AEMeasurable.nullMeasurableSet_preimage {f : α → β} {s : Set β}
    (hf : AEMeasurable f μ) (hs : MeasurableSet s) : NullMeasurableSet (f ⁻¹' s) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} {s : Set β} (hf : AEMeasurable f μ) (hs : MeasurableSet s)
Goal: MeasureTheory.NullMeasurableSet (f ⁻¹' s) μ
-- MeasureTheory.NullMeasurableSet.preimage
theorem NullMeasurableSet.preimage {ν : Measure β} {f : α → β} {t : Set β}
    (ht : NullMeasurableSet t ν) (hf : QuasiMeasurePreserving f μ ν) :
    NullMeasurableSet (f ⁻¹' t) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → β} {t : Set β} (ht : MeasureTheory.NullMeasurableSet t ν) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: MeasureTheory.NullMeasurableSet (f ⁻¹' t) μ
-- MeasureTheory.NullMeasurableSet.mono_ac
theorem NullMeasurableSet.mono_ac (h : NullMeasurableSet s μ) (hle : ν ≪ μ) :
    NullMeasurableSet s ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ) (hle : ν.AbsolutelyContinuous μ)
Goal: MeasureTheory.NullMeasurableSet s ν
-- MeasureTheory.NullMeasurableSet.mono
theorem NullMeasurableSet.mono (h : NullMeasurableSet s μ) (hle : ν ≤ μ) : NullMeasurableSet s ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ) (hle : ν ≤ μ)
Goal: MeasureTheory.NullMeasurableSet s ν
-- MeasureTheory.AEDisjoint.preimage
theorem AEDisjoint.preimage {ν : Measure β} {f : α → β} {s t : Set β} (ht : AEDisjoint ν s t)
    (hf : QuasiMeasurePreserving f μ ν) : AEDisjoint μ (f ⁻¹' s) (f ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → β} {s t : Set β} (ht : MeasureTheory.AEDisjoint ν s t) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: MeasureTheory.AEDisjoint μ (f ⁻¹' s) (f ⁻¹' t)
-- MeasureTheory.ae_eq_bot
@[simp]
theorem ae_eq_bot : ae μ = ⊥ ↔ μ = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.ae μ = ⊥ ↔ μ = 0
-- MeasureTheory.ae_neBot
@[simp]
theorem ae_neBot : (ae μ).NeBot ↔ μ ≠ 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: (MeasureTheory.ae μ).NeBot ↔ μ ≠ 0
-- MeasureTheory.ae_zero
@[simp]
theorem ae_zero {_m0 : MeasurableSpace α} : ae (0 : Measure α) = ⊥ 
Arguments: {α : Type u_1} {_m0 : MeasurableSpace α}
Goal: MeasureTheory.ae 0 = ⊥
-- MeasureTheory.ae_mono
@[mono]
theorem ae_mono (h : μ ≤ ν) : ae μ ≤ ae ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ ≤ ν)
Goal: MeasureTheory.ae μ ≤ MeasureTheory.ae ν
-- MeasureTheory.mem_ae_map_iff
theorem mem_ae_map_iff {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : MeasurableSet s) :
    s ∈ ae (μ.map f) ↔ f ⁻¹' s ∈ ae μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : MeasurableSet s)
Goal: s ∈ MeasureTheory.ae (MeasureTheory.Measure.map f μ) ↔ f ⁻¹' s ∈ MeasureTheory.ae μ
-- MeasureTheory.mem_ae_of_mem_ae_map
theorem mem_ae_of_mem_ae_map {f : α → β} (hf : AEMeasurable f μ) {s : Set β}
    (hs : s ∈ ae (μ.map f)) : f ⁻¹' s ∈ ae μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : s ∈ MeasureTheory.ae (MeasureTheory.Measure.map f μ))
Goal: f ⁻¹' s ∈ MeasureTheory.ae μ
-- MeasureTheory.ae_map_iff
theorem ae_map_iff {f : α → β} (hf : AEMeasurable f μ) {p : β → Prop}
    (hp : MeasurableSet { x | p x }) : (∀ᵐ y ∂μ.map f, p y) ↔ ∀ᵐ x ∂μ, p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) {p : β → Prop} (hp : MeasurableSet {x | p x})
Goal: (∀ᵐ (y : β) ∂MeasureTheory.Measure.map f μ, p y) ↔ ∀ᵐ (x : α) ∂μ, p (f x)
-- MeasureTheory.ae_of_ae_map
theorem ae_of_ae_map {f : α → β} (hf : AEMeasurable f μ) {p : β → Prop} (h : ∀ᵐ y ∂μ.map f, p y) :
    ∀ᵐ x ∂μ, p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) {p : β → Prop} (h : ∀ᵐ (y : β) ∂MeasureTheory.Measure.map f μ, p y)
Goal: ∀ᵐ (x : α) ∂μ, p (f x)
-- MeasureTheory.ae_map_mem_range
theorem ae_map_mem_range {m0 : MeasurableSpace α} (f : α → β) (hf : MeasurableSet (range f))
    (μ : Measure α) : ∀ᵐ x ∂μ.map f, x ∈ range f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {m0 : MeasurableSpace α} (f : α → β) (hf : MeasurableSet (Set.range f)) (μ : MeasureTheory.Measure α)
Goal: ∀ᵐ (x : β) ∂MeasureTheory.Measure.map f μ, x ∈ Set.range f
-- MeasureTheory.biSup_measure_Iic
theorem biSup_measure_Iic [Preorder α] {s : Set α} (hsc : s.Countable)
    (hst : ∀ x : α, ∃ y ∈ s, x ≤ y) (hdir : DirectedOn (· ≤ ·) s) :
    ⨆ x ∈ s, μ (Iic x) = μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Preorder α] {s : Set α} (hsc : s.Countable) (hst : ∀ (x : α), ∃ y ∈ s, x ≤ y) (hdir : DirectedOn (fun x x_1 => x ≤ x_1) s)
Goal: ⨆ x ∈ s, μ (Set.Iic x) = μ Set.univ
-- MeasureTheory.tendsto_measure_Ico_atTop
theorem tendsto_measure_Ico_atTop [SemilatticeSup α] [NoMaxOrder α]
    [(atTop : Filter α).IsCountablyGenerated] (μ : Measure α) (a : α) :
    Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a))) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [SemilatticeSup α] [NoMaxOrder α] [Filter.atTop.IsCountablyGenerated] (μ : MeasureTheory.Measure α) (a : α)
Goal: Filter.Tendsto (fun x => μ (Set.Ico a x)) Filter.atTop (nhds (μ (Set.Ici a)))
-- MeasureTheory.tendsto_measure_Ioc_atBot
theorem tendsto_measure_Ioc_atBot [SemilatticeInf α] [NoMinOrder α]
    [(atBot : Filter α).IsCountablyGenerated] (μ : Measure α) (a : α) :
    Tendsto (fun x => μ (Ioc x a)) atBot (𝓝 (μ (Iic a))) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [SemilatticeInf α] [NoMinOrder α] [Filter.atBot.IsCountablyGenerated] (μ : MeasureTheory.Measure α) (a : α)
Goal: Filter.Tendsto (fun x => μ (Set.Ioc x a)) Filter.atBot (nhds (μ (Set.Iic a)))
-- MeasureTheory.tendsto_measure_Iic_atTop
theorem tendsto_measure_Iic_atTop [SemilatticeSup α] [(atTop : Filter α).IsCountablyGenerated]
    (μ : Measure α) : Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ)) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [SemilatticeSup α] [Filter.atTop.IsCountablyGenerated] (μ : MeasureTheory.Measure α)
Goal: Filter.Tendsto (fun x => μ (Set.Iic x)) Filter.atTop (nhds (μ Set.univ))
-- MeasureTheory.tendsto_measure_Ici_atBot
theorem tendsto_measure_Ici_atBot [SemilatticeInf α] [h : (atBot : Filter α).IsCountablyGenerated]
    (μ : Measure α) : Tendsto (fun x => μ (Ici x)) atBot (𝓝 (μ univ)) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [SemilatticeInf α] [h : Filter.atBot.IsCountablyGenerated] (μ : MeasureTheory.Measure α)
Goal: Filter.Tendsto (fun x => μ (Set.Ici x)) Filter.atBot (nhds (μ Set.univ))
-- MeasureTheory.Iio_ae_eq_Iic'
theorem Iio_ae_eq_Iic' (ha : μ {a} = 0) : Iio a =ᵐ[μ] Iic a 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a : α} (ha : μ {a} = 0)
Goal: Set.Iio a =ᵐ[μ] Set.Iic a
-- MeasureTheory.Ioi_ae_eq_Ici'
theorem Ioi_ae_eq_Ici' (ha : μ {a} = 0) : Ioi a =ᵐ[μ] Ici a 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a : α} (ha : μ {a} = 0)
Goal: Set.Ioi a =ᵐ[μ] Set.Ici a
-- MeasureTheory.Ioo_ae_eq_Ioc'
theorem Ioo_ae_eq_Ioc' (hb : μ {b} = 0) : Ioo a b =ᵐ[μ] Ioc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a b : α} (hb : μ {b} = 0)
Goal: Set.Ioo a b =ᵐ[μ] Set.Ioc a b
-- MeasureTheory.Ioc_ae_eq_Icc'
theorem Ioc_ae_eq_Icc' (ha : μ {a} = 0) : Ioc a b =ᵐ[μ] Icc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0)
Goal: Set.Ioc a b =ᵐ[μ] Set.Icc a b
-- MeasureTheory.Ioo_ae_eq_Ico'
theorem Ioo_ae_eq_Ico' (ha : μ {a} = 0) : Ioo a b =ᵐ[μ] Ico a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PartialOrder α] {a b : α} (ha : μ {a} = 0)
Goal: Set.Ioo a b =ᵐ[μ] Set.Ico a b