ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.measurablySeparable_range_of_disjoint
theorem measurablySeparable_range_of_disjoint [T2Space α] [MeasurableSpace α]
    [OpensMeasurableSpace α] {f g : (ℕ → ℕ) → α} (hf : Continuous f) (hg : Continuous g)
    (h : Disjoint (range f) (range g)) : MeasurablySeparable (range f) (range g) 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] [MeasurableSpace α] [OpensMeasurableSpace α] {f g : (ℕ → ℕ) → α} (hf : Continuous f) (hg : Continuous g) (h : Disjoint (Set.range f) (Set.range g))
Goal: MeasureTheory.MeasurablySeparable (Set.range f) (Set.range g)
-- MeasureTheory.AnalyticSet.measurablySeparable
theorem AnalyticSet.measurablySeparable [T2Space α] [MeasurableSpace α] [OpensMeasurableSpace α]
    {s t : Set α} (hs : AnalyticSet s) (ht : AnalyticSet t) (h : Disjoint s t) :
    MeasurablySeparable s t 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] [MeasurableSpace α] [OpensMeasurableSpace α] {s t : Set α} (hs : MeasureTheory.AnalyticSet s) (ht : MeasureTheory.AnalyticSet t) (h : Disjoint s t)
Goal: MeasureTheory.MeasurablySeparable s t
-- MeasureTheory.AnalyticSet.measurableSet_of_compl
theorem AnalyticSet.measurableSet_of_compl [T2Space α] [MeasurableSpace α] [OpensMeasurableSpace α]
    {s : Set α} (hs : AnalyticSet s) (hsc : AnalyticSet sᶜ) : MeasurableSet s 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (hs : MeasureTheory.AnalyticSet s) (hsc : MeasureTheory.AnalyticSet sᶜ)
Goal: MeasurableSet s
-- Measurable.measurableSet_preimage_iff_of_surjective
theorem measurableSet_preimage_iff_of_surjective [CountablySeparated Z]
    {f : X → Z} (hf : Measurable f) (hsurj : Surjective f) {s : Set Z} :
    MeasurableSet (f ⁻¹' s) ↔ MeasurableSet s 
Arguments: {X : Type u_3} {Z : Type u_5} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace Z] [MeasurableSpace.CountablySeparated Z] {f : X → Z} (hf : Measurable f) (hsurj : Function.Surjective f) {s : Set Z}
Goal: MeasurableSet (f ⁻¹' s) ↔ MeasurableSet s
-- Measurable.map_measurableSpace_eq
theorem map_measurableSpace_eq [CountablySeparated Z]
    {f : X → Z} (hf : Measurable f)
    (hsurj : Surjective f) : MeasurableSpace.map f ‹MeasurableSpace X› = ‹MeasurableSpace Z› 
Arguments: {X : Type u_3} {Z : Type u_5} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace Z] [MeasurableSpace.CountablySeparated Z] {f : X → Z} (hf : Measurable f) (hsurj : Function.Surjective f)
Goal: MeasurableSpace.map f inst✝ = inst✝¹
-- Measurable.map_measurableSpace_eq_borel
theorem map_measurableSpace_eq_borel [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f)
    (hsurj : Surjective f) : MeasurableSpace.map f ‹MeasurableSpace X› = borel Y 
Arguments: {X : Type u_3} {Y : Type u_4} [MeasurableSpace X] [StandardBorelSpace X] [TopologicalSpace Y] [T0Space Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f) (hsurj : Function.Surjective f)
Goal: MeasurableSpace.map f inst✝ = borel Y
-- Measurable.borelSpace_codomain
theorem borelSpace_codomain [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f)
    (hsurj : Surjective f) : BorelSpace Y 
Arguments: {X : Type u_3} {Y : Type u_4} [MeasurableSpace X] [StandardBorelSpace X] [TopologicalSpace Y] [T0Space Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] [SecondCountableTopology Y] {f : X → Y} (hf : Measurable f) (hsurj : Function.Surjective f)
Goal: BorelSpace Y
-- Measurable.measurableSet_preimage_iff_preimage_val
theorem measurableSet_preimage_iff_preimage_val {f : X → Z} [CountablySeparated (range f)]
    (hf : Measurable f) {s : Set Z} :
    MeasurableSet (f ⁻¹' s) ↔ MeasurableSet ((↑) ⁻¹' s : Set (range f)) 
Arguments: {X : Type u_3} {Z : Type u_5} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace Z] {f : X → Z} [MeasurableSpace.CountablySeparated ↑(Set.range f)] (hf : Measurable f) {s : Set Z}
Goal: MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (Subtype.val ⁻¹' s)
-- Measurable.measurableSet_preimage_iff_inter_range
theorem measurableSet_preimage_iff_inter_range {f : X → Z} [CountablySeparated (range f)]
    (hf : Measurable f) (hr : MeasurableSet (range f)) {s : Set Z} :
    MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f) 
Arguments: {X : Type u_3} {Z : Type u_5} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace Z] {f : X → Z} [MeasurableSpace.CountablySeparated ↑(Set.range f)] (hf : Measurable f) (hr : MeasurableSet (Set.range f)) {s : Set Z}
Goal: MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ Set.range f)
-- Measurable.measurable_comp_iff_restrict
theorem measurable_comp_iff_restrict {f : X → Z}
    [CountablySeparated (range f)]
    (hf : Measurable f) {g : Z → β} : Measurable (g ∘ f) ↔ Measurable (restrict (range f) g) 
Arguments: {X : Type u_3} {Z : Type u_5} {β : Type u_6} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace β] [MeasurableSpace Z] {f : X → Z} [MeasurableSpace.CountablySeparated ↑(Set.range f)] (hf : Measurable f) {g : Z → β}
Goal: Measurable (g ∘ f) ↔ Measurable ((Set.range f).restrict g)
-- Measurable.measurable_comp_iff_of_surjective
theorem measurable_comp_iff_of_surjective [CountablySeparated Z]
    {f : X → Z} (hf : Measurable f) (hsurj : Surjective f)
    {g : Z → β} : Measurable (g ∘ f) ↔ Measurable g 
Arguments: {X : Type u_3} {Z : Type u_5} {β : Type u_6} [MeasurableSpace X] [StandardBorelSpace X] [MeasurableSpace β] [MeasurableSpace Z] [MeasurableSpace.CountablySeparated Z] {f : X → Z} (hf : Measurable f) (hsurj : Function.Surjective f) {g : Z → β}
Goal: Measurable (g ∘ f) ↔ Measurable g
-- Continuous.map_eq_borel
theorem Continuous.map_eq_borel {X Y : Type*} [TopologicalSpace X] [PolishSpace X]
    [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y]
    {f : X → Y} (hf : Continuous f) (hsurj : Surjective f) :
    MeasurableSpace.map f ‹MeasurableSpace X› = borel Y 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f)
Goal: MeasurableSpace.map f inst✝ = borel Y
-- Continuous.map_borel_eq
theorem Continuous.map_borel_eq {X Y : Type*} [TopologicalSpace X] [PolishSpace X]
    [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f)
    (hsurj : Surjective f) : MeasurableSpace.map f (borel X) = borel Y 
Arguments: {X : Type u_3} {Y : Type u_4} [TopologicalSpace X] [PolishSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f)
Goal: MeasurableSpace.map f (borel X) = borel Y
-- MeasureTheory.measurableSet_range_of_continuous_injective
theorem measurableSet_range_of_continuous_injective {β : Type*} [TopologicalSpace γ]
    [PolishSpace γ] [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β]
    {f : γ → β} (f_cont : Continuous f) (f_inj : Injective f) :
    MeasurableSet (range f) 
Arguments: {γ : Type u_3} {β : Type u_4} [TopologicalSpace γ] [PolishSpace γ] [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β] {f : γ → β} (f_cont : Continuous f) (f_inj : Function.Injective f)
Goal: MeasurableSet (Set.range f)
-- IsClosed.measurableSet_image_of_continuousOn_injOn
theorem _root_.IsClosed.measurableSet_image_of_continuousOn_injOn
    [TopologicalSpace γ] [PolishSpace γ] {β : Type*} [TopologicalSpace β] [T2Space β]
    [MeasurableSpace β] [OpensMeasurableSpace β] {s : Set γ} (hs : IsClosed s) {f : γ → β}
    (f_cont : ContinuousOn f s) (f_inj : InjOn f s) : MeasurableSet (f '' s) 
Arguments: {γ : Type u_3} [TopologicalSpace γ] [PolishSpace γ] {β : Type u_4} [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β] {s : Set γ} (hs : IsClosed s) {f : γ → β} (f_cont : ContinuousOn f s) (f_inj : Set.InjOn f s)
Goal: MeasurableSet (f '' s)
-- MeasurableSet.image_of_continuousOn_injOn
theorem _root_.MeasurableSet.image_of_continuousOn_injOn [OpensMeasurableSpace β]
    [tγ : TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ]
    (hs : MeasurableSet s)
    (f_cont : ContinuousOn f s) (f_inj : InjOn f s) : MeasurableSet (f '' s) 
Arguments: {γ : Type u_3} {β : Type u_5} [tβ : TopologicalSpace β] [T2Space β] [MeasurableSpace β] {s : Set γ} {f : γ → β} [OpensMeasurableSpace β] [tγ : TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ] (hs : MeasurableSet s) (f_cont : ContinuousOn f s) (f_inj : Set.InjOn f s)
Goal: MeasurableSet (f '' s)
-- MeasurableSet.image_of_measurable_injOn
theorem _root_.MeasurableSet.image_of_measurable_injOn {f : γ → α}
    [MeasurableSpace.CountablySeparated α]
    [MeasurableSpace γ] [StandardBorelSpace γ]
    (hs : MeasurableSet s) (f_meas : Measurable f) (f_inj : InjOn f s) :
    MeasurableSet (f '' s) 
Arguments: {γ : Type u_3} {α : Type u_4} [MeasurableSpace α] {s : Set γ} {f : γ → α} [MeasurableSpace.CountablySeparated α] [MeasurableSpace γ] [StandardBorelSpace γ] (hs : MeasurableSet s) (f_meas : Measurable f) (f_inj : Set.InjOn f s)
Goal: MeasurableSet (f '' s)
-- Continuous.measurableEmbedding
theorem _root_.Continuous.measurableEmbedding [BorelSpace β]
    [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ]
    (f_cont : Continuous f) (f_inj : Injective f) :
    MeasurableEmbedding f 
Arguments: {γ : Type u_3} {β : Type u_5} [tβ : TopologicalSpace β] [T2Space β] [MeasurableSpace β] {f : γ → β} [BorelSpace β] [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ] (f_cont : Continuous f) (f_inj : Function.Injective f)
Goal: MeasurableEmbedding f
-- ContinuousOn.measurableEmbedding
theorem _root_.ContinuousOn.measurableEmbedding [BorelSpace β]
    [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ]
    (hs : MeasurableSet s) (f_cont : ContinuousOn f s)
    (f_inj : InjOn f s) : MeasurableEmbedding (s.restrict f) 
Arguments: {γ : Type u_3} {β : Type u_5} [tβ : TopologicalSpace β] [T2Space β] [MeasurableSpace β] {s : Set γ} {f : γ → β} [BorelSpace β] [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ] (hs : MeasurableSet s) (f_cont : ContinuousOn f s) (f_inj : Set.InjOn f s)
Goal: MeasurableEmbedding (s.restrict f)
-- Measurable.measurableEmbedding
theorem _root_.Measurable.measurableEmbedding {f : γ → α}
    [MeasurableSpace.CountablySeparated α]
    [MeasurableSpace γ] [StandardBorelSpace γ]
    (f_meas : Measurable f) (f_inj : Injective f) : MeasurableEmbedding f 
Arguments: {γ : Type u_3} {α : Type u_4} [MeasurableSpace α] {f : γ → α} [MeasurableSpace.CountablySeparated α] [MeasurableSpace γ] [StandardBorelSpace γ] (f_meas : Measurable f) (f_inj : Function.Injective f)
Goal: MeasurableEmbedding f
-- MeasureTheory.borel_eq_borel_of_le
theorem borel_eq_borel_of_le {t t' : TopologicalSpace γ}
    (ht : PolishSpace (h := t)) (ht' : PolishSpace (h := t')) (hle : t ≤ t') :
    @borel _ t = @borel _ t' 
Arguments: {γ : Type u_3} {t t' : TopologicalSpace γ} (ht : PolishSpace γ) (ht' : PolishSpace γ) (hle : t ≤ t')
Goal: borel γ = borel γ
-- MeasureTheory.isClopenable_iff_measurableSet
theorem isClopenable_iff_measurableSet
    [tγ : TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ] :
    IsClopenable s ↔ MeasurableSet s 
Arguments: {γ : Type u_3} {s : Set γ} [tγ : TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ]
Goal: PolishSpace.IsClopenable s ↔ MeasurableSet s
-- MeasureTheory.measurableSet_tendsto_fun
@[measurability]
lemma measurableSet_tendsto_fun [MeasurableSpace γ] [Countable ι]
    {l : Filter ι} [l.IsCountablyGenerated]
    [TopologicalSpace γ] [SecondCountableTopology γ] [PseudoMetrizableSpace γ]
    [OpensMeasurableSpace γ]
    {f : ι → β → γ} (hf : ∀ i, Measurable (f i)) {g : β → γ} (hg : Measurable g) :
    MeasurableSet { x | Tendsto (fun n ↦ f n x) l (𝓝 (g x)) } 
Arguments: {ι : Type u_2} {γ : Type u_3} {β : Type u_5} [MeasurableSpace β] [MeasurableSpace γ] [Countable ι] {l : Filter ι} [l.IsCountablyGenerated] [TopologicalSpace γ] [SecondCountableTopology γ] [TopologicalSpace.PseudoMetrizableSpace γ] [OpensMeasurableSpace γ] {f : ι → β → γ} (hf : ∀ (i : ι), Measurable (f i)) {g : β → γ} (hg : Measurable g)
Goal: MeasurableSet {x | Filter.Tendsto (fun n => f n x) l (𝓝 (g x))}
-- MeasureTheory.measurableSet_exists_tendsto
@[measurability]
theorem measurableSet_exists_tendsto [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ]
    [hγ : OpensMeasurableSpace γ] [Countable ι] {l : Filter ι}
    [l.IsCountablyGenerated] {f : ι → β → γ} (hf : ∀ i, Measurable (f i)) :
    MeasurableSet { x | ∃ c, Tendsto (fun n => f n x) l (𝓝 c) } 
Arguments: {ι : Type u_2} {γ : Type u_3} {β : Type u_5} [MeasurableSpace β] [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [hγ : OpensMeasurableSpace γ] [Countable ι] {l : Filter ι} [l.IsCountablyGenerated] {f : ι → β → γ} (hf : ∀ (i : ι), Measurable (f i))
Goal: MeasurableSet {x | ∃ c, Filter.Tendsto (fun n => f n x) l (𝓝 c)}
-- MeasurableSet.isClopenable'
theorem _root_.MeasurableSet.isClopenable' {s : Set α} (hs : MeasurableSet s) :
    ∃ _ : TopologicalSpace α, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s 
Arguments: {α : Type u_1} [MeasurableSpace α] [StandardBorelSpace α] {s : Set α} (hs : MeasurableSet s)
Goal: ∃ x, BorelSpace α ∧ PolishSpace α ∧ IsClosed s ∧ IsOpen s
-- MeasurableSet.standardBorel
theorem _root_.MeasurableSet.standardBorel {s : Set α} (hs : MeasurableSet s) :
    StandardBorelSpace s 
Arguments: {α : Type u_1} [MeasurableSpace α] [StandardBorelSpace α] {s : Set α} (hs : MeasurableSet s)
Goal: StandardBorelSpace ↑s
-- Measurable.inner
@[aesop safe 20 apply (rule_sets := [Measurable])]
theorem Measurable.inner {_ : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E]
    [SecondCountableTopology E] {f g : α → E} (hf : Measurable f)
    (hg : Measurable g) : Measurable fun t => ⟪f t, g t⟫ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSpace E] [inst_1 : OpensMeasurableSpace E]
  [inst_2 : SecondCountableTopology E] {f g : α → E}, Measurable f → Measurable g → Measurable fun t => ⟪f t, g t⟫_𝕜
-- Measurable.const_inner
@[measurability]
theorem Measurable.const_inner {_ : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E]
    [SecondCountableTopology E] {c : E} {f : α → E} (hf : Measurable f) :
    Measurable fun t => ⟪c, f t⟫ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSpace E] [inst_1 : OpensMeasurableSpace E]
  [inst_2 : SecondCountableTopology E] {c : E} {f : α → E}, Measurable f → Measurable fun t => ⟪c, f t⟫_𝕜
-- Measurable.inner_const
@[measurability]
theorem Measurable.inner_const {_ : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E]
    [SecondCountableTopology E] {c : E} {f : α → E} (hf : Measurable f) :
    Measurable fun t => ⟪f t, c⟫ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSpace E] [inst_1 : OpensMeasurableSpace E]
  [inst_2 : SecondCountableTopology E] {c : E} {f : α → E}, Measurable f → Measurable fun t => ⟪f t, c⟫_𝕜
-- AEMeasurable.inner
@[aesop safe 20 apply (rule_sets := [Measurable])]
theorem AEMeasurable.inner {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E]
    [SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f g : α → E}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : AEMeasurable (fun x => ⟪f x, g x⟫) μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f g : α → E} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun x => ⟪f x, g x⟫_𝕜) μ
-- AEMeasurable.const_inner
@[measurability]
theorem AEMeasurable.const_inner {m : MeasurableSpace α} [MeasurableSpace E]
    [OpensMeasurableSpace E] [SecondCountableTopology E]
    {μ : MeasureTheory.Measure α} {f : α → E} {c : E} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => ⟪c, f x⟫) μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f : α → E} {c : E} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ⟪c, f x⟫_𝕜) μ
-- AEMeasurable.inner_const
@[measurability]
theorem AEMeasurable.inner_const {m : MeasurableSpace α} [MeasurableSpace E]
    [OpensMeasurableSpace E] [SecondCountableTopology E]
    {μ : MeasureTheory.Measure α} {f : α → E} {c : E} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => ⟪f x, c⟫) μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f : α → E} {c : E} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ⟪f x, c⟫_𝕜) μ
-- Real.measurable_arctan
@[measurability]
theorem measurable_arctan : Measurable arctan 
Goal: Measurable Real.arctan
-- Measurable.arctan
@[measurability]
theorem Measurable.arctan : Measurable fun x => arctan (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.arctan (f x)
-- MeasureTheory.Egorov.mem_notConvergentSeq_iff
theorem mem_notConvergentSeq_iff [Preorder ι] {x : α} :
    x ∈ notConvergentSeq f g n j ↔ ∃ k ≥ j, 1 / (n + 1 : ℝ) < dist (f k x) (g x) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} [MetricSpace β] {n : ℕ} {j : ι} {f : ι → α → β} {g : α → β} [Preorder ι] {x : α}
Goal: x ∈ MeasureTheory.Egorov.notConvergentSeq f g n j ↔ ∃ k ≥ j, 1 / (↑n + 1) < dist (f k x) (g x)
-- MeasureTheory.Egorov.notConvergentSeq_antitone
theorem notConvergentSeq_antitone [Preorder ι] : Antitone (notConvergentSeq f g n) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} [MetricSpace β] {n : ℕ} {f : ι → α → β} {g : α → β} [Preorder ι]
Goal: Antitone (MeasureTheory.Egorov.notConvergentSeq f g n)
-- MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero
theorem measure_inter_notConvergentSeq_eq_zero [SemilatticeSup ι] [Nonempty ι]
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
    μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) (n : ℕ)
Goal: μ (s ∩ ⋂ j, MeasureTheory.Egorov.notConvergentSeq f g n j) = 0
-- MeasureTheory.Egorov.notConvergentSeq_measurableSet
theorem notConvergentSeq_measurableSet [Preorder ι] [Countable ι]
    (hf : ∀ n, StronglyMeasurable[m] (f n)) (hg : StronglyMeasurable g) :
    MeasurableSet (notConvergentSeq f g n j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {n : ℕ} {j : ι} {f : ι → α → β} {g : α → β} [Preorder ι] [Countable ι] (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasurableSet (MeasureTheory.Egorov.notConvergentSeq f g n j)
-- MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero
theorem measure_notConvergentSeq_tendsto_zero [SemilatticeSup ι] [Countable ι]
    (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g) (hsm : MeasurableSet s)
    (hs : μ s ≠ ∞) (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
    Tendsto (fun j => μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Countable ι] (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) (n : ℕ)
Goal: Filter.Tendsto (fun j => μ (s ∩ MeasureTheory.Egorov.notConvergentSeq f g n j)) Filter.atTop (𝓝 0)
-- MeasureTheory.Egorov.exists_notConvergentSeq_lt
theorem exists_notConvergentSeq_lt (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
    ∃ j : ι, μ (s ∩ notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] [Countable ι] (hε : 0 < ε) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) (n : ℕ)
Goal: ∃ j, μ (s ∩ MeasureTheory.Egorov.notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n)
-- MeasureTheory.Egorov.notConvergentSeqLTIndex_spec
theorem notConvergentSeqLTIndex_spec (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
    μ (s ∩ notConvergentSeq f g n (notConvergentSeqLTIndex hε hf hg hsm hs hfg n)) ≤
      ENNReal.ofReal (ε * 2⁻¹ ^ n) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] [Countable ι] (hε : 0 < ε) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))) (n : ℕ)
Goal: μ
    (s ∩
      MeasureTheory.Egorov.notConvergentSeq f g n
        (MeasureTheory.Egorov.notConvergentSeqLTIndex hε hf hg hsm hs hfg n)) ≤
  ENNReal.ofReal (ε * 2⁻¹ ^ n)
-- MeasureTheory.Egorov.iUnionNotConvergentSeq_measurableSet
theorem iUnionNotConvergentSeq_measurableSet (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    MeasurableSet <| iUnionNotConvergentSeq hε hf hg hsm hs hfg 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] [Countable ι] (hε : 0 < ε) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: MeasurableSet (MeasureTheory.Egorov.iUnionNotConvergentSeq hε hf hg hsm hs hfg)
-- MeasureTheory.Egorov.measure_iUnionNotConvergentSeq
theorem measure_iUnionNotConvergentSeq (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
    (hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    μ (iUnionNotConvergentSeq hε hf hg hsm hs hfg) ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} [MetricSpace β] {μ : MeasureTheory.Measure α} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β} [SemilatticeSup ι] [Nonempty ι] [Countable ι] (hε : 0 < ε) (hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)) (hg : MeasureTheory.StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ⊤) (hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x)))
Goal: μ (MeasureTheory.Egorov.iUnionNotConvergentSeq hε hf hg hsm hs hfg) ≤ ENNReal.ofReal ε