ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Embedding.measurableEmbedding
protected theorem Embedding.measurableEmbedding {f : α → β} (h₁ : Embedding f)
    (h₂ : MeasurableSet (range f)) : MeasurableEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h₁ : Embedding f) (h₂ : MeasurableSet (Set.range f))
Goal: MeasurableEmbedding f
-- ClosedEmbedding.measurableEmbedding
protected theorem ClosedEmbedding.measurableEmbedding {f : α → β} (h : ClosedEmbedding f) :
    MeasurableEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h : ClosedEmbedding f)
Goal: MeasurableEmbedding f
-- OpenEmbedding.measurableEmbedding
protected theorem OpenEmbedding.measurableEmbedding {f : α → β} (h : OpenEmbedding f) :
    MeasurableEmbedding f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {f : α → β} (h : OpenEmbedding f)
Goal: MeasurableEmbedding f
-- MeasureTheory.IsSetAlgebra.empty_mem
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ 𝒜
  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (self : MeasureTheory.IsSetAlgebra 𝒜)
Goal: ∅ ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.compl_mem
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ 𝒜
  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (self : MeasureTheory.IsSetAlgebra 𝒜) ⦃s : Set α⦄
Goal: s ∈ 𝒜 → sᶜ ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.union_mem
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ 𝒜
  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (self : MeasureTheory.IsSetAlgebra 𝒜) ⦃s t : Set α⦄
Goal: s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.univ_mem
theorem univ_mem (h𝒜 : IsSetAlgebra 𝒜) : univ ∈ 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜)
Goal: Set.univ ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.inter_mem
theorem inter_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :
    s ∩ t ∈ 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} {s t : Set α} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜)
Goal: s ∩ t ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.diff_mem
theorem diff_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :
    s \ t ∈ 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} {s t : Set α} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜)
Goal: s \ t ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.isSetRing
theorem isSetRing (h𝒜 : IsSetAlgebra 𝒜) : IsSetRing 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜)
Goal: MeasureTheory.IsSetRing 𝒜
-- MeasureTheory.IsSetAlgebra.biUnion_mem
theorem biUnion_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)
    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋃ i ∈ S, s i ∈ 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} {ι : Type u_2} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι) (hs : ∀ i ∈ S, s i ∈ 𝒜)
Goal: ⋃ i ∈ S, s i ∈ 𝒜
-- MeasureTheory.IsSetAlgebra.biInter_mem
theorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)
    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} {ι : Type u_2} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι) (hs : ∀ i ∈ S, s i ∈ 𝒜)
Goal: ⋂ i ∈ S, s i ∈ 𝒜
-- MeasureTheory.isSetAlgebra_generateSetAlgebra
theorem isSetAlgebra_generateSetAlgebra :
    IsSetAlgebra (generateSetAlgebra 𝒜) 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)}
Goal: MeasureTheory.IsSetAlgebra (MeasureTheory.generateSetAlgebra 𝒜)
-- MeasureTheory.self_subset_generateSetAlgebra
theorem self_subset_generateSetAlgebra : 𝒜 ⊆ generateSetAlgebra 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)}
Goal: 𝒜 ⊆ MeasureTheory.generateSetAlgebra 𝒜
-- MeasureTheory.generateFrom_generateSetAlgebra_eq
@[simp]
theorem generateFrom_generateSetAlgebra_eq :
    generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)}
Goal: MeasurableSpace.generateFrom (MeasureTheory.generateSetAlgebra 𝒜) = MeasurableSpace.generateFrom 𝒜
-- MeasureTheory.generateSetAlgebra_mono
theorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :
    generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ 
Arguments: {α : Type u_1} {𝒜 ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
Goal: MeasureTheory.generateSetAlgebra 𝒜 ⊆ MeasureTheory.generateSetAlgebra ℬ
-- MeasureTheory.IsSetAlgebra.generateSetAlgebra_subset
theorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
    (hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ 
Arguments: {α : Type u_1} {𝒜 ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) (hℬ : MeasureTheory.IsSetAlgebra ℬ)
Goal: MeasureTheory.generateSetAlgebra 𝒜 ⊆ ℬ
-- MeasureTheory.IsSetAlgebra.generateSetAlgebra_subset_self
theorem generateSetAlgebra_subset_self (h𝒜 : IsSetAlgebra 𝒜) :
    generateSetAlgebra 𝒜 ⊆ 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜)
Goal: MeasureTheory.generateSetAlgebra 𝒜 ⊆ 𝒜
-- MeasureTheory.IsSetAlgebra.generateSetAlgebra_eq
theorem generateSetAlgebra_eq (h𝒜 : IsSetAlgebra 𝒜) : generateSetAlgebra 𝒜 = 𝒜 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (h𝒜 : MeasureTheory.IsSetAlgebra 𝒜)
Goal: MeasureTheory.generateSetAlgebra 𝒜 = 𝒜
-- MeasureTheory.mem_generateSetAlgebra_elim
theorem mem_generateSetAlgebra_elim (s_mem : s ∈ generateSetAlgebra 𝒜) :
    ∃ A : Set (Set (Set α)), A.Finite ∧ (∀ a ∈ A, a.Finite) ∧
    (∀ᵉ (a ∈ A) (t ∈ a), t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ s = ⋃ a ∈ A, ⋂ t ∈ a, t 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} {s : Set α} (s_mem : s ∈ MeasureTheory.generateSetAlgebra 𝒜)
Goal: ∃ A, A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ s = ⋃ a ∈ A, ⋂ t ∈ a, t
-- MeasureTheory.countable_generateSetAlgebra
theorem countable_generateSetAlgebra (h : 𝒜.Countable) :
    (generateSetAlgebra 𝒜).Countable 
Arguments: {α : Type u_1} {𝒜 : Set (Set α)} (h : 𝒜.Countable)
Goal: (MeasureTheory.generateSetAlgebra 𝒜).Countable
-- Real.Icc_mem_vitaliFamily_at_right
theorem Icc_mem_vitaliFamily_at_right {x y : ℝ} (hxy : x < y) :
    Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt x 
Arguments: {x y : ℝ} (hxy : x < y)
Goal: Set.Icc x y ∈ (IsUnifLocDoublingMeasure.vitaliFamily MeasureTheory.volume 1).setsAt x
-- Real.tendsto_Icc_vitaliFamily_right
theorem tendsto_Icc_vitaliFamily_right (x : ℝ) :
    Tendsto (fun y => Icc x y) (𝓝[>] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) 
Arguments: (x : ℝ)
Goal: Filter.Tendsto (fun y => Set.Icc x y) (𝓝[>] x)
  ((IsUnifLocDoublingMeasure.vitaliFamily MeasureTheory.volume 1).filterAt x)
-- Real.Icc_mem_vitaliFamily_at_left
theorem Icc_mem_vitaliFamily_at_left {x y : ℝ} (hxy : x < y) :
    Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt y 
Arguments: {x y : ℝ} (hxy : x < y)
Goal: Set.Icc x y ∈ (IsUnifLocDoublingMeasure.vitaliFamily MeasureTheory.volume 1).setsAt y
-- Real.tendsto_Icc_vitaliFamily_left
theorem tendsto_Icc_vitaliFamily_left (x : ℝ) :
    Tendsto (fun y => Icc y x) (𝓝[<] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) 
Arguments: (x : ℝ)
Goal: Filter.Tendsto (fun y => Set.Icc y x) (𝓝[<] x)
  ((IsUnifLocDoublingMeasure.vitaliFamily MeasureTheory.volume 1).filterAt x)
-- VitaliFamily.covering
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure VitaliFamily {m : MeasurableSpace α} (μ : Measure α) where
    setsAt :  α → Set (Set α)
    measurableSet : ∀ x : α, ∀ s ∈ setsAt x, MeasurableSet s
    nonempty_interior : ∀ x : α, ∀ s ∈ setsAt x, (interior s).Nonempty
    nontrivial : ∀ (x : α), ∀ ε > (0 : ℝ), ∃ s ∈ setsAt x, s ⊆ closedBall x ε
    covering : ∀ (s : Set α) (f : α → Set (Set α)),
    (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ f x, a ⊆ closedBall x ε) →
    ∃ t : Set (α × Set α), (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧
      (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
Arguments: {α : Type u_1} [MetricSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (self : VitaliFamily μ) (s : Set α) (f : α → Set (Set α))
Goal: (∀ x ∈ s, f x ⊆ self.setsAt x) →
  (∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ Metric.closedBall x ε) →
    ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
-- VitaliFamily.nonempty_interior
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure VitaliFamily {m : MeasurableSpace α} (μ : Measure α) where
    setsAt :  α → Set (Set α)
    measurableSet : ∀ x : α, ∀ s ∈ setsAt x, MeasurableSet s
    nonempty_interior : ∀ x : α, ∀ s ∈ setsAt x, (interior s).Nonempty
    nontrivial : ∀ (x : α), ∀ ε > (0 : ℝ), ∃ s ∈ setsAt x, s ⊆ closedBall x ε
    covering : ∀ (s : Set α) (f : α → Set (Set α)),
    (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ f x, a ⊆ closedBall x ε) →
    ∃ t : Set (α × Set α), (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧
      (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
Arguments: {α : Type u_1} [MetricSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (self : VitaliFamily μ) (x : α) (s : Set α)
Goal: s ∈ self.setsAt x → (interior s).Nonempty
-- VitaliFamily.measurableSet
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure VitaliFamily {m : MeasurableSpace α} (μ : Measure α) where
    setsAt :  α → Set (Set α)
    measurableSet : ∀ x : α, ∀ s ∈ setsAt x, MeasurableSet s
    nonempty_interior : ∀ x : α, ∀ s ∈ setsAt x, (interior s).Nonempty
    nontrivial : ∀ (x : α), ∀ ε > (0 : ℝ), ∃ s ∈ setsAt x, s ⊆ closedBall x ε
    covering : ∀ (s : Set α) (f : α → Set (Set α)),
    (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ f x, a ⊆ closedBall x ε) →
    ∃ t : Set (α × Set α), (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧
      (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
Arguments: {α : Type u_1} [MetricSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (self : VitaliFamily μ) (x : α) (s : Set α)
Goal: s ∈ self.setsAt x → MeasurableSet s
-- VitaliFamily.nontrivial
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure VitaliFamily {m : MeasurableSpace α} (μ : Measure α) where
    setsAt :  α → Set (Set α)
    measurableSet : ∀ x : α, ∀ s ∈ setsAt x, MeasurableSet s
    nonempty_interior : ∀ x : α, ∀ s ∈ setsAt x, (interior s).Nonempty
    nontrivial : ∀ (x : α), ∀ ε > (0 : ℝ), ∃ s ∈ setsAt x, s ⊆ closedBall x ε
    covering : ∀ (s : Set α) (f : α → Set (Set α)),
    (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ f x, a ⊆ closedBall x ε) →
    ∃ t : Set (α × Set α), (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧
      (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
Arguments: {α : Type u_1} [MetricSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (self : VitaliFamily μ) (x : α) (ε : ℝ)
Goal: ε > 0 → ∃ s ∈ self.setsAt x, s ⊆ Metric.closedBall x ε
-- VitaliFamily.FineSubfamilyOn.exists_disjoint_covering_ae
theorem exists_disjoint_covering_ae :
    ∃ t : Set (α × Set α),
      (∀ p : α × Set α, p ∈ t → p.1 ∈ s) ∧
      (t.PairwiseDisjoint fun p => p.2) ∧
      (∀ p : α × Set α, p ∈ t → p.2 ∈ v.setsAt p.1 ∩ f p.1) ∧
      μ (s \ ⋃ (p : α × Set α) (_ : p ∈ t), p.2) = 0 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s)
Goal: ∃ t,
  (∀ p ∈ t, p.1 ∈ s) ∧
    (t.PairwiseDisjoint fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ v.setsAt p.1 ∩ f p.1) ∧ μ (s \ ⋃ p ∈ t, p.2) = 0
-- VitaliFamily.FineSubfamilyOn.index_subset
theorem index_subset : ∀ p : α × Set α, p ∈ h.index → p.1 ∈ s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) (p : α × Set α)
Goal: p ∈ h.index → p.1 ∈ s
-- VitaliFamily.FineSubfamilyOn.covering_disjoint
theorem covering_disjoint : h.index.PairwiseDisjoint h.covering 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s)
Goal: h.index.PairwiseDisjoint h.covering
-- VitaliFamily.FineSubfamilyOn.covering_disjoint_subtype
theorem covering_disjoint_subtype : Pairwise (Disjoint on fun x : h.index => h.covering x) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s)
Goal: Pairwise (Disjoint on fun x => h.covering ↑x)
-- VitaliFamily.FineSubfamilyOn.covering_mem
theorem covering_mem {p : α × Set α} (hp : p ∈ h.index) : h.covering p ∈ f p.1 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) {p : α × Set α} (hp : p ∈ h.index)
Goal: h.covering p ∈ f p.1
-- VitaliFamily.FineSubfamilyOn.covering_mem_family
theorem covering_mem_family {p : α × Set α} (hp : p ∈ h.index) : h.covering p ∈ v.setsAt p.1 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) {p : α × Set α} (hp : p ∈ h.index)
Goal: h.covering p ∈ v.setsAt p.1
-- VitaliFamily.FineSubfamilyOn.measure_diff_biUnion
theorem measure_diff_biUnion : μ (s \ ⋃ p ∈ h.index, h.covering p) = 0 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s)
Goal: μ (s \ ⋃ p ∈ h.index, h.covering p) = 0
-- VitaliFamily.FineSubfamilyOn.index_countable
theorem index_countable [SecondCountableTopology α] : h.index.Countable 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) [SecondCountableTopology α]
Goal: h.index.Countable
-- VitaliFamily.FineSubfamilyOn.measurableSet_u
protected theorem measurableSet_u {p : α × Set α} (hp : p ∈ h.index) :
    MeasurableSet (h.covering p) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) {p : α × Set α} (hp : p ∈ h.index)
Goal: MeasurableSet (h.covering p)
-- VitaliFamily.FineSubfamilyOn.measure_le_tsum_of_absolutelyContinuous
theorem measure_le_tsum_of_absolutelyContinuous [SecondCountableTopology α] {ρ : Measure α}
    (hρ : ρ ≪ μ) : ρ s ≤ ∑' p : h.index, ρ (h.covering p) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) [SecondCountableTopology α] {ρ : MeasureTheory.Measure α} (hρ : ρ ≪ μ)
Goal: ρ s ≤ ∑' (p : ↑h.index), ρ (h.covering ↑p)
-- VitaliFamily.FineSubfamilyOn.measure_le_tsum
theorem measure_le_tsum [SecondCountableTopology α] : μ s ≤ ∑' x : h.index, μ (h.covering x) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {v : VitaliFamily μ} {f : α → Set (Set α)} {s : Set α} (h : v.FineSubfamilyOn f s) [SecondCountableTopology α]
Goal: μ s ≤ ∑' (x : ↑h.index), μ (h.covering ↑x)
-- Filter.HasBasis.vitaliFamily
theorem _root_.Filter.HasBasis.vitaliFamily {ι : Sort*} {p : ι → Prop} {s : ι → Set α} {x : α}
    (h : (𝓝 x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i ↦ {t ∈ v.setsAt x | t ⊆ s i}) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {ι : Sort u_2} {p : ι → Prop} {s : ι → Set α} {x : α} (h : (𝓝 x).HasBasis p s)
Goal: (v.filterAt x).HasBasis p fun i => {t | t ∈ v.setsAt x ∧ t ⊆ s i}
-- VitaliFamily.filterAt_basis_closedBall
theorem filterAt_basis_closedBall (x : α) :
    (v.filterAt x).HasBasis (0 < ·) ({a ∈ v.setsAt x | a ⊆ closedBall x ·}) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) (x : α)
Goal: (v.filterAt x).HasBasis (fun x => 0 < x) fun x_1 => {a | a ∈ v.setsAt x ∧ a ⊆ Metric.closedBall x x_1}
-- VitaliFamily.mem_filterAt_iff
theorem mem_filterAt_iff {x : α} {s : Set (Set α)} :
    s ∈ v.filterAt x ↔ ∃ ε > (0 : ℝ), ∀ a ∈ v.setsAt x, a ⊆ closedBall x ε → a ∈ s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {x : α} {s : Set (Set α)}
Goal: s ∈ v.filterAt x ↔ ∃ ε > 0, ∀ a ∈ v.setsAt x, a ⊆ Metric.closedBall x ε → a ∈ s
-- VitaliFamily.eventually_filterAt_iff
theorem eventually_filterAt_iff {x : α} {P : Set α → Prop} :
    (∀ᶠ a in v.filterAt x, P a) ↔ ∃ ε > (0 : ℝ), ∀ a ∈ v.setsAt x, a ⊆ closedBall x ε → P a 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {x : α} {P : Set α → Prop}
Goal: (∀ᶠ (a : Set α) in v.filterAt x, P a) ↔ ∃ ε > 0, ∀ a ∈ v.setsAt x, a ⊆ Metric.closedBall x ε → P a
-- VitaliFamily.tendsto_filterAt_iff
theorem tendsto_filterAt_iff {ι : Type*} {l : Filter ι} {f : ι → Set α} {x : α} :
    Tendsto f l (v.filterAt x) ↔
      (∀ᶠ i in l, f i ∈ v.setsAt x) ∧ ∀ ε > (0 : ℝ), ∀ᶠ i in l, f i ⊆ closedBall x ε 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {ι : Type u_2} {l : Filter ι} {f : ι → Set α} {x : α}
Goal: Filter.Tendsto f l (v.filterAt x) ↔
  (∀ᶠ (i : ι) in l, f i ∈ v.setsAt x) ∧ ∀ ε > 0, ∀ᶠ (i : ι) in l, f i ⊆ Metric.closedBall x ε
-- VitaliFamily.eventually_filterAt_mem_setsAt
theorem eventually_filterAt_mem_setsAt (x : α) : ∀ᶠ a in v.filterAt x, a ∈ v.setsAt x 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) (x : α)
Goal: ∀ᶠ (a : Set α) in v.filterAt x, a ∈ v.setsAt x
-- VitaliFamily.eventually_filterAt_subset_closedBall
theorem eventually_filterAt_subset_closedBall (x : α) {ε : ℝ} (hε : 0 < ε) :
    ∀ᶠ a : Set α in v.filterAt x, a ⊆ closedBall x ε 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) (x : α) {ε : ℝ} (hε : 0 < ε)
Goal: ∀ᶠ (a : Set α) in v.filterAt x, a ⊆ Metric.closedBall x ε
-- VitaliFamily.eventually_filterAt_measurableSet
theorem eventually_filterAt_measurableSet (x : α) : ∀ᶠ a in v.filterAt x, MeasurableSet a 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) (x : α)
Goal: ∀ᶠ (a : Set α) in v.filterAt x, MeasurableSet a
-- VitaliFamily.frequently_filterAt_iff
theorem frequently_filterAt_iff {x : α} {P : Set α → Prop} :
    (∃ᶠ a in v.filterAt x, P a) ↔ ∀ ε > (0 : ℝ), ∃ a ∈ v.setsAt x, a ⊆ closedBall x ε ∧ P a 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {x : α} {P : Set α → Prop}
Goal: (∃ᶠ (a : Set α) in v.filterAt x, P a) ↔ ∀ ε > 0, ∃ a ∈ v.setsAt x, a ⊆ Metric.closedBall x ε ∧ P a
-- VitaliFamily.eventually_filterAt_subset_of_nhds
theorem eventually_filterAt_subset_of_nhds {x : α} {o : Set α} (hx : o ∈ 𝓝 x) :
    ∀ᶠ a in v.filterAt x, a ⊆ o 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {x : α} {o : Set α} (hx : o ∈ 𝓝 x)
Goal: ∀ᶠ (a : Set α) in v.filterAt x, a ⊆ o
-- VitaliFamily.fineSubfamilyOn_of_frequently
theorem fineSubfamilyOn_of_frequently (v : VitaliFamily μ) (f : α → Set (Set α)) (s : Set α)
    (h : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, a ∈ f x) : v.FineSubfamilyOn f s 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) (f : α → Set (Set α)) (s : Set α) (h : ∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, a ∈ f x)
Goal: v.FineSubfamilyOn f s
-- Real.borel_eq_generateFrom_Ioo_rat
theorem borel_eq_generateFrom_Ioo_rat :
    borel ℝ = .generateFrom (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) (b : ℝ)}) 
Goal: borel ℝ = MeasurableSpace.generateFrom (⋃ a, ⋃ b, ⋃ (_ : a < b), {Set.Ioo ↑a ↑b})
-- Real.borel_eq_generateFrom_Iio_rat
theorem borel_eq_generateFrom_Iio_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iio (a : ℝ)}) 
Goal: borel ℝ = MeasurableSpace.generateFrom (⋃ a, {Set.Iio ↑a})
-- Real.borel_eq_generateFrom_Ioi_rat
theorem borel_eq_generateFrom_Ioi_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ioi (a : ℝ)}) 
Goal: borel ℝ = MeasurableSpace.generateFrom (⋃ a, {Set.Ioi ↑a})
-- Real.borel_eq_generateFrom_Iic_rat
theorem borel_eq_generateFrom_Iic_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iic (a : ℝ)}) 
Goal: borel ℝ = MeasurableSpace.generateFrom (⋃ a, {Set.Iic ↑a})
-- Real.borel_eq_generateFrom_Ici_rat
theorem borel_eq_generateFrom_Ici_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ici (a : ℝ)}) 
Goal: borel ℝ = MeasurableSpace.generateFrom (⋃ a, {Set.Ici ↑a})
-- Real.isPiSystem_Ioo_rat
theorem isPiSystem_Ioo_rat :
    IsPiSystem (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) (b : ℝ)}) 
Goal: IsPiSystem (⋃ a, ⋃ b, ⋃ (_ : a < b), {Set.Ioo ↑a ↑b})
-- Real.isPiSystem_Iio_rat
theorem isPiSystem_Iio_rat : IsPiSystem (⋃ a : ℚ, {Iio (a : ℝ)}) 
Goal: IsPiSystem (⋃ a, {Set.Iio ↑a})
-- Real.isPiSystem_Ioi_rat
theorem isPiSystem_Ioi_rat : IsPiSystem (⋃ a : ℚ, {Ioi (a : ℝ)}) 
Goal: IsPiSystem (⋃ a, {Set.Ioi ↑a})
-- Real.isPiSystem_Iic_rat
theorem isPiSystem_Iic_rat : IsPiSystem (⋃ a : ℚ, {Iic (a : ℝ)}) 
Goal: IsPiSystem (⋃ a, {Set.Iic ↑a})
-- Real.isPiSystem_Ici_rat
theorem isPiSystem_Ici_rat : IsPiSystem (⋃ a : ℚ, {Ici (a : ℝ)}) 
Goal: IsPiSystem (⋃ a, {Set.Ici ↑a})
-- Real.measure_ext_Ioo_rat
theorem measure_ext_Ioo_rat {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ]
    (h : ∀ a b : ℚ, μ (Ioo a b) = ν (Ioo a b)) : μ = ν 
Arguments: {μ ν : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] (h : ∀ (a b : ℚ), μ (Set.Ioo ↑a ↑b) = ν (Set.Ioo ↑a ↑b))
Goal: μ = ν
-- measurable_real_toNNReal
@[measurability, fun_prop]
theorem measurable_real_toNNReal : Measurable Real.toNNReal 
Goal: Measurable Real.toNNReal
-- Measurable.real_toNNReal
@[measurability, fun_prop]
theorem Measurable.real_toNNReal {f : α → ℝ} (hf : Measurable f) :
    Measurable fun x => Real.toNNReal (f x) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => (f x).toNNReal
-- AEMeasurable.real_toNNReal
@[measurability, fun_prop]
theorem AEMeasurable.real_toNNReal {f : α → ℝ} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => Real.toNNReal (f x)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => (f x).toNNReal) μ
-- measurable_coe_nnreal_real
@[measurability]
theorem measurable_coe_nnreal_real : Measurable ((↑) : ℝ≥0 → ℝ) 
Goal: Measurable NNReal.toReal
-- Measurable.coe_nnreal_real
@[measurability, fun_prop]
theorem Measurable.coe_nnreal_real {f : α → ℝ≥0} (hf : Measurable f) :
    Measurable fun x => (f x : ℝ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0} (hf : Measurable f)
Goal: Measurable fun x => ↑(f x)
-- AEMeasurable.coe_nnreal_real
@[measurability, fun_prop]
theorem AEMeasurable.coe_nnreal_real {f : α → ℝ≥0} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => (f x : ℝ)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ↑(f x)) μ
-- measurable_coe_nnreal_ennreal
@[measurability]
theorem measurable_coe_nnreal_ennreal : Measurable ((↑) : ℝ≥0 → ℝ≥0∞) 
Goal: Measurable ENNReal.ofNNReal
-- Measurable.coe_nnreal_ennreal
@[measurability, fun_prop]
theorem Measurable.coe_nnreal_ennreal {f : α → ℝ≥0} (hf : Measurable f) :
    Measurable fun x => (f x : ℝ≥0∞) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0} (hf : Measurable f)
Goal: Measurable fun x => ↑(f x)