ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.diracProba_comp_diracProbaEquiv_symm_eq_val
lemma diracProba_comp_diracProbaEquiv_symm_eq_val [T0Space X] :
    diracProba ∘ (diracProbaEquiv (X := X)).symm = fun μ ↦ μ.val 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [T0Space X]
Goal: MeasureTheory.diracProba ∘ ⇑MeasureTheory.diracProbaEquiv.symm = fun μ => ↑μ
-- MeasureTheory.tendsto_diracProbaEquivSymm_iff_tendsto
lemma tendsto_diracProbaEquivSymm_iff_tendsto [T0Space X] [CompletelyRegularSpace X]
    {μ : range (diracProba (X := X))} (F : Filter (range (diracProba (X := X)))) :
    Tendsto diracProbaEquiv.symm F (𝓝 (diracProbaEquiv.symm μ)) ↔ Tendsto id F (𝓝 μ) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [T0Space X] [CompletelyRegularSpace X] {μ : ↑(Set.range MeasureTheory.diracProba)} (F : Filter ↑(Set.range MeasureTheory.diracProba))
Goal: Filter.Tendsto (⇑MeasureTheory.diracProbaEquiv.symm) F (𝓝 (MeasureTheory.diracProbaEquiv.symm μ)) ↔
  Filter.Tendsto id F (𝓝 μ)
-- MeasureTheory.continuous_diracProbaEquiv
lemma continuous_diracProbaEquiv [T0Space X] :
    Continuous (diracProbaEquiv (X := X)) 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [T0Space X]
Goal: Continuous ⇑MeasureTheory.diracProbaEquiv
-- MeasureTheory.continuous_diracProbaEquivSymm
lemma continuous_diracProbaEquivSymm [T0Space X] [CompletelyRegularSpace X] :
    Continuous (diracProbaEquiv (X := X)).symm 
Arguments: {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [T0Space X] [CompletelyRegularSpace X]
Goal: Continuous ⇑MeasureTheory.diracProbaEquiv.symm
-- MeasureTheory.embedding_diracProba
theorem embedding_diracProba [T0Space X] [CompletelyRegularSpace X]
    [MeasurableSpace X] [OpensMeasurableSpace X] : Embedding (fun (x : X) ↦ diracProba x) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T0Space X] [CompletelyRegularSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
Goal: Embedding fun x => MeasureTheory.diracProba x
-- MeasurableSet.nullMeasurableSet
@[simp]
theorem _root_.MeasurableSet.nullMeasurableSet (h : MeasurableSet s) : NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasurableSet s)
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.nullMeasurableSet_empty
theorem nullMeasurableSet_empty : NullMeasurableSet ∅ μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.NullMeasurableSet ∅ μ
-- MeasureTheory.nullMeasurableSet_univ
theorem nullMeasurableSet_univ : NullMeasurableSet univ μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.NullMeasurableSet Set.univ μ
-- MeasureTheory.NullMeasurableSet.of_null
theorem of_null (h : μ s = 0) : NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : μ s = 0)
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.NullMeasurableSet.compl
theorem compl (h : NullMeasurableSet s μ) : NullMeasurableSet sᶜ μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.NullMeasurableSet sᶜ μ
-- MeasureTheory.NullMeasurableSet.of_compl
theorem of_compl (h : NullMeasurableSet sᶜ μ) : NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet sᶜ μ)
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.NullMeasurableSet.compl_iff
@[simp]
theorem compl_iff : NullMeasurableSet sᶜ μ ↔ NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: MeasureTheory.NullMeasurableSet sᶜ μ ↔ MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.NullMeasurableSet.of_subsingleton
@[nontriviality]
theorem of_subsingleton [Subsingleton α] : NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [Subsingleton α]
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.NullMeasurableSet.congr
protected theorem congr (hs : NullMeasurableSet s μ) (h : s =ᵐ[μ] t) : NullMeasurableSet t μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (h : s =ᵐ[μ] t)
Goal: MeasureTheory.NullMeasurableSet t μ
-- MeasureTheory.NullMeasurableSet.iUnion
protected theorem iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
    (h : ∀ i, NullMeasurableSet (s i) μ) : NullMeasurableSet (⋃ i, s i) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} [Countable ι] {s : ι → Set α} (h : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ)
Goal: MeasureTheory.NullMeasurableSet (⋃ i, s i) μ
-- MeasureTheory.NullMeasurableSet.biUnion
protected theorem biUnion {f : ι → Set α} {s : Set ι} (hs : s.Countable)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} {s : Set ι} (hs : s.Countable) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: MeasureTheory.NullMeasurableSet (⋃ b ∈ s, f b) μ
-- MeasureTheory.NullMeasurableSet.sUnion
protected theorem sUnion {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, NullMeasurableSet t μ) :
    NullMeasurableSet (⋃₀ s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (⋃₀ s) μ
-- MeasureTheory.NullMeasurableSet.iInter
protected theorem iInter {ι : Sort*} [Countable ι] {f : ι → Set α}
    (h : ∀ i, NullMeasurableSet (f i) μ) : NullMeasurableSet (⋂ i, f i) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Sort u_5} [Countable ι] {f : ι → Set α} (h : ∀ (i : ι), MeasureTheory.NullMeasurableSet (f i) μ)
Goal: MeasureTheory.NullMeasurableSet (⋂ i, f i) μ
-- MeasureTheory.NullMeasurableSet.biInter
protected theorem biInter {f : β → Set α} {s : Set β} (hs : s.Countable)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋂ b ∈ s, f b) μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : β → Set α} {s : Set β} (hs : s.Countable) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: MeasureTheory.NullMeasurableSet (⋂ b ∈ s, f b) μ
-- MeasureTheory.NullMeasurableSet.sInter
protected theorem sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, NullMeasurableSet t μ) :
    NullMeasurableSet (⋂₀ s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (⋂₀ s) μ
-- MeasureTheory.NullMeasurableSet.union
@[simp]
protected theorem union (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
    NullMeasurableSet (s ∪ t) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (s ∪ t) μ
-- MeasureTheory.NullMeasurableSet.union_null
protected theorem union_null (hs : NullMeasurableSet s μ) (ht : μ t = 0) :
    NullMeasurableSet (s ∪ t) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : μ t = 0)
Goal: MeasureTheory.NullMeasurableSet (s ∪ t) μ
-- MeasureTheory.NullMeasurableSet.inter
@[simp]
protected theorem inter (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
    NullMeasurableSet (s ∩ t) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (s ∩ t) μ
-- MeasureTheory.NullMeasurableSet.diff
@[simp]
protected theorem diff (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
    NullMeasurableSet (s \ t) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (s \ t) μ
-- MeasureTheory.NullMeasurableSet.disjointed
@[simp]
protected theorem disjointed {f : ℕ → Set α} (h : ∀ i, NullMeasurableSet (f i) μ) (n) :
    NullMeasurableSet (disjointed f n) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ℕ → Set α} (h : ∀ (i : ℕ), MeasureTheory.NullMeasurableSet (f i) μ) (n : ℕ)
Goal: MeasureTheory.NullMeasurableSet (disjointed f n) μ
-- MeasureTheory.NullMeasurableSet.const
protected theorem const (p : Prop) : NullMeasurableSet { _a : α | p } μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (p : Prop)
Goal: MeasureTheory.NullMeasurableSet {_a | p} μ
-- MeasureTheory.NullMeasurableSet.insert
protected theorem insert [MeasurableSingletonClass (NullMeasurableSpace α μ)]
    (hs : NullMeasurableSet s μ) (a : α) : NullMeasurableSet (insert a s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)] (hs : MeasureTheory.NullMeasurableSet s μ) (a : α)
Goal: MeasureTheory.NullMeasurableSet (insert a s) μ
-- MeasureTheory.NullMeasurableSet.exists_measurable_superset_ae_eq
theorem exists_measurable_superset_ae_eq (h : NullMeasurableSet s μ) :
    ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ)
Goal: ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s
-- MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq
theorem toMeasurable_ae_eq (h : NullMeasurableSet s μ) : toMeasurable μ s =ᵐ[μ] s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ)
Goal: MeasureTheory.toMeasurable μ s =ᵐ[μ] s
-- MeasureTheory.NullMeasurableSet.compl_toMeasurable_compl_ae_eq
theorem compl_toMeasurable_compl_ae_eq (h : NullMeasurableSet s μ) : (toMeasurable μ sᶜ)ᶜ =ᵐ[μ] s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ)
Goal: (MeasureTheory.toMeasurable μ sᶜ)ᶜ =ᵐ[μ] s
-- MeasureTheory.NullMeasurableSet.exists_measurable_subset_ae_eq
theorem exists_measurable_subset_ae_eq (h : NullMeasurableSet s μ) :
    ∃ t ⊆ s, MeasurableSet t ∧ t =ᵐ[μ] s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ)
Goal: ∃ t ⊆ s, MeasurableSet t ∧ t =ᵐ[μ] s
-- MeasureTheory.exists_subordinate_pairwise_disjoint
theorem exists_subordinate_pairwise_disjoint [Countable ι] {s : ι → Set α}
    (h : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s)) :
    ∃ t : ι → Set α,
      (∀ i, t i ⊆ s i) ∧
        (∀ i, s i =ᵐ[μ] t i) ∧ (∀ i, MeasurableSet (t i)) ∧ Pairwise (Disjoint on t) 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (h : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (hd : Pairwise (MeasureTheory.AEDisjoint μ on s))
Goal: ∃ t, (∀ (i : ι), t i ⊆ s i) ∧ (∀ (i : ι), s i =ᵐ[μ] t i) ∧ (∀ (i : ι), MeasurableSet (t i)) ∧ Pairwise (Disjoint on t)
-- MeasureTheory.measure_iUnion
theorem measure_iUnion {m0 : MeasurableSpace α} {μ : Measure α} [Countable ι] {f : ι → Set α}
    (hn : Pairwise (Disjoint on f)) (h : ∀ i, MeasurableSet (f i)) :
    μ (⋃ i, f i) = ∑' i, μ (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {f : ι → Set α} (hn : Pairwise (Disjoint on f)) (h : ∀ (i : ι), MeasurableSet (f i))
Goal: μ (⋃ i, f i) = ∑' (i : ι), μ (f i)
-- MeasureTheory.measure_iUnion₀
theorem measure_iUnion₀ [Countable ι] {f : ι → Set α} (hd : Pairwise (AEDisjoint μ on f))
    (h : ∀ i, NullMeasurableSet (f i) μ) : μ (⋃ i, f i) = ∑' i, μ (f i) 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {f : ι → Set α} (hd : Pairwise (MeasureTheory.AEDisjoint μ on f)) (h : ∀ (i : ι), MeasureTheory.NullMeasurableSet (f i) μ)
Goal: μ (⋃ i, f i) = ∑' (i : ι), μ (f i)
-- MeasureTheory.measure_union₀_aux
theorem measure_union₀_aux (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ)
    (hd : AEDisjoint μ s t) : μ (s ∪ t) = μ s + μ t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : MeasureTheory.NullMeasurableSet t μ) (hd : MeasureTheory.AEDisjoint μ s t)
Goal: μ (s ∪ t) = μ s + μ t
-- MeasureTheory.measure_inter_add_diff₀
theorem measure_inter_add_diff₀ (s : Set α) (ht : NullMeasurableSet t μ) :
    μ (s ∩ t) + μ (s \ t) = μ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: μ (s ∩ t) + μ (s \ t) = μ s
-- MeasureTheory.measure_union_add_inter₀
theorem measure_union_add_inter₀ (s : Set α) (ht : NullMeasurableSet t μ) :
    μ (s ∪ t) + μ (s ∩ t) = μ s + μ t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: μ (s ∪ t) + μ (s ∩ t) = μ s + μ t
-- MeasureTheory.measure_union_add_inter₀'
theorem measure_union_add_inter₀' (hs : NullMeasurableSet s μ) (t : Set α) :
    μ (s ∪ t) + μ (s ∩ t) = μ s + μ t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (t : Set α)
Goal: μ (s ∪ t) + μ (s ∩ t) = μ s + μ t
-- MeasureTheory.measure_union₀
theorem measure_union₀ (ht : NullMeasurableSet t μ) (hd : AEDisjoint μ s t) :
    μ (s ∪ t) = μ s + μ t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : MeasureTheory.NullMeasurableSet t μ) (hd : MeasureTheory.AEDisjoint μ s t)
Goal: μ (s ∪ t) = μ s + μ t
-- MeasureTheory.measure_union₀'
theorem measure_union₀' (hs : NullMeasurableSet s μ) (hd : AEDisjoint μ s t) :
    μ (s ∪ t) = μ s + μ t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (hd : MeasureTheory.AEDisjoint μ s t)
Goal: μ (s ∪ t) = μ s + μ t
-- MeasureTheory.measure_add_measure_compl₀
theorem measure_add_measure_compl₀ {s : Set α} (hs : NullMeasurableSet s μ) :
    μ s + μ sᶜ = μ univ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: μ s + μ sᶜ = μ Set.univ
-- MeasureTheory.nullMeasurableSet_singleton
theorem nullMeasurableSet_singleton (x : α) : NullMeasurableSet {x} μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)] (x : α)
Goal: MeasureTheory.NullMeasurableSet {x} μ
-- MeasureTheory.nullMeasurableSet_insert
@[simp]
theorem nullMeasurableSet_insert {a : α} {s : Set α} :
    NullMeasurableSet (insert a s) μ ↔ NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)] {a : α} {s : Set α}
Goal: MeasureTheory.NullMeasurableSet (insert a s) μ ↔ MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.nullMeasurableSet_eq
theorem nullMeasurableSet_eq {a : α} : NullMeasurableSet { x | x = a } μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)] {a : α}
Goal: MeasureTheory.NullMeasurableSet {x | x = a} μ
-- Set.Finite.nullMeasurableSet
protected theorem _root_.Set.Finite.nullMeasurableSet (hs : s.Finite) : NullMeasurableSet s μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)] (hs : s.Finite)
Goal: MeasureTheory.NullMeasurableSet s μ
-- Finset.nullMeasurableSet
protected theorem _root_.Finset.nullMeasurableSet (s : Finset α) : NullMeasurableSet (↑s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)] (s : Finset α)
Goal: MeasureTheory.NullMeasurableSet (↑s) μ
-- Set.Finite.nullMeasurableSet_biUnion
theorem _root_.Set.Finite.nullMeasurableSet_biUnion {f : ι → Set α} {s : Set ι} (hs : s.Finite)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} {s : Set ι} (hs : s.Finite) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: MeasureTheory.NullMeasurableSet (⋃ b ∈ s, f b) μ
-- Finset.nullMeasurableSet_biUnion
theorem _root_.Finset.nullMeasurableSet_biUnion {f : ι → Set α} (s : Finset ι)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} (s : Finset ι) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: MeasureTheory.NullMeasurableSet (⋃ b ∈ s, f b) μ
-- Set.Finite.nullMeasurableSet_sUnion
theorem _root_.Set.Finite.nullMeasurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)
    (h : ∀ t ∈ s, NullMeasurableSet t μ) : NullMeasurableSet (⋃₀ s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set (Set α)} (hs : s.Finite) (h : ∀ t ∈ s, MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (⋃₀ s) μ
-- Set.Finite.nullMeasurableSet_biInter
theorem _root_.Set.Finite.nullMeasurableSet_biInter {f : ι → Set α} {s : Set ι} (hs : s.Finite)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋂ b ∈ s, f b) μ 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} {s : Set ι} (hs : s.Finite) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: MeasureTheory.NullMeasurableSet (⋂ b ∈ s, f b) μ
-- Finset.nullMeasurableSet_biInter
theorem _root_.Finset.nullMeasurableSet_biInter {f : ι → Set α} (s : Finset ι)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋂ b ∈ s, f b) μ 
Arguments: {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} (s : Finset ι) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: MeasureTheory.NullMeasurableSet (⋂ b ∈ s, f b) μ
-- Set.Finite.nullMeasurableSet_sInter
theorem _root_.Set.Finite.nullMeasurableSet_sInter {s : Set (Set α)} (hs : s.Finite)
    (h : ∀ t ∈ s, NullMeasurableSet t μ) : NullMeasurableSet (⋂₀ s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set (Set α)} (hs : s.Finite) (h : ∀ t ∈ s, MeasureTheory.NullMeasurableSet t μ)
Goal: MeasureTheory.NullMeasurableSet (⋂₀ s) μ
-- MeasureTheory.nullMeasurableSet_toMeasurable
theorem nullMeasurableSet_toMeasurable : NullMeasurableSet (toMeasurable μ s) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: MeasureTheory.NullMeasurableSet (MeasureTheory.toMeasurable μ s) μ
-- Measurable.nullMeasurable
protected theorem _root_.Measurable.nullMeasurable (h : Measurable f) : NullMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : Measurable f)
Goal: MeasureTheory.NullMeasurable f μ
-- MeasureTheory.NullMeasurable.measurable'
protected theorem NullMeasurable.measurable' (h : NullMeasurable f μ) :
    @Measurable (NullMeasurableSpace α μ) β _ _ f 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : MeasureTheory.NullMeasurable f μ)
Goal: Measurable f
-- MeasureTheory.Measurable.comp_nullMeasurable
theorem Measurable.comp_nullMeasurable {g : β → γ} (hg : Measurable g) (hf : NullMeasurable f μ) :
    NullMeasurable (g ∘ f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {μ : MeasureTheory.Measure α} {g : β → γ} (hg : Measurable g) (hf : MeasureTheory.NullMeasurable f μ)
Goal: MeasureTheory.NullMeasurable (g ∘ f) μ
-- MeasureTheory.NullMeasurable.congr
theorem NullMeasurable.congr {g : α → β} (hf : NullMeasurable f μ) (hg : f =ᵐ[μ] g) :
    NullMeasurable g μ 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {g : α → β} (hf : MeasureTheory.NullMeasurable f μ) (hg : f =ᵐ[μ] g)
Goal: MeasureTheory.NullMeasurable g μ
-- MeasureTheory.Measure.IsComplete.out'
class Measure.IsComplete {_ : MeasurableSpace α} (μ : Measure α) : Prop where
  out' : ∀ s, μ s = 0 → MeasurableSet s
Arguments: {α : Type u_2}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : μ.IsComplete] (s : Set α), μ s = 0 → MeasurableSet s
-- MeasureTheory.Measure.isComplete_iff
theorem Measure.isComplete_iff : μ.IsComplete ↔ ∀ s, μ s = 0 → MeasurableSet s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.IsComplete ↔ ∀ (s : Set α), μ s = 0 → MeasurableSet s
-- MeasureTheory.Measure.IsComplete.out
theorem Measure.IsComplete.out (h : μ.IsComplete) : ∀ s, μ s = 0 → MeasurableSet s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (h : μ.IsComplete) (s : Set α)
Goal: μ s = 0 → MeasurableSet s
-- MeasureTheory.measurableSet_of_null
theorem measurableSet_of_null [μ.IsComplete] (hs : μ s = 0) : MeasurableSet s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [μ.IsComplete] (hs : μ s = 0)
Goal: MeasurableSet s
-- MeasureTheory.NullMeasurableSet.measurable_of_complete
theorem NullMeasurableSet.measurable_of_complete (hs : NullMeasurableSet s μ) [μ.IsComplete] :
    MeasurableSet s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) [μ.IsComplete]
Goal: MeasurableSet s
-- MeasureTheory.NullMeasurable.measurable_of_complete
theorem NullMeasurable.measurable_of_complete [μ.IsComplete] {_m1 : MeasurableSpace β} {f : α → β}
    (hf : NullMeasurable f μ) : Measurable f 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [μ.IsComplete] {_m1 : MeasurableSpace β} {f : α → β} (hf : MeasureTheory.NullMeasurable f μ)
Goal: Measurable f
-- Measurable.congr_ae
theorem _root_.Measurable.congr_ae {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
    [_hμ : μ.IsComplete] {f g : α → β} (hf : Measurable f) (hfg : f =ᵐ[μ] g) : Measurable g 
Arguments: {α : Type u_5} {β : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [_hμ : μ.IsComplete] {f g : α → β} (hf : Measurable f) (hfg : f =ᵐ[μ] g)
Goal: Measurable g
-- MeasureTheory.Measure.coe_completion
@[simp]
theorem coe_completion {_ : MeasurableSpace α} (μ : Measure α) : ⇑μ.completion = μ 
Arguments: {α : Type u_2}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), ⇑μ.completion = ⇑μ
-- MeasureTheory.Measure.completion_apply
theorem completion_apply {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) :
    μ.completion s = μ s 
Arguments: {α : Type u_2}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α), μ.completion s = μ s
-- MeasureTheory.Measure.ae_completion
@[simp]
theorem ae_completion {_ : MeasurableSpace α} (μ : Measure α) : ae μ.completion = ae μ 
Arguments: {α : Type u_2}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), MeasureTheory.ae μ.completion = MeasureTheory.ae μ