ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim
theorem ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hm : m ≤ m0) {f : α → E} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0) (hf : MeasureTheory.FinStronglyMeasurable f (μ.trim hm))
Goal: f =ᵐ[μ] 0
-- MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero
theorem Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E} (hf : Integrable f μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} (hf : MeasureTheory.Integrable f μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.Integrable.ae_eq_of_forall_setIntegral_eq
theorem Integrable.ae_eq_of_forall_setIntegral_eq (f g : α → E) (hf : Integrable f μ)
    (hg : Integrable g μ)
    (hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f g : α → E) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero
lemma ae_eq_zero_of_forall_setIntegral_isClosed_eq_zero {μ : Measure β} {f : β → E}
    (hf : Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ x in s, f x ∂μ = 0) :
    f =ᵐ[μ] 0 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {β : Type u_3} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {μ : MeasureTheory.Measure β} {f : β → E} (hf : MeasureTheory.Integrable f μ) (h'f : ∀ (s : Set β), IsClosed s → ∫ (x : β) in s, f x ∂μ = 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
lemma ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
    [SigmaCompactSpace β] [R1Space β] {μ : Measure β} {f : β → E} (hf : Integrable f μ)
    (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
    f =ᵐ[μ] 0 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {β : Type u_3} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [SigmaCompactSpace β] [R1Space β] {μ : MeasureTheory.Measure β} {f : β → E} (hf : MeasureTheory.Integrable f μ) (h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero'
lemma ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero'
    [SigmaCompactSpace β] [R1Space β] {μ : Measure β} {f : β → E} (hf : LocallyIntegrable f μ)
    (h'f : ∀ (s : Set β), IsCompact s → ∫ x in s, f x ∂μ = 0) :
    f =ᵐ[μ] 0 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {β : Type u_3} [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [SigmaCompactSpace β] [R1Space β] {μ : MeasureTheory.Measure β} {f : β → E} (hf : MeasureTheory.LocallyIntegrable f μ) (h'f : ∀ (s : Set β), IsCompact s → ∫ (x : β) in s, f x ∂μ = 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.AEMeasurable.ae_eq_of_forall_setLIntegral_eq
theorem AEMeasurable.ae_eq_of_forall_setLIntegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
    (hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
    f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤) (hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤) (hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.withDensity_eq_iff_of_sigmaFinite
theorem withDensity_eq_iff_of_sigmaFinite [SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g
-- MeasureTheory.withDensity_eq_iff
theorem withDensity_eq_iff {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) :
    μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g
-- MeasureTheory.exists_null_pairwise_disjoint_diff
theorem exists_null_pairwise_disjoint_diff [Countable ι] {s : ι → Set α}
    (hd : Pairwise (AEDisjoint μ on s)) : ∃ t : ι → Set α, (∀ i, MeasurableSet (t i)) ∧
    (∀ i, μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i) 
Arguments: {ι : Type u_1} {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Pairwise (MeasureTheory.AEDisjoint μ on s))
Goal: ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i)
-- MeasureTheory.AEDisjoint.eq
protected theorem eq (h : AEDisjoint μ s t) : μ (s ∩ t) = 0 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: μ (s ∩ t) = 0
-- MeasureTheory.AEDisjoint.symm
@[symm]
protected theorem symm (h : AEDisjoint μ s t) : AEDisjoint μ t s 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: MeasureTheory.AEDisjoint μ t s
-- MeasureTheory.AEDisjoint.symmetric
protected theorem symmetric : Symmetric (AEDisjoint μ) 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: Symmetric (MeasureTheory.AEDisjoint μ)
-- MeasureTheory.AEDisjoint.comm
protected theorem comm : AEDisjoint μ s t ↔ AEDisjoint μ t s 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: MeasureTheory.AEDisjoint μ s t ↔ MeasureTheory.AEDisjoint μ t s
-- Disjoint.aedisjoint
protected theorem _root_.Disjoint.aedisjoint (h : Disjoint s t) : AEDisjoint μ s t 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : Disjoint s t)
Goal: MeasureTheory.AEDisjoint μ s t
-- Pairwise.aedisjoint
protected theorem _root_.Pairwise.aedisjoint {f : ι → Set α} (hf : Pairwise (Disjoint on f)) :
    Pairwise (AEDisjoint μ on f) 
Arguments: {ι : Type u_1} {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} (hf : Pairwise (Disjoint on f))
Goal: Pairwise (MeasureTheory.AEDisjoint μ on f)
-- Set.PairwiseDisjoint.aedisjoint
protected theorem _root_.Set.PairwiseDisjoint.aedisjoint {f : ι → Set α} {s : Set ι}
    (hf : s.PairwiseDisjoint f) : s.Pairwise (AEDisjoint μ on f) 
Arguments: {ι : Type u_1} {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → Set α} {s : Set ι} (hf : s.PairwiseDisjoint f)
Goal: s.Pairwise (MeasureTheory.AEDisjoint μ on f)
-- MeasureTheory.AEDisjoint.mono_ae
theorem mono_ae (h : AEDisjoint μ s t) (hu : u ≤ᵐ[μ] s) (hv : v ≤ᵐ[μ] t) : AEDisjoint μ u v 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u v : Set α} (h : MeasureTheory.AEDisjoint μ s t) (hu : u ≤ᵐ[μ] s) (hv : v ≤ᵐ[μ] t)
Goal: MeasureTheory.AEDisjoint μ u v
-- MeasureTheory.AEDisjoint.mono
protected theorem mono (h : AEDisjoint μ s t) (hu : u ⊆ s) (hv : v ⊆ t) : AEDisjoint μ u v 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u v : Set α} (h : MeasureTheory.AEDisjoint μ s t) (hu : u ⊆ s) (hv : v ⊆ t)
Goal: MeasureTheory.AEDisjoint μ u v
-- MeasureTheory.AEDisjoint.congr
protected theorem congr (h : AEDisjoint μ s t) (hu : u =ᵐ[μ] s) (hv : v =ᵐ[μ] t) :
    AEDisjoint μ u v 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u v : Set α} (h : MeasureTheory.AEDisjoint μ s t) (hu : u =ᵐ[μ] s) (hv : v =ᵐ[μ] t)
Goal: MeasureTheory.AEDisjoint μ u v
-- MeasureTheory.AEDisjoint.iUnion_left_iff
@[simp]
theorem iUnion_left_iff [Countable ι] {s : ι → Set α} :
    AEDisjoint μ (⋃ i, s i) t ↔ ∀ i, AEDisjoint μ (s i) t 
Arguments: {ι : Type u_1} {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} [Countable ι] {s : ι → Set α}
Goal: MeasureTheory.AEDisjoint μ (⋃ i, s i) t ↔ ∀ (i : ι), MeasureTheory.AEDisjoint μ (s i) t
-- MeasureTheory.AEDisjoint.iUnion_right_iff
@[simp]
theorem iUnion_right_iff [Countable ι] {t : ι → Set α} :
    AEDisjoint μ s (⋃ i, t i) ↔ ∀ i, AEDisjoint μ s (t i) 
Arguments: {ι : Type u_1} {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [Countable ι] {t : ι → Set α}
Goal: MeasureTheory.AEDisjoint μ s (⋃ i, t i) ↔ ∀ (i : ι), MeasureTheory.AEDisjoint μ s (t i)
-- MeasureTheory.AEDisjoint.union_left_iff
@[simp]
theorem union_left_iff : AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u : Set α}
Goal: MeasureTheory.AEDisjoint μ (s ∪ t) u ↔ MeasureTheory.AEDisjoint μ s u ∧ MeasureTheory.AEDisjoint μ t u
-- MeasureTheory.AEDisjoint.union_right_iff
@[simp]
theorem union_right_iff : AEDisjoint μ s (t ∪ u) ↔ AEDisjoint μ s t ∧ AEDisjoint μ s u 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u : Set α}
Goal: MeasureTheory.AEDisjoint μ s (t ∪ u) ↔ MeasureTheory.AEDisjoint μ s t ∧ MeasureTheory.AEDisjoint μ s u
-- MeasureTheory.AEDisjoint.union_left
theorem union_left (hs : AEDisjoint μ s u) (ht : AEDisjoint μ t u) : AEDisjoint μ (s ∪ t) u 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u : Set α} (hs : MeasureTheory.AEDisjoint μ s u) (ht : MeasureTheory.AEDisjoint μ t u)
Goal: MeasureTheory.AEDisjoint μ (s ∪ t) u
-- MeasureTheory.AEDisjoint.union_right
theorem union_right (ht : AEDisjoint μ s t) (hu : AEDisjoint μ s u) : AEDisjoint μ s (t ∪ u) 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u : Set α} (ht : MeasureTheory.AEDisjoint μ s t) (hu : MeasureTheory.AEDisjoint μ s u)
Goal: MeasureTheory.AEDisjoint μ s (t ∪ u)
-- MeasureTheory.AEDisjoint.diff_ae_eq_left
theorem diff_ae_eq_left (h : AEDisjoint μ s t) : (s \ t : Set α) =ᵐ[μ] s 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: s \ t =ᵐ[μ] s
-- MeasureTheory.AEDisjoint.diff_ae_eq_right
theorem diff_ae_eq_right (h : AEDisjoint μ s t) : (t \ s : Set α) =ᵐ[μ] t 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: t \ s =ᵐ[μ] t
-- MeasureTheory.AEDisjoint.measure_diff_left
theorem measure_diff_left (h : AEDisjoint μ s t) : μ (s \ t) = μ s 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: μ (s \ t) = μ s
-- MeasureTheory.AEDisjoint.measure_diff_right
theorem measure_diff_right (h : AEDisjoint μ s t) : μ (t \ s) = μ t 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: μ (t \ s) = μ t
-- MeasureTheory.AEDisjoint.exists_disjoint_diff
theorem exists_disjoint_diff (h : AEDisjoint μ s t) :
    ∃ u, MeasurableSet u ∧ μ u = 0 ∧ Disjoint (s \ u) t 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t)
Goal: ∃ u, MeasurableSet u ∧ μ u = 0 ∧ Disjoint (s \ u) t
-- MeasureTheory.AEDisjoint.of_null_right
theorem of_null_right (h : μ t = 0) : AEDisjoint μ s t 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : μ t = 0)
Goal: MeasureTheory.AEDisjoint μ s t
-- MeasureTheory.AEDisjoint.of_null_left
theorem of_null_left (h : μ s = 0) : AEDisjoint μ s t 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : μ s = 0)
Goal: MeasureTheory.AEDisjoint μ s t
-- MeasureTheory.aedisjoint_compl_left
theorem aedisjoint_compl_left : AEDisjoint μ sᶜ s 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: MeasureTheory.AEDisjoint μ sᶜ s
-- MeasureTheory.aedisjoint_compl_right
theorem aedisjoint_compl_right : AEDisjoint μ s sᶜ 
Arguments: {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: MeasureTheory.AEDisjoint μ s sᶜ
-- MeasureTheory.Measure.trim_le
structure Measure (α : Type*) [MeasurableSpace α] extends OuterMeasure α where
  m_iUnion ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    toOuterMeasure (⋃ i, f i) = ∑' i, toOuterMeasure (f i)
  trim_le : toOuterMeasure.trim ≤ toOuterMeasure
Arguments: {α : Type u_6} [MeasurableSpace α] (self : MeasureTheory.Measure α)
Goal: self.trim ≤ self.toOuterMeasure
-- MeasureTheory.Measure.m_iUnion
structure Measure (α : Type*) [MeasurableSpace α] extends OuterMeasure α where
  m_iUnion ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    toOuterMeasure (⋃ i, f i) = ∑' i, toOuterMeasure (f i)
  trim_le : toOuterMeasure.trim ≤ toOuterMeasure
Arguments: {α : Type u_6} [MeasurableSpace α] (self : MeasureTheory.Measure α) ⦃f : ℕ → Set α⦄
Goal: (∀ (i : ℕ), MeasurableSet (f i)) →
  Pairwise (Disjoint on f) → self.toOuterMeasure (⋃ i, f i) = ∑' (i : ℕ), self.toOuterMeasure (f i)
-- MeasureTheory.Measure.toOuterMeasure_injective
theorem Measure.toOuterMeasure_injective [MeasurableSpace α] :
    Injective (toOuterMeasure : Measure α → OuterMeasure α)
  
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Function.Injective MeasureTheory.Measure.toOuterMeasure
-- MeasureTheory.Measure.trimmed
theorem trimmed (μ : Measure α) : μ.toOuterMeasure.trim = μ.toOuterMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: μ.trim = μ.toOuterMeasure
-- MeasureTheory.Measure.ofMeasurable_apply
theorem ofMeasurable_apply {m : ∀ s : Set α, MeasurableSet s → ℝ≥0∞}
    {m0 : m ∅ MeasurableSet.empty = 0}
    {mU :
      ∀ ⦃f : ℕ → Set α⦄ (h : ∀ i, MeasurableSet (f i)),
        Pairwise (Disjoint on f) → m (⋃ i, f i) (MeasurableSet.iUnion h) = ∑' i, m (f i) (h i)}
    (s : Set α) (hs : MeasurableSet s) : ofMeasurable m m0 mU s = m s hs 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ℝ≥0∞} {m0 : m ∅ ⋯ = 0} {mU :
  ∀ ⦃f : ℕ → Set α⦄ (h : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯} (s : Set α) (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.ofMeasurable m m0 mU) s = m s hs
-- MeasureTheory.Measure.ext
@[ext]
theorem ext (h : ∀ s, MeasurableSet s → μ₁ s = μ₂ s) : μ₁ = μ₂ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ₁ μ₂ : MeasureTheory.Measure α} (h : ∀ (s : Set α), MeasurableSet s → μ₁ s = μ₂ s)
Goal: μ₁ = μ₂
-- MeasureTheory.Measure.ext_iff
theorem ext_iff : μ₁ = μ₂ ↔ ∀ s, MeasurableSet s → μ₁ s = μ₂ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ₁ μ₂ : MeasureTheory.Measure α}
Goal: μ₁ = μ₂ ↔ ∀ (s : Set α), MeasurableSet s → μ₁ s = μ₂ s
-- MeasureTheory.Measure.ext_iff'
theorem ext_iff' : μ₁ = μ₂ ↔ ∀ s, μ₁ s = μ₂ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ₁ μ₂ : MeasureTheory.Measure α}
Goal: μ₁ = μ₂ ↔ ∀ (s : Set α), μ₁ s = μ₂ s
-- MeasureTheory.Measure.outerMeasure_le_iff
theorem outerMeasure_le_iff {m : OuterMeasure α} : m ≤ μ.1 ↔ ∀ s, MeasurableSet s → m s ≤ μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {m : MeasureTheory.OuterMeasure α}
Goal: m ≤ μ.toOuterMeasure ↔ ∀ (s : Set α), MeasurableSet s → m s ≤ μ s
-- MeasureTheory.Measure.coe_toOuterMeasure
@[simp] theorem Measure.coe_toOuterMeasure (μ : Measure α) : ⇑μ.toOuterMeasure = μ 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: ⇑μ.toOuterMeasure = ⇑μ
-- MeasureTheory.Measure.toOuterMeasure_apply
theorem Measure.toOuterMeasure_apply (μ : Measure α) (s : Set α) :
    μ.toOuterMeasure s = μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: μ.toOuterMeasure s = μ s
-- MeasureTheory.measure_eq_trim
theorem measure_eq_trim (s : Set α) : μ s = μ.toOuterMeasure.trim s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Set α)
Goal: μ s = μ.trim s
-- MeasureTheory.measure_eq_iInf
theorem measure_eq_iInf (s : Set α) : μ s = ⨅ (t) (_ : s ⊆ t) (_ : MeasurableSet t), μ t 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Set α)
Goal: μ s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), μ t
-- MeasureTheory.measure_eq_iInf'
theorem measure_eq_iInf' (μ : Measure α) (s : Set α) :
    μ s = ⨅ t : { t // s ⊆ t ∧ MeasurableSet t }, μ t 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: μ s = ⨅ t, μ ↑t
-- MeasureTheory.measure_eq_inducedOuterMeasure
theorem measure_eq_inducedOuterMeasure :
    μ s = inducedOuterMeasure (fun s _ => μ s) MeasurableSet.empty μ.empty s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α}
Goal: μ s = (MeasureTheory.inducedOuterMeasure (fun s x => μ s) ⋯ ⋯) s
-- MeasureTheory.toOuterMeasure_eq_inducedOuterMeasure
theorem toOuterMeasure_eq_inducedOuterMeasure :
    μ.toOuterMeasure = inducedOuterMeasure (fun s _ => μ s) MeasurableSet.empty μ.empty 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
Goal: μ.toOuterMeasure = MeasureTheory.inducedOuterMeasure (fun s x => μ s) ⋯ ⋯
-- MeasureTheory.measure_eq_extend
theorem measure_eq_extend (hs : MeasurableSet s) :
    μ s = extend (fun t (_ht : MeasurableSet t) => μ t) s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: μ s = MeasureTheory.extend (fun t _ht => μ t) s
-- MeasureTheory.nonempty_of_measure_ne_zero
theorem nonempty_of_measure_ne_zero (h : μ s ≠ 0) : s.Nonempty 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (h : μ s ≠ 0)
Goal: s.Nonempty
-- MeasureTheory.measure_mono_top
theorem measure_mono_top (h : s₁ ⊆ s₂) (h₁ : μ s₁ = ∞) : μ s₂ = ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α} (h : s₁ ⊆ s₂) (h₁ : μ s₁ = ⊤)
Goal: μ s₂ = ⊤
-- MeasureTheory.measure_le_measure_union_left
@[simp, mono]
theorem measure_le_measure_union_left : μ s ≤ μ (s ∪ t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: μ s ≤ μ (s ∪ t)
-- MeasureTheory.measure_le_measure_union_right
@[simp, mono]
theorem measure_le_measure_union_right : μ t ≤ μ (s ∪ t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: μ t ≤ μ (s ∪ t)
-- MeasureTheory.exists_measurable_superset
theorem exists_measurable_superset (μ : Measure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s
-- MeasureTheory.exists_measurable_superset_forall_eq
theorem exists_measurable_superset_forall_eq [Countable ι] (μ : ι → Measure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = μ i s 
Arguments: {α : Type u_1} {ι : Sort u_5} [MeasurableSpace α] [Countable ι] (μ : ι → MeasureTheory.Measure α) (s : Set α)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), (μ i) t = (μ i) s
-- MeasureTheory.exists_measurable_superset₂
theorem exists_measurable_superset₂ (μ ν : Measure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ ν : MeasureTheory.Measure α) (s : Set α)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s
-- MeasureTheory.exists_measurable_superset_of_null
theorem exists_measurable_superset_of_null (h : μ s = 0) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (h : μ s = 0)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0
-- MeasureTheory.exists_measurable_superset_iff_measure_eq_zero
theorem exists_measurable_superset_iff_measure_eq_zero :
    (∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0) ↔ μ s = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α}
Goal: (∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0) ↔ μ s = 0
-- MeasureTheory.measure_biUnion_lt_top
theorem measure_biUnion_lt_top {s : Set β} {f : β → Set α} (hs : s.Finite)
    (hfin : ∀ i ∈ s, μ (f i) ≠ ∞) : μ (⋃ i ∈ s, f i) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set β} {f : β → Set α} (hs : s.Finite) (hfin : ∀ i ∈ s, μ (f i) ≠ ⊤)
Goal: μ (⋃ i ∈ s, f i) < ⊤
-- MeasureTheory.measure_iUnion_null_iff'
@[deprecated measure_iUnion_null_iff (since := "2024-01-14")]
theorem measure_iUnion_null_iff' {ι : Prop} {s : ι → Set α} : μ (⋃ i, s i) = 0 ↔ ∀ i, μ (s i) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {ι : Prop} {s : ι → Set α}
Goal: μ (⋃ (i : ι), s i) = 0 ↔ ∀ (i : ι), μ (s i) = 0
-- MeasureTheory.measure_union_lt_top
theorem measure_union_lt_top (hs : μ s < ∞) (ht : μ t < ∞) : μ (s ∪ t) < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} (hs : μ s < ⊤) (ht : μ t < ⊤)
Goal: μ (s ∪ t) < ⊤
-- MeasureTheory.measure_union_lt_top_iff
@[simp]
theorem measure_union_lt_top_iff : μ (s ∪ t) < ∞ ↔ μ s < ∞ ∧ μ t < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: μ (s ∪ t) < ⊤ ↔ μ s < ⊤ ∧ μ t < ⊤
-- MeasureTheory.measure_union_ne_top
theorem measure_union_ne_top (hs : μ s ≠ ∞) (ht : μ t ≠ ∞) : μ (s ∪ t) ≠ ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} (hs : μ s ≠ ⊤) (ht : μ t ≠ ⊤)
Goal: μ (s ∪ t) ≠ ⊤
-- MeasureTheory.measure_symmDiff_ne_top
theorem measure_symmDiff_ne_top (hs : μ s ≠ ∞) (ht : μ t ≠ ∞) : μ (s ∆ t) ≠ ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} (hs : μ s ≠ ⊤) (ht : μ t ≠ ⊤)
Goal: μ (symmDiff s t) ≠ ⊤
-- MeasureTheory.measure_union_eq_top_iff
@[simp]
theorem measure_union_eq_top_iff : μ (s ∪ t) = ∞ ↔ μ s = ∞ ∨ μ t = ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: μ (s ∪ t) = ⊤ ↔ μ s = ⊤ ∨ μ t = ⊤
-- MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null
theorem exists_measure_pos_of_not_measure_iUnion_null [Countable ι] {s : ι → Set α}
    (hs : μ (⋃ n, s n) ≠ 0) : ∃ n, 0 < μ (s n) 
Arguments: {α : Type u_1} {ι : Sort u_5} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hs : μ (⋃ n, s n) ≠ 0)
Goal: ∃ n, 0 < μ (s n)
-- MeasureTheory.measure_lt_top_of_subset
theorem measure_lt_top_of_subset (hst : t ⊆ s) (hs : μ s ≠ ∞) : μ t < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} (hst : t ⊆ s) (hs : μ s ≠ ⊤)
Goal: μ t < ⊤
-- MeasureTheory.measure_inter_lt_top_of_left_ne_top
theorem measure_inter_lt_top_of_left_ne_top (hs_finite : μ s ≠ ∞) : μ (s ∩ t) < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} (hs_finite : μ s ≠ ⊤)
Goal: μ (s ∩ t) < ⊤