ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsFiniteMeasure.measure_univ_lt_top
class IsFiniteMeasure (μ : Measure α) : Prop where
  measure_univ_lt_top : μ univ < ∞
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.IsFiniteMeasure μ]
Goal: μ Set.univ < ⊤
-- MeasureTheory.not_isFiniteMeasure_iff
theorem not_isFiniteMeasure_iff : ¬IsFiniteMeasure μ ↔ μ Set.univ = ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: ¬MeasureTheory.IsFiniteMeasure μ ↔ μ Set.univ = ⊤
-- MeasureTheory.measure_lt_top
theorem measure_lt_top (μ : Measure α) [IsFiniteMeasure μ] (s : Set α) : μ s < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (s : Set α)
Goal: μ s < ⊤
-- MeasureTheory.measure_ne_top
theorem measure_ne_top (μ : Measure α) [IsFiniteMeasure μ] (s : Set α) : μ s ≠ ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (s : Set α)
Goal: μ s ≠ ⊤
-- MeasureTheory.measure_compl_le_add_of_le_add
theorem measure_compl_le_add_of_le_add [IsFiniteMeasure μ] (hs : MeasurableSet s)
    (ht : MeasurableSet t) {ε : ℝ≥0∞} (h : μ s ≤ μ t + ε) : μ tᶜ ≤ μ sᶜ + ε 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [MeasureTheory.IsFiniteMeasure μ] (hs : MeasurableSet s) (ht : MeasurableSet t) {ε : ℝ≥0∞} (h : μ s ≤ μ t + ε)
Goal: μ tᶜ ≤ μ sᶜ + ε
-- MeasureTheory.measure_compl_le_add_iff
theorem measure_compl_le_add_iff [IsFiniteMeasure μ] (hs : MeasurableSet s) (ht : MeasurableSet t)
    {ε : ℝ≥0∞} : μ sᶜ ≤ μ tᶜ + ε ↔ μ t ≤ μ s + ε 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [MeasureTheory.IsFiniteMeasure μ] (hs : MeasurableSet s) (ht : MeasurableSet t) {ε : ℝ≥0∞}
Goal: μ sᶜ ≤ μ tᶜ + ε ↔ μ t ≤ μ s + ε
-- MeasureTheory.coe_measureUnivNNReal
@[simp]
theorem coe_measureUnivNNReal (μ : Measure α) [IsFiniteMeasure μ] :
    ↑(measureUnivNNReal μ) = μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ]
Goal: ↑(MeasureTheory.measureUnivNNReal μ) = μ Set.univ
-- MeasureTheory.measureUnivNNReal_zero
@[simp]
theorem measureUnivNNReal_zero : measureUnivNNReal (0 : Measure α) = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α}
Goal: MeasureTheory.measureUnivNNReal 0 = 0
-- MeasureTheory.isFiniteMeasure_of_le
theorem isFiniteMeasure_of_le (μ : Measure α) [IsFiniteMeasure μ] (h : ν ≤ μ) : IsFiniteMeasure ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {ν : MeasureTheory.Measure α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (h : ν ≤ μ)
Goal: MeasureTheory.IsFiniteMeasure ν
-- MeasureTheory.Measure.isFiniteMeasure_map
@[instance]
theorem Measure.isFiniteMeasure_map {m : MeasurableSpace α} (μ : Measure α) [IsFiniteMeasure μ]
    (f : α → β) : IsFiniteMeasure (μ.map f) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → β)
Goal: MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.map f μ)
-- MeasureTheory.measureUnivNNReal_eq_zero
@[simp]
theorem measureUnivNNReal_eq_zero [IsFiniteMeasure μ] : measureUnivNNReal μ = 0 ↔ μ = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.measureUnivNNReal μ = 0 ↔ μ = 0
-- MeasureTheory.measureUnivNNReal_pos
theorem measureUnivNNReal_pos [IsFiniteMeasure μ] (hμ : μ ≠ 0) : 0 < measureUnivNNReal μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (hμ : μ ≠ 0)
Goal: 0 < MeasureTheory.measureUnivNNReal μ
-- MeasureTheory.Measure.le_of_add_le_add_left
theorem Measure.le_of_add_le_add_left [IsFiniteMeasure μ] (A2 : μ + ν₁ ≤ μ + ν₂) : ν₁ ≤ ν₂ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν₁ ν₂ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (A2 : μ + ν₁ ≤ μ + ν₂)
Goal: ν₁ ≤ ν₂
-- MeasureTheory.summable_measure_toReal
theorem summable_measure_toReal [hμ : IsFiniteMeasure μ] {f : ℕ → Set α}
    (hf₁ : ∀ i : ℕ, MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) :
    Summable fun x => (μ (f x)).toReal 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [hμ : MeasureTheory.IsFiniteMeasure μ] {f : ℕ → Set α} (hf₁ : ∀ (i : ℕ), MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f))
Goal: Summable fun x => (μ (f x)).toReal
-- MeasureTheory.ae_eq_univ_iff_measure_eq
theorem ae_eq_univ_iff_measure_eq [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ) :
    s =ᵐ[μ] univ ↔ μ s = μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsFiniteMeasure μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: s =ᵐ[μ] Set.univ ↔ μ s = μ Set.univ
-- MeasureTheory.ae_iff_measure_eq
theorem ae_iff_measure_eq [IsFiniteMeasure μ] {p : α → Prop}
    (hp : NullMeasurableSet { a | p a } μ) : (∀ᵐ a ∂μ, p a) ↔ μ { a | p a } = μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {p : α → Prop} (hp : MeasureTheory.NullMeasurableSet {a | p a} μ)
Goal: (∀ᵐ (a : α) ∂μ, p a) ↔ μ {a | p a} = μ Set.univ
-- MeasureTheory.ae_mem_iff_measure_eq
theorem ae_mem_iff_measure_eq [IsFiniteMeasure μ] {s : Set α} (hs : NullMeasurableSet s μ) :
    (∀ᵐ a ∂μ, a ∈ s) ↔ μ s = μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: (∀ᵐ (a : α) ∂μ, a ∈ s) ↔ μ s = μ Set.univ
-- MeasureTheory.tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint
lemma tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint
    {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
    {Es : ℕ → Set X} (Es_mble : ∀ i, MeasurableSet (Es i))
    (Es_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)) :
    Tendsto (μ ∘ fun n ↦ ⋃ i ≥ n, Es i) atTop (𝓝 0) 
Arguments: {X : Type u_5} [MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasure μ] {Es : ℕ → Set X} (Es_mble : ∀ (i : ℕ), MeasurableSet (Es i)) (Es_disj : Pairwise fun n m => Disjoint (Es n) (Es m))
Goal: Filter.Tendsto (⇑μ ∘ fun n => ⋃ i, ⋃ (_ : i ≥ n), Es i) Filter.atTop (𝓝 0)
-- MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff'
theorem abs_toReal_measure_sub_le_measure_symmDiff'
    (hs : MeasurableSet s) (ht : MeasurableSet t) (hs' : μ s ≠ ∞) (ht' : μ t ≠ ∞) :
    |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (hs' : μ s ≠ ⊤) (ht' : μ t ≠ ⊤)
Goal: |(μ s).toReal - (μ t).toReal| ≤ (μ (symmDiff s t)).toReal
-- MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff
theorem abs_toReal_measure_sub_le_measure_symmDiff [IsFiniteMeasure μ]
    (hs : MeasurableSet s) (ht : MeasurableSet t) :
    |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [MeasureTheory.IsFiniteMeasure μ] (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: |(μ s).toReal - (μ t).toReal| ≤ (μ (symmDiff s t)).toReal
-- MeasureTheory.IsProbabilityMeasure.measure_univ
class IsProbabilityMeasure (μ : Measure α) : Prop where
  measure_univ : μ univ = 1
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.IsProbabilityMeasure μ]
Goal: μ Set.univ = 1
-- MeasureTheory.isProbabilityMeasure_iff
lemma isProbabilityMeasure_iff : IsProbabilityMeasure μ ↔ μ univ = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.IsProbabilityMeasure μ ↔ μ Set.univ = 1
-- MeasureTheory.IsProbabilityMeasure.ne_zero
theorem IsProbabilityMeasure.ne_zero (μ : Measure α) [IsProbabilityMeasure μ] : μ ≠ 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ]
Goal: μ ≠ 0
-- MeasureTheory.IsProbabilityMeasure.ae_neBot
theorem IsProbabilityMeasure.ae_neBot [IsProbabilityMeasure μ] : NeBot (ae μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ]
Goal: (MeasureTheory.ae μ).NeBot
-- MeasureTheory.prob_add_prob_compl
theorem prob_add_prob_compl [IsProbabilityMeasure μ] (h : MeasurableSet s) : μ s + μ sᶜ = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (h : MeasurableSet s)
Goal: μ s + μ sᶜ = 1
-- MeasureTheory.prob_le_one
theorem prob_le_one [IsProbabilityMeasure μ] : μ s ≤ 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ]
Goal: μ s ≤ 1
-- MeasureTheory.isProbabilityMeasure_map
theorem isProbabilityMeasure_map {f : α → β} (hf : AEMeasurable f μ) :
    IsProbabilityMeasure (map f μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ] {f : α → β} (hf : AEMeasurable f μ)
Goal: MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map f μ)
-- MeasureTheory.one_le_prob_iff
@[simp]
theorem one_le_prob_iff : 1 ≤ μ s ↔ μ s = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ]
Goal: 1 ≤ μ s ↔ μ s = 1
-- MeasureTheory.prob_compl_eq_one_sub₀
lemma prob_compl_eq_one_sub₀ (h : NullMeasurableSet s μ) : μ sᶜ = 1 - μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (h : MeasureTheory.NullMeasurableSet s μ)
Goal: μ sᶜ = 1 - μ s
-- MeasureTheory.prob_compl_eq_one_sub
theorem prob_compl_eq_one_sub (hs : MeasurableSet s) : μ sᶜ = 1 - μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasurableSet s)
Goal: μ sᶜ = 1 - μ s
-- MeasureTheory.prob_compl_lt_one_sub_of_lt_prob
lemma prob_compl_lt_one_sub_of_lt_prob {p : ℝ≥0∞} (hμs : p < μ s) (s_mble : MeasurableSet s) :
    μ sᶜ < 1 - p 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] {p : ℝ≥0∞} (hμs : p < μ s) (s_mble : MeasurableSet s)
Goal: μ sᶜ < 1 - p
-- MeasureTheory.prob_compl_le_one_sub_of_le_prob
lemma prob_compl_le_one_sub_of_le_prob {p : ℝ≥0∞} (hμs : p ≤ μ s) (s_mble : MeasurableSet s) :
    μ sᶜ ≤ 1 - p 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] {p : ℝ≥0∞} (hμs : p ≤ μ s) (s_mble : MeasurableSet s)
Goal: μ sᶜ ≤ 1 - p
-- MeasureTheory.prob_compl_eq_zero_iff₀
@[simp] lemma prob_compl_eq_zero_iff₀ (hs : NullMeasurableSet s μ) : μ sᶜ = 0 ↔ μ s = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: μ sᶜ = 0 ↔ μ s = 1
-- MeasureTheory.prob_compl_eq_zero_iff
@[simp] lemma prob_compl_eq_zero_iff (hs : MeasurableSet s) : μ sᶜ = 0 ↔ μ s = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasurableSet s)
Goal: μ sᶜ = 0 ↔ μ s = 1
-- MeasureTheory.prob_compl_eq_one_iff₀
@[simp] lemma prob_compl_eq_one_iff₀ (hs : NullMeasurableSet s μ) : μ sᶜ = 1 ↔ μ s = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: μ sᶜ = 1 ↔ μ s = 0
-- MeasureTheory.prob_compl_eq_one_iff
@[simp] lemma prob_compl_eq_one_iff (hs : MeasurableSet s) : μ sᶜ = 1 ↔ μ s = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasurableSet s)
Goal: μ sᶜ = 1 ↔ μ s = 0
-- MeasureTheory.mem_ae_iff_prob_eq_one₀
lemma mem_ae_iff_prob_eq_one₀ (hs : NullMeasurableSet s μ) : s ∈ ae μ ↔ μ s = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: s ∈ MeasureTheory.ae μ ↔ μ s = 1
-- MeasureTheory.mem_ae_iff_prob_eq_one
lemma mem_ae_iff_prob_eq_one (hs : MeasurableSet s) : s ∈ ae μ ↔ μ s = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.IsProbabilityMeasure μ] (hs : MeasurableSet s)
Goal: s ∈ MeasureTheory.ae μ ↔ μ s = 1
-- MeasureTheory.ae_iff_prob_eq_one
lemma ae_iff_prob_eq_one (hp : Measurable p) : (∀ᵐ a ∂μ, p a) ↔ μ {a | p a} = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ] {p : α → Prop} (hp : Measurable p)
Goal: (∀ᵐ (a : α) ∂μ, p a) ↔ μ {a | p a} = 1
-- MeasureTheory.isProbabilityMeasure_comap
lemma isProbabilityMeasure_comap (hf : Injective f) (hf' : ∀ᵐ a ∂μ, a ∈ range f)
    (hf'' : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) :
    IsProbabilityMeasure (μ.comap f) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ] {f : β → α} (hf : Function.Injective f) (hf' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range f) (hf'' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (f '' s))
Goal: MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.comap f μ)
-- MeasurableEmbedding.isProbabilityMeasure_comap
protected lemma _root_.MeasurableEmbedding.isProbabilityMeasure_comap (hf : MeasurableEmbedding f)
    (hf' : ∀ᵐ a ∂μ, a ∈ range f) : IsProbabilityMeasure (μ.comap f) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ] {f : β → α} (hf : MeasurableEmbedding f) (hf' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range f)
Goal: MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.comap f μ)
-- MeasureTheory.NoAtoms.measure_singleton
class NoAtoms {m0 : MeasurableSpace α} (μ : Measure α) : Prop where
  measure_singleton : ∀ x, μ {x} = 0
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.NoAtoms μ] (x : α)
Goal: μ {x} = 0
-- Set.Subsingleton.measure_zero
theorem _root_.Set.Subsingleton.measure_zero (hs : s.Subsingleton) (μ : Measure α) [NoAtoms μ] :
    μ s = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} (hs : s.Subsingleton) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: μ s = 0
-- MeasureTheory.Measure.restrict_singleton'
theorem Measure.restrict_singleton' {a : α} : μ.restrict {a} = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] {a : α}
Goal: μ.restrict {a} = 0
-- Set.Countable.measure_zero
theorem _root_.Set.Countable.measure_zero (h : s.Countable) (μ : Measure α) [NoAtoms μ] :
    μ s = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} (h : s.Countable) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: μ s = 0
-- Set.Countable.ae_not_mem
theorem _root_.Set.Countable.ae_not_mem (h : s.Countable) (μ : Measure α) [NoAtoms μ] :
    ∀ᵐ x ∂μ, x ∉ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} (h : s.Countable) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: ∀ᵐ (x : α) ∂μ, x ∉ s
-- Set.Countable.measure_restrict_compl
lemma _root_.Set.Countable.measure_restrict_compl (h : s.Countable) (μ : Measure α) [NoAtoms μ] :
    μ.restrict sᶜ = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} (h : s.Countable) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: μ.restrict sᶜ = μ
-- MeasureTheory.restrict_compl_singleton
@[simp]
lemma restrict_compl_singleton (a : α) : μ.restrict ({a}ᶜ) = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] (a : α)
Goal: μ.restrict {a}ᶜ = μ
-- Set.Finite.measure_zero
theorem _root_.Set.Finite.measure_zero (h : s.Finite) (μ : Measure α) [NoAtoms μ] : μ s = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} (h : s.Finite) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: μ s = 0
-- Finset.measure_zero
theorem _root_.Finset.measure_zero (s : Finset α) (μ : Measure α) [NoAtoms μ] : μ s = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (s : Finset α) (μ : MeasureTheory.Measure α) [MeasureTheory.NoAtoms μ]
Goal: μ ↑s = 0
-- MeasureTheory.insert_ae_eq_self
theorem insert_ae_eq_self (a : α) (s : Set α) : (insert a s : Set α) =ᵐ[μ] s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] (a : α) (s : Set α)
Goal: insert a s =ᵐ[μ] s
-- MeasureTheory.Iio_ae_eq_Iic
theorem Iio_ae_eq_Iic : Iio a =ᵐ[μ] Iic a 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a : α}
Goal: Set.Iio a =ᵐ[μ] Set.Iic a
-- MeasureTheory.Ioi_ae_eq_Ici
theorem Ioi_ae_eq_Ici : Ioi a =ᵐ[μ] Ici a 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a : α}
Goal: Set.Ioi a =ᵐ[μ] Set.Ici a
-- MeasureTheory.Ioo_ae_eq_Ioc
theorem Ioo_ae_eq_Ioc : Ioo a b =ᵐ[μ] Ioc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: Set.Ioo a b =ᵐ[μ] Set.Ioc a b
-- MeasureTheory.Ioc_ae_eq_Icc
theorem Ioc_ae_eq_Icc : Ioc a b =ᵐ[μ] Icc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: Set.Ioc a b =ᵐ[μ] Set.Icc a b
-- MeasureTheory.Ioo_ae_eq_Ico
theorem Ioo_ae_eq_Ico : Ioo a b =ᵐ[μ] Ico a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: Set.Ioo a b =ᵐ[μ] Set.Ico a b
-- MeasureTheory.Ioo_ae_eq_Icc
theorem Ioo_ae_eq_Icc : Ioo a b =ᵐ[μ] Icc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: Set.Ioo a b =ᵐ[μ] Set.Icc a b
-- MeasureTheory.Ico_ae_eq_Icc
theorem Ico_ae_eq_Icc : Ico a b =ᵐ[μ] Icc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: Set.Ico a b =ᵐ[μ] Set.Icc a b
-- MeasureTheory.Ico_ae_eq_Ioc
theorem Ico_ae_eq_Ioc : Ico a b =ᵐ[μ] Ioc a b 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: Set.Ico a b =ᵐ[μ] Set.Ioc a b
-- MeasureTheory.restrict_Iio_eq_restrict_Iic
theorem restrict_Iio_eq_restrict_Iic : μ.restrict (Iio a) = μ.restrict (Iic a) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a : α}
Goal: μ.restrict (Set.Iio a) = μ.restrict (Set.Iic a)
-- MeasureTheory.restrict_Ioi_eq_restrict_Ici
theorem restrict_Ioi_eq_restrict_Ici : μ.restrict (Ioi a) = μ.restrict (Ici a) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a : α}
Goal: μ.restrict (Set.Ioi a) = μ.restrict (Set.Ici a)
-- MeasureTheory.restrict_Ioo_eq_restrict_Ioc
theorem restrict_Ioo_eq_restrict_Ioc : μ.restrict (Ioo a b) = μ.restrict (Ioc a b) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: μ.restrict (Set.Ioo a b) = μ.restrict (Set.Ioc a b)
-- MeasureTheory.restrict_Ioc_eq_restrict_Icc
theorem restrict_Ioc_eq_restrict_Icc : μ.restrict (Ioc a b) = μ.restrict (Icc a b) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: μ.restrict (Set.Ioc a b) = μ.restrict (Set.Icc a b)
-- MeasureTheory.restrict_Ioo_eq_restrict_Ico
theorem restrict_Ioo_eq_restrict_Ico : μ.restrict (Ioo a b) = μ.restrict (Ico a b) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: μ.restrict (Set.Ioo a b) = μ.restrict (Set.Ico a b)
-- MeasureTheory.restrict_Ioo_eq_restrict_Icc
theorem restrict_Ioo_eq_restrict_Icc : μ.restrict (Ioo a b) = μ.restrict (Icc a b) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: μ.restrict (Set.Ioo a b) = μ.restrict (Set.Icc a b)
-- MeasureTheory.restrict_Ico_eq_restrict_Icc
theorem restrict_Ico_eq_restrict_Icc : μ.restrict (Ico a b) = μ.restrict (Icc a b) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: μ.restrict (Set.Ico a b) = μ.restrict (Set.Icc a b)
-- MeasureTheory.restrict_Ico_eq_restrict_Ioc
theorem restrict_Ico_eq_restrict_Ioc : μ.restrict (Ico a b) = μ.restrict (Ioc a b) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [PartialOrder α] {a b : α}
Goal: μ.restrict (Set.Ico a b) = μ.restrict (Set.Ioc a b)
-- MeasureTheory.uIoc_ae_eq_interval
theorem uIoc_ae_eq_interval [LinearOrder α] {a b : α} : Ι a b =ᵐ[μ] [[a, b]] 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.NoAtoms μ] [LinearOrder α] {a b : α}
Goal: Ι a b =ᵐ[μ] Set.uIcc a b
-- MeasureTheory.ite_ae_eq_of_measure_zero
theorem ite_ae_eq_of_measure_zero {γ} (f : α → γ) (g : α → γ) (s : Set α) [DecidablePred (· ∈ s)]
    (hs_zero : μ s = 0) :
    (fun x => ite (x ∈ s) (f x) (g x)) =ᵐ[μ] g 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {γ : Type u_5} (f g : α → γ) (s : Set α) [DecidablePred fun x => x ∈ s] (hs_zero : μ s = 0)
Goal: (fun x => if x ∈ s then f x else g x) =ᵐ[μ] g
-- MeasureTheory.ite_ae_eq_of_measure_compl_zero
theorem ite_ae_eq_of_measure_compl_zero {γ} (f : α → γ) (g : α → γ)
    (s : Set α) [DecidablePred (· ∈ s)] (hs_zero : μ sᶜ = 0) :
    (fun x => ite (x ∈ s) (f x) (g x)) =ᵐ[μ] f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {γ : Type u_5} (f g : α → γ) (s : Set α) [DecidablePred fun x => x ∈ s] (hs_zero : μ sᶜ = 0)
Goal: (fun x => if x ∈ s then f x else g x) =ᵐ[μ] f
-- MeasureTheory.Measure.finiteAtFilter_of_finite
theorem finiteAtFilter_of_finite {_m0 : MeasurableSpace α} (μ : Measure α) [IsFiniteMeasure μ]
    (f : Filter α) : μ.FiniteAtFilter f 
Arguments: {α : Type u_1} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : Filter α)
Goal: μ.FiniteAtFilter f
-- MeasureTheory.Measure.FiniteAtFilter.exists_mem_basis
theorem FiniteAtFilter.exists_mem_basis {f : Filter α} (hμ : FiniteAtFilter μ f) {p : ι → Prop}
    {s : ι → Set α} (hf : f.HasBasis p s) : ∃ i, p i ∧ μ (s i) < ∞ 
Arguments: {α : Type u_1} {ι : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : Filter α} (hμ : μ.FiniteAtFilter f) {p : ι → Prop} {s : ι → Set α} (hf : f.HasBasis p s)
Goal: ∃ i, p i ∧ μ (s i) < ⊤
-- MeasureTheory.Measure.finiteAtBot
theorem finiteAtBot {m0 : MeasurableSpace α} (μ : Measure α) : μ.FiniteAtFilter ⊥ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ.FiniteAtFilter ⊥
-- MeasureTheory.Measure.FiniteSpanningSetsIn.finite
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure FiniteSpanningSetsIn {m0 : MeasurableSpace α} (μ : Measure α) (C : Set (Set α)) where
  protected set : ℕ → Set α
  protected set_mem : ∀ i, set i ∈ C
  protected finite : ∀ i, μ (set i) < ∞
  protected spanning : ⋃ i, set i = univ
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {C : Set (Set α)} (self : μ.FiniteSpanningSetsIn C) (i : ℕ)
Goal: μ (self.set i) < ⊤