ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.FiniteSpanningSetsIn.spanning
-- 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)
Goal: ⋃ i, self.set i = Set.univ
-- MeasureTheory.Measure.FiniteSpanningSetsIn.set_mem
-- 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: self.set i ∈ C
-- MeasureTheory.SFinite.out'
class SFinite (μ : Measure α) : Prop where
  out' : ∃ m : ℕ → Measure α, (∀ n, IsFiniteMeasure (m n)) ∧ μ = Measure.sum m
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.SFinite μ]
Goal: ∃ m, (∀ (n : ℕ), MeasureTheory.IsFiniteMeasure (m n)) ∧ μ = MeasureTheory.Measure.sum m
-- MeasureTheory.sum_sFiniteSeq
lemma sum_sFiniteSeq (μ : Measure α) [h : SFinite μ] : sum (sFiniteSeq μ) = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [h : MeasureTheory.SFinite μ]
Goal: MeasureTheory.Measure.sum (MeasureTheory.sFiniteSeq μ) = μ
-- MeasureTheory.sFiniteSeq_zero
@[simp]
lemma sFiniteSeq_zero (n : ℕ) : sFiniteSeq (0 : Measure α) n = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (n : ℕ)
Goal: MeasureTheory.sFiniteSeq 0 n = 0
-- MeasureTheory.sfinite_sum_of_countable
lemma sfinite_sum_of_countable [Countable ι]
    (m : ι → Measure α) [∀ n, IsFiniteMeasure (m n)] : SFinite (Measure.sum m) 
Arguments: {α : Type u_1} {ι : Type u_4} {m0 : MeasurableSpace α} [Countable ι] (m : ι → MeasureTheory.Measure α) [∀ (n : ι), MeasureTheory.IsFiniteMeasure (m n)]
Goal: MeasureTheory.SFinite (MeasureTheory.Measure.sum m)
-- MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure
theorem exists_absolutelyContinuous_isFiniteMeasure [SFinite μ] :
    ∃ ν : Measure α, IsFiniteMeasure ν ∧ μ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: ∃ ν, MeasureTheory.IsFiniteMeasure ν ∧ μ ≪ ν
-- MeasureTheory.SigmaFinite.out'
class SigmaFinite {m0 : MeasurableSpace α} (μ : Measure α) : Prop where
  out' : Nonempty (μ.FiniteSpanningSetsIn univ)
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.SigmaFinite μ]
Goal: Nonempty (μ.FiniteSpanningSetsIn Set.univ)
-- MeasureTheory.sigmaFinite_iff
theorem sigmaFinite_iff : SigmaFinite μ ↔ Nonempty (μ.FiniteSpanningSetsIn univ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.SigmaFinite μ ↔ Nonempty (μ.FiniteSpanningSetsIn Set.univ)
-- MeasureTheory.SigmaFinite.out
theorem SigmaFinite.out (h : SigmaFinite μ) : Nonempty (μ.FiniteSpanningSetsIn univ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.SigmaFinite μ)
Goal: Nonempty (μ.FiniteSpanningSetsIn Set.univ)
-- MeasureTheory.monotone_spanningSets
theorem monotone_spanningSets (μ : Measure α) [SigmaFinite μ] : Monotone (spanningSets μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: Monotone (MeasureTheory.spanningSets μ)
-- MeasureTheory.measurable_spanningSets
theorem measurable_spanningSets (μ : Measure α) [SigmaFinite μ] (i : ℕ) :
    MeasurableSet (spanningSets μ i) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (i : ℕ)
Goal: MeasurableSet (MeasureTheory.spanningSets μ i)
-- MeasureTheory.measure_spanningSets_lt_top
theorem measure_spanningSets_lt_top (μ : Measure α) [SigmaFinite μ] (i : ℕ) :
    μ (spanningSets μ i) < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (i : ℕ)
Goal: μ (MeasureTheory.spanningSets μ i) < ⊤
-- MeasureTheory.iUnion_spanningSets
theorem iUnion_spanningSets (μ : Measure α) [SigmaFinite μ] : ⋃ i : ℕ, spanningSets μ i = univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ⋃ i, MeasureTheory.spanningSets μ i = Set.univ
-- MeasureTheory.isCountablySpanning_spanningSets
theorem isCountablySpanning_spanningSets (μ : Measure α) [SigmaFinite μ] :
    IsCountablySpanning (range (spanningSets μ)) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: IsCountablySpanning (Set.range (MeasureTheory.spanningSets μ))
-- MeasureTheory.measurable_spanningSetsIndex
theorem measurable_spanningSetsIndex (μ : Measure α) [SigmaFinite μ] :
    Measurable (spanningSetsIndex μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: Measurable (MeasureTheory.spanningSetsIndex μ)
-- MeasureTheory.preimage_spanningSetsIndex_singleton
theorem preimage_spanningSetsIndex_singleton (μ : Measure α) [SigmaFinite μ] (n : ℕ) :
    spanningSetsIndex μ ⁻¹' {n} = disjointed (spanningSets μ) n 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (n : ℕ)
Goal: MeasureTheory.spanningSetsIndex μ ⁻¹' {n} = disjointed (MeasureTheory.spanningSets μ) n
-- MeasureTheory.spanningSetsIndex_eq_iff
theorem spanningSetsIndex_eq_iff (μ : Measure α) [SigmaFinite μ] {x : α} {n : ℕ} :
    spanningSetsIndex μ x = n ↔ x ∈ disjointed (spanningSets μ) n 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] {x : α} {n : ℕ}
Goal: MeasureTheory.spanningSetsIndex μ x = n ↔ x ∈ disjointed (MeasureTheory.spanningSets μ) n
-- MeasureTheory.mem_disjointed_spanningSetsIndex
theorem mem_disjointed_spanningSetsIndex (μ : Measure α) [SigmaFinite μ] (x : α) :
    x ∈ disjointed (spanningSets μ) (spanningSetsIndex μ x) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (x : α)
Goal: x ∈ disjointed (MeasureTheory.spanningSets μ) (MeasureTheory.spanningSetsIndex μ x)
-- MeasureTheory.mem_spanningSetsIndex
theorem mem_spanningSetsIndex (μ : Measure α) [SigmaFinite μ] (x : α) :
    x ∈ spanningSets μ (spanningSetsIndex μ x) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (x : α)
Goal: x ∈ MeasureTheory.spanningSets μ (MeasureTheory.spanningSetsIndex μ x)
-- MeasureTheory.mem_spanningSets_of_index_le
theorem mem_spanningSets_of_index_le (μ : Measure α) [SigmaFinite μ] (x : α) {n : ℕ}
    (hn : spanningSetsIndex μ x ≤ n) : x ∈ spanningSets μ n 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (x : α) {n : ℕ} (hn : MeasureTheory.spanningSetsIndex μ x ≤ n)
Goal: x ∈ MeasureTheory.spanningSets μ n
-- MeasureTheory.eventually_mem_spanningSets
theorem eventually_mem_spanningSets (μ : Measure α) [SigmaFinite μ] (x : α) :
    ∀ᶠ n in atTop, x ∈ spanningSets μ n 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (x : α)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, x ∈ MeasureTheory.spanningSets μ n
-- MeasureTheory.sum_restrict_disjointed_spanningSets
theorem sum_restrict_disjointed_spanningSets (μ ν : Measure α) [SigmaFinite ν] :
    sum (fun n ↦ μ.restrict (disjointed (spanningSets ν) n)) = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν]
Goal: (MeasureTheory.Measure.sum fun n => μ.restrict (disjointed (MeasureTheory.spanningSets ν) n)) = μ
-- MeasureTheory.Measure.forall_measure_inter_spanningSets_eq_zero
theorem forall_measure_inter_spanningSets_eq_zero [MeasurableSpace α] {μ : Measure α}
    [SigmaFinite μ] (s : Set α) : (∀ n, μ (s ∩ spanningSets μ n) = 0) ↔ μ s = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (s : Set α)
Goal: (∀ (n : ℕ), μ (s ∩ MeasureTheory.spanningSets μ n) = 0) ↔ μ s = 0
-- MeasureTheory.Measure.exists_measure_inter_spanningSets_pos
theorem exists_measure_inter_spanningSets_pos [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ]
    (s : Set α) : (∃ n, 0 < μ (s ∩ spanningSets μ n)) ↔ 0 < μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (s : Set α)
Goal: (∃ n, 0 < μ (s ∩ MeasureTheory.spanningSets μ n)) ↔ 0 < μ s
-- MeasureTheory.Measure.finite_const_le_meas_of_disjoint_iUnion₀
theorem finite_const_le_meas_of_disjoint_iUnion₀ {ι : Type*} [MeasurableSpace α] (μ : Measure α)
    {ε : ℝ≥0∞} (ε_pos : 0 < ε) {As : ι → Set α} (As_mble : ∀ i : ι, NullMeasurableSet (As i) μ)
    (As_disj : Pairwise (AEDisjoint μ on As)) (Union_As_finite : μ (⋃ i, As i) ≠ ∞) :
    Set.Finite { i : ι | ε ≤ μ (As i) } 
Arguments: {α : Type u_1} {ι : Type u_5} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {ε : ℝ≥0∞} (ε_pos : 0 < ε) {As : ι → Set α} (As_mble : ∀ (i : ι), MeasureTheory.NullMeasurableSet (As i) μ) (As_disj : Pairwise (MeasureTheory.AEDisjoint μ on As)) (Union_As_finite : μ (⋃ i, As i) ≠ ⊤)
Goal: {i | ε ≤ μ (As i)}.Finite
-- MeasureTheory.Measure.finite_const_le_meas_of_disjoint_iUnion
theorem finite_const_le_meas_of_disjoint_iUnion {ι : Type*} [MeasurableSpace α] (μ : Measure α)
    {ε : ℝ≥0∞} (ε_pos : 0 < ε) {As : ι → Set α} (As_mble : ∀ i : ι, MeasurableSet (As i))
    (As_disj : Pairwise (Disjoint on As)) (Union_As_finite : μ (⋃ i, As i) ≠ ∞) :
    Set.Finite { i : ι | ε ≤ μ (As i) } 
Arguments: {α : Type u_1} {ι : Type u_5} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {ε : ℝ≥0∞} (ε_pos : 0 < ε) {As : ι → Set α} (As_mble : ∀ (i : ι), MeasurableSet (As i)) (As_disj : Pairwise (Disjoint on As)) (Union_As_finite : μ (⋃ i, As i) ≠ ⊤)
Goal: {i | ε ≤ μ (As i)}.Finite
-- Set.Infinite.meas_eq_top
theorem _root_.Set.Infinite.meas_eq_top [MeasurableSingletonClass α]
    {s : Set α} (hs : s.Infinite) (h' : ∃ ε, ε ≠ 0 ∧ ∀ x ∈ s, ε ≤ μ {x}) : μ s = ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] {s : Set α} (hs : s.Infinite) (h' : ∃ ε, ε ≠ 0 ∧ ∀ x ∈ s, ε ≤ μ {x})
Goal: μ s = ⊤
-- MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top₀
theorem countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top₀ {ι : Type*} {_ : MeasurableSpace α}
    (μ : Measure α) {As : ι → Set α} (As_mble : ∀ i : ι, NullMeasurableSet (As i) μ)
    (As_disj : Pairwise (AEDisjoint μ on As)) (Union_As_finite : μ (⋃ i, As i) ≠ ∞) :
    Set.Countable { i : ι | 0 < μ (As i) } 
Arguments: {α : Type u_1} {ι : Type u_5}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) {As : ι → Set α},
  (∀ (i : ι), MeasureTheory.NullMeasurableSet (As i) μ) →
    Pairwise (MeasureTheory.AEDisjoint μ on As) → μ (⋃ i, As i) ≠ ⊤ → {i | 0 < μ (As i)}.Countable
-- MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top
theorem countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top {ι : Type*} {_ : MeasurableSpace α}
    (μ : Measure α) {As : ι → Set α} (As_mble : ∀ i : ι, MeasurableSet (As i))
    (As_disj : Pairwise (Disjoint on As)) (Union_As_finite : μ (⋃ i, As i) ≠ ∞) :
    Set.Countable { i : ι | 0 < μ (As i) } 
Arguments: {α : Type u_1} {ι : Type u_5}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) {As : ι → Set α},
  (∀ (i : ι), MeasurableSet (As i)) → Pairwise (Disjoint on As) → μ (⋃ i, As i) ≠ ⊤ → {i | 0 < μ (As i)}.Countable
-- MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion₀
theorem countable_meas_pos_of_disjoint_iUnion₀ {ι : Type*} {_ : MeasurableSpace α} {μ : Measure α}
    [SFinite μ] {As : ι → Set α} (As_mble : ∀ i : ι, NullMeasurableSet (As i) μ)
    (As_disj : Pairwise (AEDisjoint μ on As)) :
    Set.Countable { i : ι | 0 < μ (As i) } 
Arguments: {α : Type u_1} {ι : Type u_5}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SFinite μ] {As : ι → Set α},
  (∀ (i : ι), MeasureTheory.NullMeasurableSet (As i) μ) →
    Pairwise (MeasureTheory.AEDisjoint μ on As) → {i | 0 < μ (As i)}.Countable
-- MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion
theorem countable_meas_pos_of_disjoint_iUnion {ι : Type*} {_ : MeasurableSpace α} {μ : Measure α}
    [SFinite μ] {As : ι → Set α} (As_mble : ∀ i : ι, MeasurableSet (As i))
    (As_disj : Pairwise (Disjoint on As)) : Set.Countable { i : ι | 0 < μ (As i) } 
Arguments: {α : Type u_1} {ι : Type u_5}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SFinite μ] {As : ι → Set α},
  (∀ (i : ι), MeasurableSet (As i)) → Pairwise (Disjoint on As) → {i | 0 < μ (As i)}.Countable
-- MeasureTheory.Measure.countable_meas_level_set_pos₀
theorem countable_meas_level_set_pos₀ {α β : Type*} {_ : MeasurableSpace α} {μ : Measure α}
    [SFinite μ] [MeasurableSpace β] [MeasurableSingletonClass β] {g : α → β}
    (g_mble : NullMeasurable g μ) : Set.Countable { t : β | 0 < μ { a : α | g a = t } } 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SFinite μ] [inst : MeasurableSpace β]
  [inst_1 : MeasurableSingletonClass β] {g : α → β},
  MeasureTheory.NullMeasurable g μ → {t | 0 < μ {a | g a = t}}.Countable
-- MeasureTheory.Measure.countable_meas_level_set_pos
theorem countable_meas_level_set_pos {α β : Type*} {_ : MeasurableSpace α} {μ : Measure α}
    [SFinite μ] [MeasurableSpace β] [MeasurableSingletonClass β] {g : α → β}
    (g_mble : Measurable g) : Set.Countable { t : β | 0 < μ { a : α | g a = t } } 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SFinite μ] [inst : MeasurableSpace β]
  [inst_1 : MeasurableSingletonClass β] {g : α → β}, Measurable g → {t | 0 < μ {a | g a = t}}.Countable
-- MeasureTheory.Measure.measure_toMeasurable_inter_of_sum
theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) {t : Set α} {m : ℕ → MeasureTheory.Measure α} (hv : ∀ (n : ℕ), (m n) t ≠ ⊤) (hμ : μ = MeasureTheory.Measure.sum m)
Goal: μ (MeasureTheory.toMeasurable μ t ∩ s) = μ (t ∩ s)
-- MeasureTheory.Measure.measure_toMeasurable_inter_of_cover
theorem measure_toMeasurable_inter_of_cover {s : Set α} (hs : MeasurableSet s) {t : Set α}
    {v : ℕ → Set α} (hv : t ⊆ ⋃ n, v n) (h'v : ∀ n, μ (t ∩ v n) ≠ ∞) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) {t : Set α} {v : ℕ → Set α} (hv : t ⊆ ⋃ n, v n) (h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ⊤)
Goal: μ (MeasureTheory.toMeasurable μ t ∩ s) = μ (t ∩ s)
-- MeasureTheory.Measure.restrict_toMeasurable_of_cover
theorem restrict_toMeasurable_of_cover {s : Set α} {v : ℕ → Set α} (hv : s ⊆ ⋃ n, v n)
    (h'v : ∀ n, μ (s ∩ v n) ≠ ∞) : μ.restrict (toMeasurable μ s) = μ.restrict s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {v : ℕ → Set α} (hv : s ⊆ ⋃ n, v n) (h'v : ∀ (n : ℕ), μ (s ∩ v n) ≠ ⊤)
Goal: μ.restrict (MeasureTheory.toMeasurable μ s) = μ.restrict s
-- MeasureTheory.Measure.measure_toMeasurable_inter_of_sFinite
theorem measure_toMeasurable_inter_of_sFinite [SFinite μ] {s : Set α} (hs : MeasurableSet s)
    (t : Set α) : μ (toMeasurable μ t ∩ s) = μ (t ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {s : Set α} (hs : MeasurableSet s) (t : Set α)
Goal: μ (MeasureTheory.toMeasurable μ t ∩ s) = μ (t ∩ s)
-- MeasureTheory.Measure.restrict_toMeasurable_of_sFinite
@[simp]
theorem restrict_toMeasurable_of_sFinite [SFinite μ] (s : Set α) :
    μ.restrict (toMeasurable μ s) = μ.restrict s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (s : Set α)
Goal: μ.restrict (MeasureTheory.toMeasurable μ s) = μ.restrict s
-- MeasureTheory.Measure.iSup_restrict_spanningSets_of_measurableSet
theorem iSup_restrict_spanningSets_of_measurableSet [SigmaFinite μ] (hs : MeasurableSet s) :
    ⨆ i, μ.restrict (spanningSets μ i) s = μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.SigmaFinite μ] (hs : MeasurableSet s)
Goal: ⨆ i, (μ.restrict (MeasureTheory.spanningSets μ i)) s = μ s
-- MeasureTheory.Measure.iSup_restrict_spanningSets
theorem iSup_restrict_spanningSets [SigmaFinite μ] (s : Set α) :
    ⨆ i, μ.restrict (spanningSets μ i) s = μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (s : Set α)
Goal: ⨆ i, (μ.restrict (MeasureTheory.spanningSets μ i)) s = μ s
-- MeasureTheory.Measure.exists_subset_measure_lt_top
theorem exists_subset_measure_lt_top [SigmaFinite μ] {r : ℝ≥0∞} (hs : MeasurableSet s)
    (h's : r < μ s) : ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < μ t ∧ μ t < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.SigmaFinite μ] {r : ℝ≥0∞} (hs : MeasurableSet s) (h's : r < μ s)
Goal: ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < μ t ∧ μ t < ⊤
-- MeasureTheory.Measure.FiniteSpanningSetsIn.sigmaFinite
protected theorem sigmaFinite (h : μ.FiniteSpanningSetsIn C) : SigmaFinite μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {C : Set (Set α)} (h : μ.FiniteSpanningSetsIn C)
Goal: MeasureTheory.SigmaFinite μ
-- MeasureTheory.Measure.FiniteSpanningSetsIn.ext
protected theorem ext {ν : Measure α} {C : Set (Set α)} (hA : ‹_› = generateFrom C)
    (hC : IsPiSystem C) (h : μ.FiniteSpanningSetsIn C) (h_eq : ∀ s ∈ C, μ s = ν s) : μ = ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {C : Set (Set α)} (hA : m0 = MeasurableSpace.generateFrom C) (hC : IsPiSystem C) (h : μ.FiniteSpanningSetsIn C) (h_eq : ∀ s ∈ C, μ s = ν s)
Goal: μ = ν
-- MeasureTheory.Measure.FiniteSpanningSetsIn.isCountablySpanning
protected theorem isCountablySpanning (h : μ.FiniteSpanningSetsIn C) : IsCountablySpanning C 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {C : Set (Set α)} (h : μ.FiniteSpanningSetsIn C)
Goal: IsCountablySpanning C
-- MeasureTheory.Measure.sigmaFinite_of_countable
theorem sigmaFinite_of_countable {S : Set (Set α)} (hc : S.Countable) (hμ : ∀ s ∈ S, μ s < ∞)
    (hU : ⋃₀ S = univ) : SigmaFinite μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {S : Set (Set α)} (hc : S.Countable) (hμ : ∀ s ∈ S, μ s < ⊤) (hU : ⋃₀ S = Set.univ)
Goal: MeasureTheory.SigmaFinite μ
-- MeasureTheory.Measure.sigmaFinite_of_le
theorem sigmaFinite_of_le (μ : Measure α) [hs : SigmaFinite μ] (h : ν ≤ μ) : SigmaFinite ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {ν : MeasureTheory.Measure α} (μ : MeasureTheory.Measure α) [hs : MeasureTheory.SigmaFinite μ] (h : ν ≤ μ)
Goal: MeasureTheory.SigmaFinite ν
-- MeasureTheory.Measure.add_right_inj
@[simp] lemma add_right_inj (μ ν₁ ν₂ : Measure α) [SigmaFinite μ] :
    μ + ν₁ = μ + ν₂ ↔ ν₁ = ν₂ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ ν₁ ν₂ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: μ + ν₁ = μ + ν₂ ↔ ν₁ = ν₂
-- MeasureTheory.Measure.add_left_inj
@[simp] lemma add_left_inj (μ ν₁ ν₂ : Measure α) [SigmaFinite μ] :
    ν₁ + μ = ν₂ + μ ↔ ν₁ = ν₂ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ ν₁ ν₂ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ν₁ + μ = ν₂ + μ ↔ ν₁ = ν₂
-- MeasureTheory.sigmaFinite_bot_iff
theorem sigmaFinite_bot_iff (μ : @Measure α ⊥) : SigmaFinite μ ↔ IsFiniteMeasure μ 
Arguments: {α : Type u_1} (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.SigmaFinite μ ↔ MeasureTheory.IsFiniteMeasure μ
-- MeasureTheory.SigmaFinite.of_map
theorem SigmaFinite.of_map (μ : Measure α) {f : α → β} (hf : AEMeasurable f μ)
    (h : SigmaFinite (μ.map f)) : SigmaFinite μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] (μ : MeasureTheory.Measure α) {f : α → β} (hf : AEMeasurable f μ) (h : MeasureTheory.SigmaFinite (MeasureTheory.Measure.map f μ))
Goal: MeasureTheory.SigmaFinite μ
-- MeasurableEmbedding.sigmaFinite_map
lemma _root_.MeasurableEmbedding.sigmaFinite_map {f : α → β} (hf : MeasurableEmbedding f)
    [SigmaFinite μ] :
    SigmaFinite (μ.map f) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : MeasurableEmbedding f) [MeasureTheory.SigmaFinite μ]
Goal: MeasureTheory.SigmaFinite (MeasureTheory.Measure.map f μ)
-- MeasurableEquiv.sigmaFinite_map
theorem _root_.MeasurableEquiv.sigmaFinite_map (f : α ≃ᵐ β) [SigmaFinite μ] :
    SigmaFinite (μ.map f) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} (f : α ≃ᵐ β) [MeasureTheory.SigmaFinite μ]
Goal: MeasureTheory.SigmaFinite (MeasureTheory.Measure.map (⇑f) μ)
-- MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict'
theorem ae_of_forall_measure_lt_top_ae_restrict' {μ : Measure α} (ν : Measure α) [SigmaFinite μ]
    [SigmaFinite ν] (P : α → Prop)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ν s < ∞ → ∀ᵐ x ∂μ.restrict s, P x) : ∀ᵐ x ∂μ, P x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (P : α → Prop) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x)
Goal: ∀ᵐ (x : α) ∂μ, P x
-- MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict
theorem ae_of_forall_measure_lt_top_ae_restrict {μ : Measure α} [SigmaFinite μ] (P : α → Prop)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → ∀ᵐ x ∂μ.restrict s, P x) : ∀ᵐ x ∂μ, P x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (P : α → Prop) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x)
Goal: ∀ᵐ (x : α) ∂μ, P x
-- MeasureTheory.IsLocallyFiniteMeasure.finiteAtNhds
class IsLocallyFiniteMeasure [TopologicalSpace α] (μ : Measure α) : Prop where
  finiteAtNhds : ∀ x, μ.FiniteAtFilter (𝓝 x)
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : MeasureTheory.IsLocallyFiniteMeasure μ] (x : α)
Goal: μ.FiniteAtFilter (𝓝 x)
-- MeasureTheory.Measure.finiteAt_nhds
theorem Measure.finiteAt_nhds [TopologicalSpace α] (μ : Measure α) [IsLocallyFiniteMeasure μ]
    (x : α) : μ.FiniteAtFilter (𝓝 x) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (x : α)
Goal: μ.FiniteAtFilter (𝓝 x)
-- MeasureTheory.Measure.smul_finite
theorem Measure.smul_finite (μ : Measure α) [IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) :
    IsFiniteMeasure (c • μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: MeasureTheory.IsFiniteMeasure (c • μ)
-- MeasureTheory.Measure.exists_isOpen_measure_lt_top
theorem Measure.exists_isOpen_measure_lt_top [TopologicalSpace α] (μ : Measure α)
    [IsLocallyFiniteMeasure μ] (x : α) : ∃ s : Set α, x ∈ s ∧ IsOpen s ∧ μ s < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (x : α)
Goal: ∃ s, x ∈ s ∧ IsOpen s ∧ μ s < ⊤