ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.restrict_inter_add_diff₀
theorem restrict_inter_add_diff₀ (s : Set α) (ht : NullMeasurableSet t μ) :
    μ.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict 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.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict s
-- MeasureTheory.Measure.restrict_inter_add_diff
theorem restrict_inter_add_diff (s : Set α) (ht : MeasurableSet t) :
    μ.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t)
Goal: μ.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict s
-- MeasureTheory.Measure.restrict_union_add_inter₀
theorem restrict_union_add_inter₀ (s : Set α) (ht : NullMeasurableSet t μ) :
    μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict 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.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t
-- MeasureTheory.Measure.restrict_union_add_inter
theorem restrict_union_add_inter (s : Set α) (ht : MeasurableSet t) :
    μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t)
Goal: μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t
-- MeasureTheory.Measure.restrict_union_add_inter'
theorem restrict_union_add_inter' (hs : MeasurableSet s) (t : Set α) :
    μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (t : Set α)
Goal: μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t
-- MeasureTheory.Measure.restrict_union₀
theorem restrict_union₀ (h : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
    μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t
-- MeasureTheory.Measure.restrict_union
theorem restrict_union (h : Disjoint s t) (ht : MeasurableSet t) :
    μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : Disjoint s t) (ht : MeasurableSet t)
Goal: μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t
-- MeasureTheory.Measure.restrict_union'
theorem restrict_union' (h : Disjoint s t) (hs : MeasurableSet s) :
    μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h : Disjoint s t) (hs : MeasurableSet s)
Goal: μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t
-- MeasureTheory.Measure.restrict_add_restrict_compl
@[simp]
theorem restrict_add_restrict_compl (hs : MeasurableSet s) :
    μ.restrict s + μ.restrict sᶜ = μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: μ.restrict s + μ.restrict sᶜ = μ
-- MeasureTheory.Measure.restrict_compl_add_restrict
@[simp]
theorem restrict_compl_add_restrict (hs : MeasurableSet s) : μ.restrict sᶜ + μ.restrict s = μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: μ.restrict sᶜ + μ.restrict s = μ
-- MeasureTheory.Measure.restrict_union_le
theorem restrict_union_le (s s' : Set α) : μ.restrict (s ∪ s') ≤ μ.restrict s + μ.restrict s' 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s s' : Set α)
Goal: μ.restrict (s ∪ s') ≤ μ.restrict s + μ.restrict s'
-- MeasureTheory.Measure.restrict_iUnion_apply_ae
theorem restrict_iUnion_apply_ae [Countable ι] {s : ι → Set α} (hd : Pairwise (AEDisjoint μ on s))
    (hm : ∀ i, NullMeasurableSet (s i) μ) {t : Set α} (ht : MeasurableSet t) :
    μ.restrict (⋃ i, s i) t = ∑' i, μ.restrict (s i) t 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Pairwise (MeasureTheory.AEDisjoint μ on s)) (hm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) {t : Set α} (ht : MeasurableSet t)
Goal: (μ.restrict (⋃ i, s i)) t = ∑' (i : ι), (μ.restrict (s i)) t
-- MeasureTheory.Measure.restrict_iUnion_apply
theorem restrict_iUnion_apply [Countable ι] {s : ι → Set α} (hd : Pairwise (Disjoint on s))
    (hm : ∀ i, MeasurableSet (s i)) {t : Set α} (ht : MeasurableSet t) :
    μ.restrict (⋃ i, s i) t = ∑' i, μ.restrict (s i) t 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Pairwise (Disjoint on s)) (hm : ∀ (i : ι), MeasurableSet (s i)) {t : Set α} (ht : MeasurableSet t)
Goal: (μ.restrict (⋃ i, s i)) t = ∑' (i : ι), (μ.restrict (s i)) t
-- MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup
theorem restrict_iUnion_apply_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s)
    {t : Set α} (ht : MeasurableSet t) : μ.restrict (⋃ i, s i) t = ⨆ i, μ.restrict (s i) t 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Directed (fun x x_1 => x ⊆ x_1) s) {t : Set α} (ht : MeasurableSet t)
Goal: (μ.restrict (⋃ i, s i)) t = ⨆ i, (μ.restrict (s i)) t
-- MeasureTheory.Measure.restrict_map
theorem restrict_map {f : α → β} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) :
    (μ.map f).restrict s = (μ.restrict <| f ⁻¹' s).map f 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.map f μ).restrict s = MeasureTheory.Measure.map f (μ.restrict (f ⁻¹' s))
-- MeasureTheory.Measure.restrict_toMeasurable
theorem restrict_toMeasurable (h : μ s ≠ ∞) : μ.restrict (toMeasurable μ s) = μ.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : μ s ≠ ⊤)
Goal: μ.restrict (MeasureTheory.toMeasurable μ s) = μ.restrict s
-- MeasureTheory.Measure.restrict_eq_self_of_ae_mem
theorem restrict_eq_self_of_ae_mem {_m0 : MeasurableSpace α} ⦃s : Set α⦄ ⦃μ : Measure α⦄
    (hs : ∀ᵐ x ∂μ, x ∈ s) : μ.restrict s = μ 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} ⦃s : Set α⦄ ⦃μ : MeasureTheory.Measure α⦄ (hs : ∀ᵐ (x : α) ∂μ, x ∈ s)
Goal: μ.restrict s = μ
-- MeasureTheory.Measure.restrict_congr_meas
theorem restrict_congr_meas (hs : MeasurableSet s) :
    μ.restrict s = ν.restrict s ↔ ∀ t ⊆ s, MeasurableSet t → μ t = ν t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: μ.restrict s = ν.restrict s ↔ ∀ t ⊆ s, MeasurableSet t → μ t = ν t
-- MeasureTheory.Measure.restrict_congr_mono
theorem restrict_congr_mono (hs : s ⊆ t) (h : μ.restrict t = ν.restrict t) :
    μ.restrict s = ν.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s t : Set α} (hs : s ⊆ t) (h : μ.restrict t = ν.restrict t)
Goal: μ.restrict s = ν.restrict s
-- MeasureTheory.Measure.restrict_union_congr
theorem restrict_union_congr :
    μ.restrict (s ∪ t) = ν.restrict (s ∪ t) ↔
      μ.restrict s = ν.restrict s ∧ μ.restrict t = ν.restrict t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s t : Set α}
Goal: μ.restrict (s ∪ t) = ν.restrict (s ∪ t) ↔ μ.restrict s = ν.restrict s ∧ μ.restrict t = ν.restrict t
-- MeasureTheory.Measure.restrict_finset_biUnion_congr
theorem restrict_finset_biUnion_congr {s : Finset ι} {t : ι → Set α} :
    μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔
      ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s : Finset ι} {t : ι → Set α}
Goal: μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔ ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i)
-- MeasureTheory.Measure.restrict_iUnion_congr
theorem restrict_iUnion_congr [Countable ι] {s : ι → Set α} :
    μ.restrict (⋃ i, s i) = ν.restrict (⋃ i, s i) ↔ ∀ i, μ.restrict (s i) = ν.restrict (s i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α}
Goal: μ.restrict (⋃ i, s i) = ν.restrict (⋃ i, s i) ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
-- MeasureTheory.Measure.restrict_biUnion_congr
theorem restrict_biUnion_congr {s : Set ι} {t : ι → Set α} (hc : s.Countable) :
    μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔
      ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s : Set ι} {t : ι → Set α} (hc : s.Countable)
Goal: μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔ ∀ i ∈ s, μ.restrict (t i) = ν.restrict (t i)
-- MeasureTheory.Measure.restrict_sUnion_congr
theorem restrict_sUnion_congr {S : Set (Set α)} (hc : S.Countable) :
    μ.restrict (⋃₀ S) = ν.restrict (⋃₀ S) ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {S : Set (Set α)} (hc : S.Countable)
Goal: μ.restrict (⋃₀ S) = ν.restrict (⋃₀ S) ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s
-- MeasureTheory.Measure.restrict_sInf_eq_sInf_restrict
theorem restrict_sInf_eq_sInf_restrict {m0 : MeasurableSpace α} {m : Set (Measure α)}
    (hm : m.Nonempty) (ht : MeasurableSet t) :
    (sInf m).restrict t = sInf ((fun μ : Measure α => μ.restrict t) '' m) 
Arguments: {α : Type u_2} {t : Set α} {m0 : MeasurableSpace α} {m : Set (MeasureTheory.Measure α)} (hm : m.Nonempty) (ht : MeasurableSet t)
Goal: (sInf m).restrict t = sInf ((fun μ => μ.restrict t) '' m)
-- MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae
theorem exists_mem_of_measure_ne_zero_of_ae (hs : μ s ≠ 0) {p : α → Prop}
    (hp : ∀ᵐ x ∂μ.restrict s, p x) : ∃ x, x ∈ s ∧ p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : μ s ≠ 0) {p : α → Prop} (hp : ∀ᵐ (x : α) ∂μ.restrict s, p x)
Goal: ∃ x ∈ s, p x
-- MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ
theorem ext_iff_of_iUnion_eq_univ [Countable ι] {s : ι → Set α} (hs : ⋃ i, s i = univ) :
    μ = ν ↔ ∀ i, μ.restrict (s i) = ν.restrict (s i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hs : ⋃ i, s i = Set.univ)
Goal: μ = ν ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
-- MeasureTheory.Measure.ext_iff_of_biUnion_eq_univ
theorem ext_iff_of_biUnion_eq_univ {S : Set ι} {s : ι → Set α} (hc : S.Countable)
    (hs : ⋃ i ∈ S, s i = univ) : μ = ν ↔ ∀ i ∈ S, μ.restrict (s i) = ν.restrict (s i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {S : Set ι} {s : ι → Set α} (hc : S.Countable) (hs : ⋃ i ∈ S, s i = Set.univ)
Goal: μ = ν ↔ ∀ i ∈ S, μ.restrict (s i) = ν.restrict (s i)
-- MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ
theorem ext_iff_of_sUnion_eq_univ {S : Set (Set α)} (hc : S.Countable) (hs : ⋃₀ S = univ) :
    μ = ν ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {S : Set (Set α)} (hc : S.Countable) (hs : ⋃₀ S = Set.univ)
Goal: μ = ν ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s
-- MeasureTheory.Measure.ext_of_generateFrom_of_cover
theorem ext_of_generateFrom_of_cover {S T : Set (Set α)} (h_gen : ‹_› = generateFrom S)
    (hc : T.Countable) (h_inter : IsPiSystem S) (hU : ⋃₀ T = univ) (htop : ∀ t ∈ T, μ t ≠ ∞)
    (ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)) (T_eq : ∀ t ∈ T, μ t = ν t) : μ = ν 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {S T : Set (Set α)} (h_gen : m0 = MeasurableSpace.generateFrom S) (hc : T.Countable) (h_inter : IsPiSystem S) (hU : ⋃₀ T = Set.univ) (htop : ∀ t ∈ T, μ t ≠ ⊤) (ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)) (T_eq : ∀ t ∈ T, μ t = ν t)
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_generateFrom_of_cover_subset
theorem ext_of_generateFrom_of_cover_subset {S T : Set (Set α)} (h_gen : ‹_› = generateFrom S)
    (h_inter : IsPiSystem S) (h_sub : T ⊆ S) (hc : T.Countable) (hU : ⋃₀ T = univ)
    (htop : ∀ s ∈ T, μ s ≠ ∞) (h_eq : ∀ s ∈ S, μ s = ν s) : μ = ν 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {S T : Set (Set α)} (h_gen : m0 = MeasurableSpace.generateFrom S) (h_inter : IsPiSystem S) (h_sub : T ⊆ S) (hc : T.Countable) (hU : ⋃₀ T = Set.univ) (htop : ∀ s ∈ T, μ s ≠ ⊤) (h_eq : ∀ s ∈ S, μ s = ν s)
Goal: μ = ν
-- MeasureTheory.Measure.ext_of_generateFrom_of_iUnion
theorem ext_of_generateFrom_of_iUnion (C : Set (Set α)) (B : ℕ → Set α) (hA : ‹_› = generateFrom C)
    (hC : IsPiSystem C) (h1B : ⋃ i, B i = univ) (h2B : ∀ i, B i ∈ C) (hμB : ∀ i, μ (B i) ≠ ∞)
    (h_eq : ∀ s ∈ C, μ s = ν s) : μ = ν 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (C : Set (Set α)) (B : ℕ → Set α) (hA : m0 = MeasurableSpace.generateFrom C) (hC : IsPiSystem C) (h1B : ⋃ i, B i = Set.univ) (h2B : ∀ (i : ℕ), B i ∈ C) (hμB : ∀ (i : ℕ), μ (B i) ≠ ⊤) (h_eq : ∀ s ∈ C, μ s = ν s)
Goal: μ = ν
-- MeasureTheory.Measure.restrict_sum
@[simp]
theorem restrict_sum (μ : ι → Measure α) {s : Set α} (hs : MeasurableSet s) :
    (sum μ).restrict s = sum fun i => (μ i).restrict s 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} (μ : ι → MeasureTheory.Measure α) {s : Set α} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.sum μ).restrict s = MeasureTheory.Measure.sum fun i => (μ i).restrict s
-- MeasureTheory.Measure.restrict_sum_of_countable
@[simp]
theorem restrict_sum_of_countable [Countable ι] (μ : ι → Measure α) (s : Set α) :
    (sum μ).restrict s = sum fun i => (μ i).restrict s 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} [Countable ι] (μ : ι → MeasureTheory.Measure α) (s : Set α)
Goal: (MeasureTheory.Measure.sum μ).restrict s = MeasureTheory.Measure.sum fun i => (μ i).restrict s
-- MeasureTheory.Measure.AbsolutelyContinuous.restrict
lemma AbsolutelyContinuous.restrict (h : μ ≪ ν) (s : Set α) : μ.restrict s ≪ ν.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ ≪ ν) (s : Set α)
Goal: μ.restrict s ≪ ν.restrict s
-- MeasureTheory.Measure.restrict_iUnion_ae
theorem restrict_iUnion_ae [Countable ι] {s : ι → Set α} (hd : Pairwise (AEDisjoint μ on s))
    (hm : ∀ i, NullMeasurableSet (s i) μ) : μ.restrict (⋃ i, s i) = sum fun i => μ.restrict (s i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Pairwise (MeasureTheory.AEDisjoint μ on s)) (hm : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ)
Goal: μ.restrict (⋃ i, s i) = MeasureTheory.Measure.sum fun i => μ.restrict (s i)
-- MeasureTheory.Measure.restrict_iUnion
theorem restrict_iUnion [Countable ι] {s : ι → Set α} (hd : Pairwise (Disjoint on s))
    (hm : ∀ i, MeasurableSet (s i)) : μ.restrict (⋃ i, s i) = sum fun i => μ.restrict (s i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Pairwise (Disjoint on s)) (hm : ∀ (i : ι), MeasurableSet (s i))
Goal: μ.restrict (⋃ i, s i) = MeasureTheory.Measure.sum fun i => μ.restrict (s i)
-- MeasureTheory.Measure.restrict_iUnion_le
theorem restrict_iUnion_le [Countable ι] {s : ι → Set α} :
    μ.restrict (⋃ i, s i) ≤ sum fun i => μ.restrict (s i) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α}
Goal: μ.restrict (⋃ i, s i) ≤ MeasureTheory.Measure.sum fun i => μ.restrict (s i)
-- MeasureTheory.ae_restrict_iUnion_eq
@[simp]
theorem ae_restrict_iUnion_eq [Countable ι] (s : ι → Set α) :
    ae (μ.restrict (⋃ i, s i)) = ⨆ i, ae (μ.restrict (s i)) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] (s : ι → Set α)
Goal: MeasureTheory.ae (μ.restrict (⋃ i, s i)) = ⨆ i, MeasureTheory.ae (μ.restrict (s i))
-- MeasureTheory.ae_restrict_union_eq
@[simp]
theorem ae_restrict_union_eq (s t : Set α) :
    ae (μ.restrict (s ∪ t)) = ae (μ.restrict s) ⊔ ae (μ.restrict t) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α)
Goal: MeasureTheory.ae (μ.restrict (s ∪ t)) = MeasureTheory.ae (μ.restrict s) ⊔ MeasureTheory.ae (μ.restrict t)
-- MeasureTheory.ae_restrict_biUnion_eq
theorem ae_restrict_biUnion_eq (s : ι → Set α) {t : Set ι} (ht : t.Countable) :
    ae (μ.restrict (⋃ i ∈ t, s i)) = ⨆ i ∈ t, ae (μ.restrict (s i)) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ι → Set α) {t : Set ι} (ht : t.Countable)
Goal: MeasureTheory.ae (μ.restrict (⋃ i ∈ t, s i)) = ⨆ i ∈ t, MeasureTheory.ae (μ.restrict (s i))
-- MeasureTheory.ae_restrict_biUnion_finset_eq
theorem ae_restrict_biUnion_finset_eq (s : ι → Set α) (t : Finset ι) :
    ae (μ.restrict (⋃ i ∈ t, s i)) = ⨆ i ∈ t, ae (μ.restrict (s i)) 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ι → Set α) (t : Finset ι)
Goal: MeasureTheory.ae (μ.restrict (⋃ i ∈ t, s i)) = ⨆ i ∈ t, MeasureTheory.ae (μ.restrict (s i))
-- MeasureTheory.ae_restrict_iUnion_iff
theorem ae_restrict_iUnion_iff [Countable ι] (s : ι → Set α) (p : α → Prop) :
    (∀ᵐ x ∂μ.restrict (⋃ i, s i), p x) ↔ ∀ i, ∀ᵐ x ∂μ.restrict (s i), p x 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] (s : ι → Set α) (p : α → Prop)
Goal: (∀ᵐ (x : α) ∂μ.restrict (⋃ i, s i), p x) ↔ ∀ (i : ι), ∀ᵐ (x : α) ∂μ.restrict (s i), p x
-- MeasureTheory.ae_restrict_union_iff
theorem ae_restrict_union_iff (s t : Set α) (p : α → Prop) :
    (∀ᵐ x ∂μ.restrict (s ∪ t), p x) ↔ (∀ᵐ x ∂μ.restrict s, p x) ∧ ∀ᵐ x ∂μ.restrict t, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α) (p : α → Prop)
Goal: (∀ᵐ (x : α) ∂μ.restrict (s ∪ t), p x) ↔ (∀ᵐ (x : α) ∂μ.restrict s, p x) ∧ ∀ᵐ (x : α) ∂μ.restrict t, p x
-- MeasureTheory.ae_restrict_biUnion_iff
theorem ae_restrict_biUnion_iff (s : ι → Set α) {t : Set ι} (ht : t.Countable) (p : α → Prop) :
    (∀ᵐ x ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ x ∂μ.restrict (s i), p x 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ι → Set α) {t : Set ι} (ht : t.Countable) (p : α → Prop)
Goal: (∀ᵐ (x : α) ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ (x : α) ∂μ.restrict (s i), p x
-- MeasureTheory.ae_restrict_biUnion_finset_iff
@[simp]
theorem ae_restrict_biUnion_finset_iff (s : ι → Set α) (t : Finset ι) (p : α → Prop) :
    (∀ᵐ x ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ x ∂μ.restrict (s i), p x 
Arguments: {α : Type u_2} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ι → Set α) (t : Finset ι) (p : α → Prop)
Goal: (∀ᵐ (x : α) ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ (x : α) ∂μ.restrict (s i), p x
-- MeasureTheory.ae_eq_restrict_iUnion_iff
theorem ae_eq_restrict_iUnion_iff [Countable ι] (s : ι → Set α) (f g : α → δ) :
    f =ᵐ[μ.restrict (⋃ i, s i)] g ↔ ∀ i, f =ᵐ[μ.restrict (s i)] g 
Arguments: {α : Type u_2} {δ : Type u_4} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] (s : ι → Set α) (f g : α → δ)
Goal: f =ᵐ[μ.restrict (⋃ i, s i)] g ↔ ∀ (i : ι), f =ᵐ[μ.restrict (s i)] g
-- MeasureTheory.ae_eq_restrict_biUnion_iff
theorem ae_eq_restrict_biUnion_iff (s : ι → Set α) {t : Set ι} (ht : t.Countable) (f g : α → δ) :
    f =ᵐ[μ.restrict (⋃ i ∈ t, s i)] g ↔ ∀ i ∈ t, f =ᵐ[μ.restrict (s i)] g 
Arguments: {α : Type u_2} {δ : Type u_4} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ι → Set α) {t : Set ι} (ht : t.Countable) (f g : α → δ)
Goal: f =ᵐ[μ.restrict (⋃ i ∈ t, s i)] g ↔ ∀ i ∈ t, f =ᵐ[μ.restrict (s i)] g
-- MeasureTheory.ae_eq_restrict_biUnion_finset_iff
theorem ae_eq_restrict_biUnion_finset_iff (s : ι → Set α) (t : Finset ι) (f g : α → δ) :
    f =ᵐ[μ.restrict (⋃ i ∈ t, s i)] g ↔ ∀ i ∈ t, f =ᵐ[μ.restrict (s i)] g 
Arguments: {α : Type u_2} {δ : Type u_4} {ι : Type u_6} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ι → Set α) (t : Finset ι) (f g : α → δ)
Goal: f =ᵐ[μ.restrict (⋃ i ∈ t, s i)] g ↔ ∀ i ∈ t, f =ᵐ[μ.restrict (s i)] g
-- MeasureTheory.ae_restrict_uIoc_eq
theorem ae_restrict_uIoc_eq [LinearOrder α] (a b : α) :
    ae (μ.restrict (Ι a b)) = ae (μ.restrict (Ioc a b)) ⊔ ae (μ.restrict (Ioc b a)) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [LinearOrder α] (a b : α)
Goal: MeasureTheory.ae (μ.restrict (Ι a b)) =
  MeasureTheory.ae (μ.restrict (Set.Ioc a b)) ⊔ MeasureTheory.ae (μ.restrict (Set.Ioc b a))
-- MeasureTheory.ae_restrict_uIoc_iff
theorem ae_restrict_uIoc_iff [LinearOrder α] {a b : α} {P : α → Prop} :
    (∀ᵐ x ∂μ.restrict (Ι a b), P x) ↔
      (∀ᵐ x ∂μ.restrict (Ioc a b), P x) ∧ ∀ᵐ x ∂μ.restrict (Ioc b a), P x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [LinearOrder α] {a b : α} {P : α → Prop}
Goal: (∀ᵐ (x : α) ∂μ.restrict (Ι a b), P x) ↔
  (∀ᵐ (x : α) ∂μ.restrict (Set.Ioc a b), P x) ∧ ∀ᵐ (x : α) ∂μ.restrict (Set.Ioc b a), P x
-- MeasureTheory.ae_restrict_iff₀
theorem ae_restrict_iff₀ {p : α → Prop} (hp : NullMeasurableSet { x | p x } (μ.restrict s)) :
    (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {p : α → Prop} (hp : MeasureTheory.NullMeasurableSet {x | p x} (μ.restrict s))
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x
-- MeasureTheory.ae_restrict_iff
theorem ae_restrict_iff {p : α → Prop} (hp : MeasurableSet { x | p x }) :
    (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {p : α → Prop} (hp : MeasurableSet {x | p x})
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x
-- MeasureTheory.ae_imp_of_ae_restrict
theorem ae_imp_of_ae_restrict {s : Set α} {p : α → Prop} (h : ∀ᵐ x ∂μ.restrict s, p x) :
    ∀ᵐ x ∂μ, x ∈ s → p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {p : α → Prop} (h : ∀ᵐ (x : α) ∂μ.restrict s, p x)
Goal: ∀ᵐ (x : α) ∂μ, x ∈ s → p x
-- MeasureTheory.ae_restrict_iff'₀
theorem ae_restrict_iff'₀ {p : α → Prop} (hs : NullMeasurableSet s μ) :
    (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {p : α → Prop} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x
-- MeasureTheory.ae_restrict_iff'
theorem ae_restrict_iff' {p : α → Prop} (hs : MeasurableSet s) :
    (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {p : α → Prop} (hs : MeasurableSet s)
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x
-- Filter.EventuallyEq.restrict
theorem _root_.Filter.EventuallyEq.restrict {f g : α → δ} {s : Set α} (hfg : f =ᵐ[μ] g) :
    f =ᵐ[μ.restrict s] g 
Arguments: {α : Type u_2} {δ : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → δ} {s : Set α} (hfg : f =ᵐ[μ] g)
Goal: f =ᵐ[μ.restrict s] g
-- MeasureTheory.ae_restrict_mem₀
theorem ae_restrict_mem₀ (hs : NullMeasurableSet s μ) : ∀ᵐ x ∂μ.restrict s, x ∈ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: ∀ᵐ (x : α) ∂μ.restrict s, x ∈ s