ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IsLocallyFiniteMeasure.withDensity_ofReal
lemma IsLocallyFiniteMeasure.withDensity_ofReal {f : α → ℝ} (hf : Continuous f) :
    IsLocallyFiniteMeasure (μ.withDensity fun x ↦ .ofReal (f x)) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [OpensMeasurableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → ℝ} (hf : Continuous f)
Goal: MeasureTheory.IsLocallyFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal (f x))
-- MeasureTheory.ae_uIoc_iff
theorem ae_uIoc_iff [LinearOrder α] {a b : α} {P : α → Prop} :
    (∀ᵐ x ∂μ, x ∈ Ι a b → P x) ↔ (∀ᵐ x ∂μ, x ∈ Ioc a b → P x) ∧ ∀ᵐ x ∂μ, x ∈ Ioc b a → P x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [LinearOrder α] {a b : α} {P : α → Prop}
Goal: (∀ᵐ (x : α) ∂μ, x ∈ Ι a b → P x) ↔ (∀ᵐ (x : α) ∂μ, x ∈ Set.Ioc a b → P x) ∧ ∀ᵐ (x : α) ∂μ, x ∈ Set.Ioc b a → P x
-- MeasureTheory.measure_union
theorem measure_union (hd : Disjoint s₁ s₂) (h : MeasurableSet s₂) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h : MeasurableSet s₂)
Goal: μ (s₁ ∪ s₂) = μ s₁ + μ s₂
-- MeasureTheory.measure_union'
theorem measure_union' (hd : Disjoint s₁ s₂) (h : MeasurableSet s₁) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h : MeasurableSet s₁)
Goal: μ (s₁ ∪ s₂) = μ s₁ + μ s₂
-- MeasureTheory.measure_inter_add_diff
theorem measure_inter_add_diff (s : Set α) (ht : MeasurableSet t) : μ (s ∩ t) + μ (s \ t) = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t)
Goal: μ (s ∩ t) + μ (s \ t) = μ s
-- MeasureTheory.measure_diff_add_inter
theorem measure_diff_add_inter (s : Set α) (ht : MeasurableSet t) : μ (s \ t) + μ (s ∩ t) = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t)
Goal: μ (s \ t) + μ (s ∩ t) = μ s
-- MeasureTheory.measure_union_add_inter
theorem measure_union_add_inter (s : Set α) (ht : MeasurableSet t) :
    μ (s ∪ t) + μ (s ∩ t) = μ s + μ t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (ht : MeasurableSet t)
Goal: μ (s ∪ t) + μ (s ∩ t) = μ s + μ t
-- MeasureTheory.measure_union_add_inter'
theorem measure_union_add_inter' (hs : MeasurableSet s) (t : Set α) :
    μ (s ∪ t) + μ (s ∩ t) = μ s + μ t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (t : Set α)
Goal: μ (s ∪ t) + μ (s ∩ t) = μ s + μ t
-- MeasureTheory.measure_symmDiff_eq
lemma measure_symmDiff_eq (hs : MeasurableSet s) (ht : MeasurableSet t) :
    μ (s ∆ t) = μ (s \ t) + μ (t \ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: μ (symmDiff s t) = μ (s \ t) + μ (t \ s)
-- MeasureTheory.measure_symmDiff_le
lemma measure_symmDiff_le (s t u : Set α) :
    μ (s ∆ u) ≤ μ (s ∆ t) + μ (t ∆ u) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t u : Set α)
Goal: μ (symmDiff s u) ≤ μ (symmDiff s t) + μ (symmDiff t u)
-- MeasureTheory.measure_add_measure_compl
theorem measure_add_measure_compl (h : MeasurableSet s) : μ s + μ sᶜ = μ univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasurableSet s)
Goal: μ s + μ sᶜ = μ Set.univ
-- MeasureTheory.measure_biUnion₀
theorem measure_biUnion₀ {s : Set β} {f : β → Set α} (hs : s.Countable)
    (hd : s.Pairwise (AEDisjoint μ on f)) (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) :
    μ (⋃ b ∈ s, f b) = ∑' p : s, μ (f p) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.Pairwise (MeasureTheory.AEDisjoint μ on f)) (h : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: μ (⋃ b ∈ s, f b) = ∑' (p : ↑s), μ (f ↑p)
-- MeasureTheory.measure_biUnion
theorem measure_biUnion {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.PairwiseDisjoint f)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑' p : s, μ (f p) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set β} {f : β → Set α} (hs : s.Countable) (hd : s.PairwiseDisjoint f) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: μ (⋃ b ∈ s, f b) = ∑' (p : ↑s), μ (f ↑p)
-- MeasureTheory.measure_sUnion₀
theorem measure_sUnion₀ {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise (AEDisjoint μ))
    (h : ∀ s ∈ S, NullMeasurableSet s μ) : μ (⋃₀ S) = ∑' s : S, μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise (MeasureTheory.AEDisjoint μ)) (h : ∀ s ∈ S, MeasureTheory.NullMeasurableSet s μ)
Goal: μ (⋃₀ S) = ∑' (s : ↑S), μ ↑s
-- MeasureTheory.measure_sUnion
theorem measure_sUnion {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise Disjoint)
    (h : ∀ s ∈ S, MeasurableSet s) : μ (⋃₀ S) = ∑' s : S, μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise Disjoint) (h : ∀ s ∈ S, MeasurableSet s)
Goal: μ (⋃₀ S) = ∑' (s : ↑S), μ ↑s
-- MeasureTheory.measure_biUnion_finset₀
theorem measure_biUnion_finset₀ {s : Finset ι} {f : ι → Set α}
    (hd : Set.Pairwise (↑s) (AEDisjoint μ on f)) (hm : ∀ b ∈ s, NullMeasurableSet (f b) μ) :
    μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset ι} {f : ι → Set α} (hd : (↑s).Pairwise (MeasureTheory.AEDisjoint μ on f)) (hm : ∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ)
Goal: μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p)
-- MeasureTheory.measure_biUnion_finset
theorem measure_biUnion_finset {s : Finset ι} {f : ι → Set α} (hd : PairwiseDisjoint (↑s) f)
    (hm : ∀ b ∈ s, MeasurableSet (f b)) : μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset ι} {f : ι → Set α} (hd : (↑s).PairwiseDisjoint f) (hm : ∀ b ∈ s, MeasurableSet (f b))
Goal: μ (⋃ b ∈ s, f b) = ∑ p ∈ s, μ (f p)
-- MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint₀
theorem tsum_meas_le_meas_iUnion_of_disjoint₀ {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)) : (∑' i, μ (As i)) ≤ μ (⋃ i, As i) 
Arguments: {α : Type u_1} {ι : Type u_8} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {As : ι → Set α} (As_mble : ∀ (i : ι), MeasureTheory.NullMeasurableSet (As i) μ) (As_disj : Pairwise (MeasureTheory.AEDisjoint μ on As))
Goal: ∑' (i : ι), μ (As i) ≤ μ (⋃ i, As i)
-- MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint
theorem tsum_meas_le_meas_iUnion_of_disjoint {ι : Type*} [MeasurableSpace α] (μ : Measure α)
    {As : ι → Set α} (As_mble : ∀ i : ι, MeasurableSet (As i))
    (As_disj : Pairwise (Disjoint on As)) : (∑' i, μ (As i)) ≤ μ (⋃ i, As i) 
Arguments: {α : Type u_1} {ι : Type u_8} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {As : ι → Set α} (As_mble : ∀ (i : ι), MeasurableSet (As i)) (As_disj : Pairwise (Disjoint on As))
Goal: ∑' (i : ι), μ (As i) ≤ μ (⋃ i, As i)
-- MeasureTheory.tsum_measure_preimage_singleton
theorem tsum_measure_preimage_singleton {s : Set β} (hs : s.Countable) {f : α → β}
    (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : (∑' b : s, μ (f ⁻¹' {↑b})) = μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set β} (hs : s.Countable) {f : α → β} (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}))
Goal: ∑' (b : ↑s), μ (f ⁻¹' {↑b}) = μ (f ⁻¹' s)
-- MeasureTheory.measure_preimage_eq_zero_iff_of_countable
lemma measure_preimage_eq_zero_iff_of_countable {s : Set β} {f : α → β} (hs : s.Countable) :
    μ (f ⁻¹' s) = 0 ↔ ∀ x ∈ s, μ (f ⁻¹' {x}) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set β} {f : α → β} (hs : s.Countable)
Goal: μ (f ⁻¹' s) = 0 ↔ ∀ x ∈ s, μ (f ⁻¹' {x}) = 0
-- MeasureTheory.sum_measure_preimage_singleton
theorem sum_measure_preimage_singleton (s : Finset β) {f : α → β}
    (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : (∑ b ∈ s, μ (f ⁻¹' {b})) = μ (f ⁻¹' ↑s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : α → β} (hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}))
Goal: ∑ b ∈ s, μ (f ⁻¹' {b}) = μ (f ⁻¹' ↑s)
-- MeasureTheory.measure_diff_null'
theorem measure_diff_null' (h : μ (s₁ ∩ s₂) = 0) : μ (s₁ \ s₂) = μ s₁ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α} (h : μ (s₁ ∩ s₂) = 0)
Goal: μ (s₁ \ s₂) = μ s₁
-- MeasureTheory.measure_add_diff
theorem measure_add_diff (hs : MeasurableSet s) (t : Set α) : μ s + μ (t \ s) = μ (s ∪ t) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (t : Set α)
Goal: μ s + μ (t \ s) = μ (s ∪ t)
-- MeasureTheory.measure_diff'
theorem measure_diff' (s : Set α) (hm : MeasurableSet t) (h_fin : μ t ≠ ∞) :
    μ (s \ t) = μ (s ∪ t) - μ t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α) (hm : MeasurableSet t) (h_fin : μ t ≠ ⊤)
Goal: μ (s \ t) = μ (s ∪ t) - μ t
-- MeasureTheory.measure_diff
theorem measure_diff (h : s₂ ⊆ s₁) (h₂ : MeasurableSet s₂) (h_fin : μ s₂ ≠ ∞) :
    μ (s₁ \ s₂) = μ s₁ - μ s₂ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α} (h : s₂ ⊆ s₁) (h₂ : MeasurableSet s₂) (h_fin : μ s₂ ≠ ⊤)
Goal: μ (s₁ \ s₂) = μ s₁ - μ s₂
-- MeasureTheory.le_measure_diff
theorem le_measure_diff : μ s₁ - μ s₂ ≤ μ (s₁ \ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α}
Goal: μ s₁ - μ s₂ ≤ μ (s₁ \ s₂)
-- MeasureTheory.measure_eq_top_iff_of_symmDiff
theorem measure_eq_top_iff_of_symmDiff (hμst : μ (s ∆ t) ≠ ∞) : μ s = ∞ ↔ μ t = ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hμst : μ (symmDiff s t) ≠ ⊤)
Goal: μ s = ⊤ ↔ μ t = ⊤
-- MeasureTheory.measure_ne_top_iff_of_symmDiff
theorem measure_ne_top_iff_of_symmDiff (hμst : μ (s ∆ t) ≠ ∞) : μ s ≠ ∞ ↔ μ t ≠ ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hμst : μ (symmDiff s t) ≠ ⊤)
Goal: μ s ≠ ⊤ ↔ μ t ≠ ⊤
-- MeasureTheory.measure_diff_lt_of_lt_add
theorem measure_diff_lt_of_lt_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞}
    (h : μ t < μ s + ε) : μ (t \ s) < ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ⊤) {ε : ENNReal} (h : μ t < μ s + ε)
Goal: μ (t \ s) < ε
-- MeasureTheory.measure_diff_le_iff_le_add
theorem measure_diff_le_iff_le_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞} :
    μ (t \ s) ≤ ε ↔ μ t ≤ μ s + ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ⊤) {ε : ENNReal}
Goal: μ (t \ s) ≤ ε ↔ μ t ≤ μ s + ε
-- MeasureTheory.measure_eq_measure_of_null_diff
theorem measure_eq_measure_of_null_diff {s t : Set α} (hst : s ⊆ t) (h_nulldiff : μ (t \ s) = 0) :
    μ s = μ t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hst : s ⊆ t) (h_nulldiff : μ (t \ s) = 0)
Goal: μ s = μ t
-- MeasureTheory.measure_eq_measure_of_between_null_diff
theorem measure_eq_measure_of_between_null_diff {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃)
    (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₁ = μ s₂ ∧ μ s₂ = μ s₃ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0)
Goal: μ s₁ = μ s₂ ∧ μ s₂ = μ s₃
-- MeasureTheory.measure_eq_measure_smaller_of_between_null_diff
theorem measure_eq_measure_smaller_of_between_null_diff {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂)
    (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₁ = μ s₂ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0)
Goal: μ s₁ = μ s₂
-- MeasureTheory.measure_eq_measure_larger_of_between_null_diff
theorem measure_eq_measure_larger_of_between_null_diff {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂)
    (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0) : μ s₂ = μ s₃ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α} (h12 : s₁ ⊆ s₂) (h23 : s₂ ⊆ s₃) (h_nulldiff : μ (s₃ \ s₁) = 0)
Goal: μ s₂ = μ s₃
-- MeasureTheory.measure_compl₀
lemma measure_compl₀ (h : NullMeasurableSet s μ) (hs : μ s ≠ ∞) :
    μ sᶜ = μ Set.univ - μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.NullMeasurableSet s μ) (hs : μ s ≠ ⊤)
Goal: μ sᶜ = μ Set.univ - μ s
-- MeasureTheory.measure_compl
theorem measure_compl (h₁ : MeasurableSet s) (h_fin : μ s ≠ ∞) : μ sᶜ = μ univ - μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h₁ : MeasurableSet s) (h_fin : μ s ≠ ⊤)
Goal: μ sᶜ = μ Set.univ - μ s
-- MeasureTheory.measure_inter_conull'
lemma measure_inter_conull' (ht : μ (s \ t) = 0) : μ (s ∩ t) = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : μ (s \ t) = 0)
Goal: μ (s ∩ t) = μ s
-- MeasureTheory.measure_inter_conull
lemma measure_inter_conull (ht : μ tᶜ = 0) : μ (s ∩ t) = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (ht : μ tᶜ = 0)
Goal: μ (s ∩ t) = μ s
-- MeasureTheory.union_ae_eq_left_iff_ae_subset
@[simp]
theorem union_ae_eq_left_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] s ↔ t ≤ᵐ[μ] s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: s ∪ t =ᵐ[μ] s ↔ t ≤ᵐ[μ] s
-- MeasureTheory.union_ae_eq_right_iff_ae_subset
@[simp]
theorem union_ae_eq_right_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] t ↔ s ≤ᵐ[μ] t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: s ∪ t =ᵐ[μ] t ↔ s ≤ᵐ[μ] t
-- MeasureTheory.ae_eq_of_ae_subset_of_measure_ge
theorem ae_eq_of_ae_subset_of_measure_ge (h₁ : s ≤ᵐ[μ] t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s)
    (ht : μ t ≠ ∞) : s =ᵐ[μ] t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h₁ : s ≤ᵐ[μ] t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s) (ht : μ t ≠ ⊤)
Goal: s =ᵐ[μ] t
-- MeasureTheory.ae_eq_of_subset_of_measure_ge
theorem ae_eq_of_subset_of_measure_ge (h₁ : s ⊆ t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s)
    (ht : μ t ≠ ∞) : s =ᵐ[μ] t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (h₁ : s ⊆ t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s) (ht : μ t ≠ ⊤)
Goal: s =ᵐ[μ] t
-- MeasureTheory.measure_iUnion_congr_of_subset
theorem measure_iUnion_congr_of_subset [Countable β] {s : β → Set α} {t : β → Set α}
    (hsub : ∀ b, s b ⊆ t b) (h_le : ∀ b, μ (t b) ≤ μ (s b)) : μ (⋃ b, s b) = μ (⋃ b, t b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] {s t : β → Set α} (hsub : ∀ (b : β), s b ⊆ t b) (h_le : ∀ (b : β), μ (t b) ≤ μ (s b))
Goal: μ (⋃ b, s b) = μ (⋃ b, t b)
-- MeasureTheory.measure_union_congr_of_subset
theorem measure_union_congr_of_subset {t₁ t₂ : Set α} (hs : s₁ ⊆ s₂) (hsμ : μ s₂ ≤ μ s₁)
    (ht : t₁ ⊆ t₂) (htμ : μ t₂ ≤ μ t₁) : μ (s₁ ∪ t₁) = μ (s₂ ∪ t₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ t₁ t₂ : Set α} (hs : s₁ ⊆ s₂) (hsμ : μ s₂ ≤ μ s₁) (ht : t₁ ⊆ t₂) (htμ : μ t₂ ≤ μ t₁)
Goal: μ (s₁ ∪ t₁) = μ (s₂ ∪ t₂)
-- MeasureTheory.measure_iUnion_toMeasurable
@[simp]
theorem measure_iUnion_toMeasurable [Countable β] (s : β → Set α) :
    μ (⋃ b, toMeasurable μ (s b)) = μ (⋃ b, s b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable β] (s : β → Set α)
Goal: μ (⋃ b, MeasureTheory.toMeasurable μ (s b)) = μ (⋃ b, s b)
-- MeasureTheory.measure_biUnion_toMeasurable
theorem measure_biUnion_toMeasurable {I : Set β} (hc : I.Countable) (s : β → Set α) :
    μ (⋃ b ∈ I, toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {I : Set β} (hc : I.Countable) (s : β → Set α)
Goal: μ (⋃ b ∈ I, MeasureTheory.toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b)
-- MeasureTheory.measure_toMeasurable_union
@[simp]
theorem measure_toMeasurable_union : μ (toMeasurable μ s ∪ t) = μ (s ∪ t) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: μ (MeasureTheory.toMeasurable μ s ∪ t) = μ (s ∪ t)
-- MeasureTheory.measure_union_toMeasurable
@[simp]
theorem measure_union_toMeasurable : μ (s ∪ toMeasurable μ t) = μ (s ∪ t) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: μ (s ∪ MeasureTheory.toMeasurable μ t) = μ (s ∪ t)
-- MeasureTheory.sum_measure_le_measure_univ
theorem sum_measure_le_measure_univ {s : Finset ι} {t : ι → Set α}
    (h : ∀ i ∈ s, NullMeasurableSet (t i) μ) (H : Set.Pairwise s (AEDisjoint μ on t)) :
    (∑ i ∈ s, μ (t i)) ≤ μ (univ : Set α) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasureTheory.NullMeasurableSet (t i) μ) (H : (↑s).Pairwise (MeasureTheory.AEDisjoint μ on t))
Goal: ∑ i ∈ s, μ (t i) ≤ μ Set.univ
-- MeasureTheory.tsum_measure_le_measure_univ
theorem tsum_measure_le_measure_univ {s : ι → Set α} (hs : ∀ i, NullMeasurableSet (s i) μ)
    (H : Pairwise (AEDisjoint μ on s)) : ∑' i, μ (s i) ≤ μ (univ : Set α) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : ι → Set α} (hs : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (H : Pairwise (MeasureTheory.AEDisjoint μ on s))
Goal: ∑' (i : ι), μ (s i) ≤ μ Set.univ
-- MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure
theorem exists_nonempty_inter_of_measure_univ_lt_tsum_measure {m : MeasurableSpace α}
    (μ : Measure α) {s : ι → Set α} (hs : ∀ i, NullMeasurableSet (s i) μ)
    (H : μ (univ : Set α) < ∑' i, μ (s i)) : ∃ i j, i ≠ j ∧ (s i ∩ s j).Nonempty 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s : ι → Set α} (hs : ∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) (H : μ Set.univ < ∑' (i : ι), μ (s i))
Goal: ∃ i j, i ≠ j ∧ (s i ∩ s j).Nonempty
-- MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_sum_measure
theorem exists_nonempty_inter_of_measure_univ_lt_sum_measure {m : MeasurableSpace α} (μ : Measure α)
    {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, NullMeasurableSet (t i) μ)
    (H : μ (univ : Set α) < ∑ i ∈ s, μ (t i)) :
    ∃ i ∈ s, ∃ j ∈ s, ∃ _h : i ≠ j, (t i ∩ t j).Nonempty 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasureTheory.NullMeasurableSet (t i) μ) (H : μ Set.univ < ∑ i ∈ s, μ (t i))
Goal: ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), (t i ∩ t j).Nonempty
-- MeasureTheory.nonempty_inter_of_measure_lt_add
theorem nonempty_inter_of_measure_lt_add {m : MeasurableSpace α} (μ : Measure α) {s t u : Set α}
    (ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) :
    (s ∩ t).Nonempty 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s t u : Set α} (ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t)
Goal: (s ∩ t).Nonempty
-- MeasureTheory.nonempty_inter_of_measure_lt_add'
theorem nonempty_inter_of_measure_lt_add' {m : MeasurableSpace α} (μ : Measure α) {s t u : Set α}
    (hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) :
    (s ∩ t).Nonempty 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s t u : Set α} (hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t)
Goal: (s ∩ t).Nonempty
-- MeasureTheory.measure_iUnion_eq_iSup
theorem measure_iUnion_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s) :
    μ (⋃ i, s i) = ⨆ i, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (hd : Directed (fun x x_1 => x ⊆ x_1) s)
Goal: μ (⋃ i, s i) = ⨆ i, μ (s i)
-- MeasureTheory.measure_iUnion_eq_iSup'
theorem measure_iUnion_eq_iSup' {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
    [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
    {f : ι → Set α} : μ (⋃ i, f i) = ⨆ i, μ (Accumulate f i) 
Arguments: {α : Type u_8} {ι : Type u_9} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Countable ι] [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {f : ι → Set α}
Goal: μ (⋃ i, f i) = ⨆ i, μ (Set.Accumulate f i)
-- MeasureTheory.measure_biUnion_eq_iSup
theorem measure_biUnion_eq_iSup {s : ι → Set α} {t : Set ι} (ht : t.Countable)
    (hd : DirectedOn ((· ⊆ ·) on s) t) : μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : ι → Set α} {t : Set ι} (ht : t.Countable) (hd : DirectedOn ((fun x x_1 => x ⊆ x_1) on s) t)
Goal: μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i)
-- MeasureTheory.measure_iInter_eq_iInf
theorem measure_iInter_eq_iInf [Countable ι] {s : ι → Set α} (h : ∀ i, MeasurableSet (s i))
    (hd : Directed (· ⊇ ·) s) (hfin : ∃ i, μ (s i) ≠ ∞) : μ (⋂ i, s i) = ⨅ i, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (h : ∀ (i : ι), MeasurableSet (s i)) (hd : Directed (fun x x_1 => x ⊇ x_1) s) (hfin : ∃ i, μ (s i) ≠ ⊤)
Goal: μ (⋂ i, s i) = ⨅ i, μ (s i)
-- MeasureTheory.measure_iInter_eq_iInf'
theorem measure_iInter_eq_iInf' {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
    [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
    {f : ι → Set α} (h : ∀ i, MeasurableSet (f i)) (hfin : ∃ i, μ (f i) ≠ ∞) :
    μ (⋂ i, f i) = ⨅ i, μ (⋂ j ≤ i, f j) 
Arguments: {α : Type u_8} {ι : Type u_9} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Countable ι] [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {f : ι → Set α} (h : ∀ (i : ι), MeasurableSet (f i)) (hfin : ∃ i, μ (f i) ≠ ⊤)
Goal: μ (⋂ i, f i) = ⨅ i, μ (⋂ j, ⋂ (_ : j ≤ i), f j)
-- MeasureTheory.tendsto_measure_iUnion
theorem tendsto_measure_iUnion [Preorder ι] [IsDirected ι (· ≤ ·)] [Countable ι]
    {s : ι → Set α} (hm : Monotone s) : Tendsto (μ ∘ s) atTop (𝓝 (μ (⋃ n, s n))) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] [Countable ι] {s : ι → Set α} (hm : Monotone s)
Goal: Filter.Tendsto (⇑μ ∘ s) Filter.atTop (nhds (μ (⋃ n, s n)))
-- MeasureTheory.tendsto_measure_iUnion'
theorem tendsto_measure_iUnion' {α ι : Type*} [MeasurableSpace α] {μ : Measure α} [Countable ι]
    [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} :
    Tendsto (fun i ↦ μ (Accumulate f i)) atTop (𝓝 (μ (⋃ i, f i))) 
Arguments: {α : Type u_8} {ι : Type u_9} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Countable ι] [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {f : ι → Set α}
Goal: Filter.Tendsto (fun i => μ (Set.Accumulate f i)) Filter.atTop (nhds (μ (⋃ i, f i)))
-- MeasureTheory.tendsto_measure_iInter
theorem tendsto_measure_iInter [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {s : ι → Set α}
    (hs : ∀ n, MeasurableSet (s n)) (hm : Antitone s) (hf : ∃ i, μ (s i) ≠ ∞) :
    Tendsto (μ ∘ s) atTop (𝓝 (μ (⋂ n, s n))) 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {s : ι → Set α} (hs : ∀ (n : ι), MeasurableSet (s n)) (hm : Antitone s) (hf : ∃ i, μ (s i) ≠ ⊤)
Goal: Filter.Tendsto (⇑μ ∘ s) Filter.atTop (nhds (μ (⋂ n, s n)))