ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm'_of_norm
theorem eLpNorm_le_eLpNorm_mul_eLpNorm'_of_norm {p q r : ℝ≥0∞} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖ ≤ ‖f x‖ * ‖g x‖) (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f q μ * eLpNorm g r μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {m : MeasurableSpace α} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] {μ : MeasureTheory.Measure α} {f : α → E} {g : α → F} {p q r : ℝ≥0∞} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (b : E → F → G) (h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖ ≤ ‖f x‖ * ‖g x‖) (hpqr : 1 / p = 1 / q + 1 / r)
Goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f q μ * MeasureTheory.eLpNorm g r μ
-- MeasureTheory.eLpNorm_smul_le_eLpNorm_top_mul_eLpNorm
theorem eLpNorm_smul_le_eLpNorm_top_mul_eLpNorm (p : ℝ≥0∞) (hf : AEStronglyMeasurable f μ)
    (φ : α → 𝕜) : eLpNorm (φ • f) p μ ≤ eLpNorm φ ∞ μ * eLpNorm f p μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {f : α → E} (p : ℝ≥0∞) (hf : MeasureTheory.AEStronglyMeasurable f μ) (φ : α → 𝕜)
Goal: MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ ⊤ μ * MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_smul_le_eLpNorm_mul_eLpNorm_top
theorem eLpNorm_smul_le_eLpNorm_mul_eLpNorm_top (p : ℝ≥0∞) (f : α → E) {φ : α → 𝕜}
    (hφ : AEStronglyMeasurable φ μ) : eLpNorm (φ • f) p μ ≤ eLpNorm φ p μ * eLpNorm f ∞ μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] (p : ℝ≥0∞) (f : α → E) {φ : α → 𝕜} (hφ : MeasureTheory.AEStronglyMeasurable φ μ)
Goal: MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ p μ * MeasureTheory.eLpNorm f ⊤ μ
-- MeasureTheory.eLpNorm'_smul_le_mul_eLpNorm'
theorem eLpNorm'_smul_le_mul_eLpNorm' {p q r : ℝ} {f : α → E} (hf : AEStronglyMeasurable f μ)
    {φ : α → 𝕜} (hφ : AEStronglyMeasurable φ μ) (hp0_lt : 0 < p) (hpq : p < q)
    (hpqr : 1 / p = 1 / q + 1 / r) : eLpNorm' (φ • f) p μ ≤ eLpNorm' φ q μ * eLpNorm' f r μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {p q r : ℝ} {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) {φ : α → 𝕜} (hφ : MeasureTheory.AEStronglyMeasurable φ μ) (hp0_lt : 0 < p) (hpq : p < q) (hpqr : 1 / p = 1 / q + 1 / r)
Goal: MeasureTheory.eLpNorm' (φ • f) p μ ≤ MeasureTheory.eLpNorm' φ q μ * MeasureTheory.eLpNorm' f r μ
-- MeasureTheory.eLpNorm_smul_le_mul_eLpNorm
theorem eLpNorm_smul_le_mul_eLpNorm {p q r : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ)
    {φ : α → 𝕜} (hφ : AEStronglyMeasurable φ μ) (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm (φ • f) p μ ≤ eLpNorm φ q μ * eLpNorm f r μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {p q r : ℝ≥0∞} {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) {φ : α → 𝕜} (hφ : MeasureTheory.AEStronglyMeasurable φ μ) (hpqr : 1 / p = 1 / q + 1 / r)
Goal: MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ q μ * MeasureTheory.eLpNorm f r μ
-- MeasureTheory.Memℒp.smul
theorem Memℒp.smul {p q r : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : Memℒp f r μ) (hφ : Memℒp φ q μ)
    (hpqr : 1 / p = 1 / q + 1 / r) : Memℒp (φ • f) p μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {p q r : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : MeasureTheory.Memℒp f r μ) (hφ : MeasureTheory.Memℒp φ q μ) (hpqr : 1 / p = 1 / q + 1 / r)
Goal: MeasureTheory.Memℒp (φ • f) p μ
-- MeasureTheory.Memℒp.smul_of_top_right
theorem Memℒp.smul_of_top_right {p : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : Memℒp f p μ)
    (hφ : Memℒp φ ∞ μ) : Memℒp (φ • f) p μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {p : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : MeasureTheory.Memℒp f p μ) (hφ : MeasureTheory.Memℒp φ ⊤ μ)
Goal: MeasureTheory.Memℒp (φ • f) p μ
-- MeasureTheory.Memℒp.smul_of_top_left
theorem Memℒp.smul_of_top_left {p : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : Memℒp f ∞ μ)
    (hφ : Memℒp φ p μ) : Memℒp (φ • f) p μ 
Arguments: {𝕜 : Type u_1} {α : Type u_2} {E : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedRing 𝕜] [NormedAddCommGroup E] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {p : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : MeasureTheory.Memℒp f ⊤ μ) (hφ : MeasureTheory.Memℒp φ p μ)
Goal: MeasureTheory.Memℒp (φ • f) p μ
-- MeasureTheory.measure_empty
@[simp]
theorem measure_empty : μ ∅ = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F}
Goal: μ ∅ = 0
-- MeasureTheory.measure_mono
@[mono, gcongr]
theorem measure_mono (h : s ⊆ t) : μ s ≤ μ t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s ⊆ t)
Goal: μ s ≤ μ t
-- MeasureTheory.measure_mono_null
theorem measure_mono_null (h : s ⊆ t) (ht : μ t = 0) : μ s = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s ⊆ t) (ht : μ t = 0)
Goal: μ s = 0
-- MeasureTheory.measure_pos_of_superset
theorem measure_pos_of_superset (h : s ⊆ t) (hs : μ s ≠ 0) : 0 < μ t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (h : s ⊆ t) (hs : μ s ≠ 0)
Goal: 0 < μ t
-- MeasureTheory.measure_iUnion_le
theorem measure_iUnion_le [Countable ι] (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} [Countable ι] (s : ι → Set α)
Goal: μ (⋃ i, s i) ≤ ∑' (i : ι), μ (s i)
-- MeasureTheory.measure_biUnion_le
theorem measure_biUnion_le {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α) :
    μ (⋃ i ∈ I, s i) ≤ ∑' i : I, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α)
Goal: μ (⋃ i ∈ I, s i) ≤ ∑' (i : ↑I), μ (s ↑i)
-- MeasureTheory.measure_biUnion_finset_le
theorem measure_biUnion_finset_le (I : Finset ι) (s : ι → Set α) :
    μ (⋃ i ∈ I, s i) ≤ ∑ i ∈ I, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} (I : Finset ι) (s : ι → Set α)
Goal: μ (⋃ i ∈ I, s i) ≤ ∑ i ∈ I, μ (s i)
-- MeasureTheory.measure_iUnion_fintype_le
theorem measure_iUnion_fintype_le [Fintype ι] (μ : F) (s : ι → Set α) :
    μ (⋃ i, s i) ≤ ∑ i, μ (s i) 
Arguments: {α : Type u_1} {ι : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] [Fintype ι] (μ : F) (s : ι → Set α)
Goal: μ (⋃ i, s i) ≤ ∑ i : ι, μ (s i)
-- MeasureTheory.measure_union_le
theorem measure_union_le (s t : Set α) : μ (s ∪ t) ≤ μ s + μ t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} (s t : Set α)
Goal: μ (s ∪ t) ≤ μ s + μ t
-- MeasureTheory.measure_le_inter_add_diff
theorem measure_le_inter_add_diff (μ : F) (s t : Set α) : μ s ≤ μ (s ∩ t) + μ (s \ t) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] (μ : F) (s t : Set α)
Goal: μ s ≤ μ (s ∩ t) + μ (s \ t)
-- MeasureTheory.measure_diff_null
theorem measure_diff_null (ht : μ t = 0) : μ (s \ t) = μ s 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (ht : μ t = 0)
Goal: μ (s \ t) = μ s
-- MeasureTheory.measure_biUnion_null_iff
theorem measure_biUnion_null_iff {I : Set ι} (hI : I.Countable) {s : ι → Set α} :
    μ (⋃ i ∈ I, s i) = 0 ↔ ∀ i ∈ I, μ (s i) = 0 
Arguments: {α : Type u_1} {ι : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {I : Set ι} (hI : I.Countable) {s : ι → Set α}
Goal: μ (⋃ i ∈ I, s i) = 0 ↔ ∀ i ∈ I, μ (s i) = 0
-- MeasureTheory.measure_sUnion_null_iff
theorem measure_sUnion_null_iff {S : Set (Set α)} (hS : S.Countable) :
    μ (⋃₀ S) = 0 ↔ ∀ s ∈ S, μ s = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {S : Set (Set α)} (hS : S.Countable)
Goal: μ (⋃₀ S) = 0 ↔ ∀ s ∈ S, μ s = 0
-- MeasureTheory.measure_iUnion_null_iff
@[simp]
theorem measure_iUnion_null_iff {ι : Sort*} [Countable ι] {s : ι → Set α} :
    μ (⋃ i, s i) = 0 ↔ ∀ i, μ (s i) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {ι : Sort u_4} [Countable ι] {s : ι → Set α}
Goal: μ (⋃ i, s i) = 0 ↔ ∀ (i : ι), μ (s i) = 0
-- MeasureTheory.measure_union_null_iff
@[simp]
theorem measure_union_null_iff : μ (s ∪ t) = 0 ↔ μ s = 0 ∧ μ t = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: μ (s ∪ t) = 0 ↔ μ s = 0 ∧ μ t = 0
-- MeasureTheory.measure_union_null
theorem measure_union_null (hs : μ s = 0) (ht : μ t = 0) : μ (s ∪ t) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α} (hs : μ s = 0) (ht : μ t = 0)
Goal: μ (s ∪ t) = 0
-- MeasureTheory.measure_null_iff_singleton
lemma measure_null_iff_singleton (hs : s.Countable) : μ s = 0 ↔ ∀ x ∈ s, μ {x} = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α} (hs : s.Countable)
Goal: μ s = 0 ↔ ∀ x ∈ s, μ {x} = 0
-- MeasureTheory.measure_iUnion_of_tendsto_zero
theorem measure_iUnion_of_tendsto_zero {ι} (μ : F) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => μ ((⋃ n, s n) \ s k)) l (𝓝 0)) : μ (⋃ n, s n) = ⨆ n, μ (s n) 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {ι : Type u_4} (μ : F) {s : ι → Set α} (l : Filter ι) [l.NeBot] (h0 : Filter.Tendsto (fun k => μ ((⋃ n, s n) \ s k)) l (𝓝 0))
Goal: μ (⋃ n, s n) = ⨆ n, μ (s n)
-- MeasureTheory.measure_null_of_locally_null
theorem measure_null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α]
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, μ u = 0) : μ s = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} [TopologicalSpace α] [SecondCountableTopology α] (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, μ u = 0)
Goal: μ s = 0
-- MeasureTheory.exists_mem_forall_mem_nhdsWithin_pos_measure
theorem exists_mem_forall_mem_nhdsWithin_pos_measure [TopologicalSpace α]
    [SecondCountableTopology α] {s : Set α} (hs : μ s ≠ 0) :
    ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} [TopologicalSpace α] [SecondCountableTopology α] {s : Set α} (hs : μ s ≠ 0)
Goal: ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t
-- MeasureTheory.OuterMeasure.empty'
@[deprecated measure_empty (since := "2024-05-14")]
theorem empty' (m : OuterMeasure α) : m ∅ = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: m ∅ = 0
-- MeasureTheory.OuterMeasure.mono'
@[deprecated measure_mono (since := "2024-05-14")]
theorem mono' (m : OuterMeasure α) {s₁ s₂} (h : s₁ ⊆ s₂) : m s₁ ≤ m s₂ 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {s₁ s₂ : Set α} (h : s₁ ⊆ s₂)
Goal: m s₁ ≤ m s₂
-- MeasureTheory.OuterMeasure.mono_null
@[deprecated measure_mono_null (since := "2024-05-14")]
theorem mono_null (m : OuterMeasure α) {s t} (h : s ⊆ t) (ht : m t = 0) : m s = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {s t : Set α} (h : s ⊆ t) (ht : m t = 0)
Goal: m s = 0
-- MeasureTheory.OuterMeasure.pos_of_subset_ne_zero
@[deprecated measure_pos_of_superset (since := "2024-05-14")]
theorem pos_of_subset_ne_zero (m : OuterMeasure α) {a b : Set α} (hs : a ⊆ b) (hnz : m a ≠ 0) :
    0 < m b 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {a b : Set α} (hs : a ⊆ b) (hnz : m a ≠ 0)
Goal: 0 < m b
-- MeasureTheory.OuterMeasure.iUnion
@[deprecated measure_iUnion_le (since := "2024-05-14")]
protected theorem iUnion (m : OuterMeasure α) {β} [Countable β] (s : β → Set α) :
    m (⋃ i, s i) ≤ ∑' i, m (s i) 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {β : Type u_3} [Countable β] (s : β → Set α)
Goal: m (⋃ i, s i) ≤ ∑' (i : β), m (s i)
-- MeasureTheory.OuterMeasure.biUnion_null_iff
@[deprecated measure_biUnion_null_iff (since := "2024-05-14")]
theorem biUnion_null_iff (m : OuterMeasure α) {s : Set β} (hs : s.Countable) {t : β → Set α} :
    m (⋃ i ∈ s, t i) = 0 ↔ ∀ i ∈ s, m (t i) = 0 
Arguments: {α : Type u_1} {β : Type u_2} (m : MeasureTheory.OuterMeasure α) {s : Set β} (hs : s.Countable) {t : β → Set α}
Goal: m (⋃ i ∈ s, t i) = 0 ↔ ∀ i ∈ s, m (t i) = 0
-- MeasureTheory.OuterMeasure.sUnion_null_iff
@[deprecated measure_sUnion_null_iff (since := "2024-05-14")]
theorem sUnion_null_iff (m : OuterMeasure α) {S : Set (Set α)} (hS : S.Countable) :
    m (⋃₀ S) = 0 ↔ ∀ s ∈ S, m s = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {S : Set (Set α)} (hS : S.Countable)
Goal: m (⋃₀ S) = 0 ↔ ∀ s ∈ S, m s = 0
-- MeasureTheory.OuterMeasure.iUnion_null_iff
@[deprecated measure_iUnion_null_iff (since := "2024-05-14")]
theorem iUnion_null_iff {ι : Sort*} [Countable ι] (m : OuterMeasure α) {s : ι → Set α} :
    m (⋃ i, s i) = 0 ↔ ∀ i, m (s i) = 0 
Arguments: {α : Type u_1} {ι : Sort u_3} [Countable ι] (m : MeasureTheory.OuterMeasure α) {s : ι → Set α}
Goal: m (⋃ i, s i) = 0 ↔ ∀ (i : ι), m (s i) = 0
-- MeasureTheory.OuterMeasure.iUnion_null_iff'
@[deprecated (since := "2024-01-14")]
theorem iUnion_null_iff' (m : OuterMeasure α) {ι : Prop} {s : ι → Set α} :
    m (⋃ i, s i) = 0 ↔ ∀ i, m (s i) = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {ι : Prop} {s : ι → Set α}
Goal: m (⋃ (i : ι), s i) = 0 ↔ ∀ (i : ι), m (s i) = 0
-- MeasureTheory.OuterMeasure.iUnion_finset
@[deprecated measure_biUnion_finset_le (since := "2024-05-14")]
protected theorem iUnion_finset (m : OuterMeasure α) (s : β → Set α) (t : Finset β) :
    m (⋃ i ∈ t, s i) ≤ ∑ i ∈ t, m (s i) 
Arguments: {α : Type u_1} {β : Type u_2} (m : MeasureTheory.OuterMeasure α) (s : β → Set α) (t : Finset β)
Goal: m (⋃ i ∈ t, s i) ≤ ∑ i ∈ t, m (s i)
-- MeasureTheory.OuterMeasure.union
@[deprecated measure_union_le (since := "2024-05-14")]
protected theorem union (m : OuterMeasure α) (s₁ s₂ : Set α) : m (s₁ ∪ s₂) ≤ m s₁ + m s₂ 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) (s₁ s₂ : Set α)
Goal: m (s₁ ∪ s₂) ≤ m s₁ + m s₂
-- MeasureTheory.OuterMeasure.null_of_locally_null
@[deprecated measure_null_of_locally_null (since := "2024-05-14")]
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 
Arguments: {α : Type u_1} [TopologicalSpace α] [SecondCountableTopology α] (m : MeasureTheory.OuterMeasure α) (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0)
Goal: m s = 0
-- MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos
@[deprecated exists_mem_forall_mem_nhdsWithin_pos_measure (since := "2024-05-14")]
theorem exists_mem_forall_mem_nhds_within_pos [TopologicalSpace α] [SecondCountableTopology α]
    (m : OuterMeasure α) {s : Set α} (hs : m s ≠ 0) : ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t 
Arguments: {α : Type u_1} [TopologicalSpace α] [SecondCountableTopology α] (m : MeasureTheory.OuterMeasure α) {s : Set α} (hs : m s ≠ 0)
Goal: ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t
-- MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) 
Arguments: {α : Type u_1} {ι : Type u_3} (m : MeasureTheory.OuterMeasure α) {s : ι → Set α} (l : Filter ι) [l.NeBot] (h0 : Filter.Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0))
Goal: m (⋃ n, s n) = ⨆ n, m (s n)
-- MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞) :
    m (⋃ n, s n) = ⨆ n, m (s n) 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α} (h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)) (h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤)
Goal: m (⋃ n, s n) = ⨆ n, m (s n)
-- MeasureTheory.OuterMeasure.le_inter_add_diff
@[deprecated measure_le_inter_add_diff (since := "2024-05-14")]
theorem le_inter_add_diff {m : OuterMeasure α} {t : Set α} (s : Set α) :
    m t ≤ m (t ∩ s) + m (t \ s) 
Arguments: {α : Type u_1} {m : MeasureTheory.OuterMeasure α} {t : Set α} (s : Set α)
Goal: m t ≤ m (t ∩ s) + m (t \ s)
-- MeasureTheory.OuterMeasure.diff_null
@[deprecated measure_diff_null (since := "2024-05-14")]
theorem diff_null (m : OuterMeasure α) (s : Set α) {t : Set α} (ht : m t = 0) : m (s \ t) = m s 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) (s : Set α) {t : Set α} (ht : m t = 0)
Goal: m (s \ t) = m s
-- MeasureTheory.OuterMeasure.union_null
@[deprecated measure_union_null (since := "2024-05-14")]
theorem union_null (m : OuterMeasure α) {s₁ s₂ : Set α} (h₁ : m s₁ = 0) (h₂ : m s₂ = 0) :
    m (s₁ ∪ s₂) = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {s₁ s₂ : Set α} (h₁ : m s₁ = 0) (h₂ : m s₂ = 0)
Goal: m (s₁ ∪ s₂) = 0
-- MeasureTheory.OuterMeasure.coe_fn_injective
theorem coe_fn_injective : Injective fun (μ : OuterMeasure α) (s : Set α) => μ s 
Arguments: {α : Type u_1}
Goal: Function.Injective fun μ s => μ s
-- MeasureTheory.OuterMeasure.ext
@[ext]
theorem ext {μ₁ μ₂ : OuterMeasure α} (h : ∀ s, μ₁ s = μ₂ s) : μ₁ = μ₂ 
Arguments: {α : Type u_1} {μ₁ μ₂ : MeasureTheory.OuterMeasure α} (h : ∀ (s : Set α), μ₁ s = μ₂ s)
Goal: μ₁ = μ₂
-- MeasureTheory.OuterMeasure.ext_nonempty
theorem ext_nonempty {μ₁ μ₂ : OuterMeasure α} (h : ∀ s : Set α, s.Nonempty → μ₁ s = μ₂ s) :
    μ₁ = μ₂ 
Arguments: {α : Type u_1} {μ₁ μ₂ : MeasureTheory.OuterMeasure α} (h : ∀ (s : Set α), s.Nonempty → μ₁ s = μ₂ s)
Goal: μ₁ = μ₂
-- MeasureTheory.mem_ae_iff
theorem mem_ae_iff {s : Set α} : s ∈ ae μ ↔ μ sᶜ = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α}
Goal: s ∈ MeasureTheory.ae μ ↔ μ sᶜ = 0
-- MeasureTheory.ae_iff
theorem ae_iff {p : α → Prop} : (∀ᵐ a ∂μ, p a) ↔ μ { a | ¬p a } = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {p : α → Prop}
Goal: (∀ᵐ (a : α) ∂μ, p a) ↔ μ {a | ¬p a} = 0
-- MeasureTheory.compl_mem_ae_iff
theorem compl_mem_ae_iff {s : Set α} : sᶜ ∈ ae μ ↔ μ s = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α}
Goal: sᶜ ∈ MeasureTheory.ae μ ↔ μ s = 0
-- MeasureTheory.frequently_ae_iff
theorem frequently_ae_iff {p : α → Prop} : (∃ᵐ a ∂μ, p a) ↔ μ { a | p a } ≠ 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {p : α → Prop}
Goal: (∃ᵐ (a : α) ∂μ, p a) ↔ μ {a | p a} ≠ 0
-- MeasureTheory.frequently_ae_mem_iff
theorem frequently_ae_mem_iff {s : Set α} : (∃ᵐ a ∂μ, a ∈ s) ↔ μ s ≠ 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α}
Goal: (∃ᵐ (a : α) ∂μ, a ∈ s) ↔ μ s ≠ 0
-- MeasureTheory.measure_zero_iff_ae_nmem
theorem measure_zero_iff_ae_nmem {s : Set α} : μ s = 0 ↔ ∀ᵐ a ∂μ, a ∉ s 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α}
Goal: μ s = 0 ↔ ∀ᵐ (a : α) ∂μ, a ∉ s
-- MeasureTheory.ae_of_all
theorem ae_of_all {p : α → Prop} (μ : F) : (∀ a, p a) → ∀ᵐ a ∂μ, p a 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {p : α → Prop} (μ : F)
Goal: (∀ (a : α), p a) → ∀ᵐ (a : α) ∂μ, p a
-- MeasureTheory.ae_all_iff
theorem ae_all_iff {ι : Sort*} [Countable ι] {p : α → ι → Prop} :
    (∀ᵐ a ∂μ, ∀ i, p a i) ↔ ∀ i, ∀ᵐ a ∂μ, p a i 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {ι : Sort u_4} [Countable ι] {p : α → ι → Prop}
Goal: (∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i) ↔ ∀ (i : ι), ∀ᵐ (a : α) ∂μ, p a i
-- MeasureTheory.all_ae_of
theorem all_ae_of {ι : Sort*} {p : α → ι → Prop} (hp : ∀ᵐ a ∂μ, ∀ i, p a i) (i : ι) :
    ∀ᵐ a ∂μ, p a i 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {ι : Sort u_4} {p : α → ι → Prop} (hp : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i) (i : ι)
Goal: ∀ᵐ (a : α) ∂μ, p a i
-- MeasureTheory.ae_iff_of_countable
lemma ae_iff_of_countable [Countable α] {p : α → Prop} : (∀ᵐ x ∂μ, p x) ↔ ∀ x, μ {x} ≠ 0 → p x 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} [Countable α] {p : α → Prop}
Goal: (∀ᵐ (x : α) ∂μ, p x) ↔ ∀ (x : α), μ {x} ≠ 0 → p x
-- MeasureTheory.ae_ball_iff
theorem ae_ball_iff {ι : Type*} {S : Set ι} (hS : S.Countable) {p : α → ∀ i ∈ S, Prop} :
    (∀ᵐ x ∂μ, ∀ i (hi : i ∈ S), p x i hi) ↔ ∀ i (hi : i ∈ S), ∀ᵐ x ∂μ, p x i hi 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {ι : Type u_4} {S : Set ι} (hS : S.Countable) {p : α → (i : ι) → i ∈ S → Prop}
Goal: (∀ᵐ (x : α) ∂μ, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᵐ (x : α) ∂μ, p x i hi
-- MeasureTheory.ae_eq_refl
theorem ae_eq_refl (f : α → β) : f =ᵐ[μ] f 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} (f : α → β)
Goal: f =ᵐ[μ] f
-- MeasureTheory.ae_eq_symm
theorem ae_eq_symm {f g : α → β} (h : f =ᵐ[μ] g) : g =ᵐ[μ] f 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {f g : α → β} (h : f =ᵐ[μ] g)
Goal: g =ᵐ[μ] f
-- MeasureTheory.ae_eq_trans
theorem ae_eq_trans {f g h : α → β} (h₁ : f =ᵐ[μ] g) (h₂ : g =ᵐ[μ] h) : f =ᵐ[μ] h 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {f g h : α → β} (h₁ : f =ᵐ[μ] g) (h₂ : g =ᵐ[μ] h)
Goal: f =ᵐ[μ] h
-- MeasureTheory.ae_le_of_ae_lt
theorem ae_le_of_ae_lt {β : Type*} [Preorder β] {f g : α → β} (h : ∀ᵐ x ∂μ, f x < g x) :
    f ≤ᵐ[μ] g 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {β : Type u_4} [Preorder β] {f g : α → β} (h : ∀ᵐ (x : α) ∂μ, f x < g x)
Goal: f ≤ᵐ[μ] g
-- MeasureTheory.ae_eq_empty
@[simp]
theorem ae_eq_empty : s =ᵐ[μ] (∅ : Set α) ↔ μ s = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α}
Goal: s =ᵐ[μ] ∅ ↔ μ s = 0
-- MeasureTheory.ae_eq_univ
@[simp high]
theorem ae_eq_univ : s =ᵐ[μ] (univ : Set α) ↔ μ sᶜ = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s : Set α}
Goal: s =ᵐ[μ] Set.univ ↔ μ sᶜ = 0
-- MeasureTheory.ae_le_set
theorem ae_le_set : s ≤ᵐ[μ] t ↔ μ (s \ t) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [FunLike F (Set α) ℝ≥0∞] [MeasureTheory.OuterMeasureClass F α] {μ : F} {s t : Set α}
Goal: s ≤ᵐ[μ] t ↔ μ (s \ t) = 0