ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ENNReal.count_const_le_le_of_tsum_le
theorem _root_.ENNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0∞}
    (a_mble : Measurable a) {c : ℝ≥0∞} (tsum_le_c : ∑' i, a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0)
    (ε_ne_top : ε ≠ ∞) : Measure.count { i : α | ε ≤ a i } ≤ c / ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α → ℝ≥0∞} (a_mble : Measurable a) {c : ℝ≥0∞} (tsum_le_c : ∑' (i : α), a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0) (ε_ne_top : ε ≠ ⊤)
Goal: MeasureTheory.Measure.count {i | ε ≤ a i} ≤ c / ε
-- NNReal.count_const_le_le_of_tsum_le
theorem _root_.NNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0}
    (a_mble : Measurable a) (a_summable : Summable a) {c : ℝ≥0} (tsum_le_c : ∑' i, a i ≤ c)
    {ε : ℝ≥0} (ε_ne_zero : ε ≠ 0) : Measure.count { i : α | ε ≤ a i } ≤ c / ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α → ℝ≥0} (a_mble : Measurable a) (a_summable : Summable a) {c : ℝ≥0} (tsum_le_c : ∑' (i : α), a i ≤ c) {ε : ℝ≥0} (ε_ne_zero : ε ≠ 0)
Goal: MeasureTheory.Measure.count {i | ε ≤ a i} ≤ ↑c / ↑ε
-- MeasureTheory.lintegral_countable'
theorem lintegral_countable' [Countable α] [MeasurableSingletonClass α] (f : α → ℝ≥0∞) :
    ∫⁻ a, f a ∂μ = ∑' a, f a * μ {a} 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable α] [MeasurableSingletonClass α] (f : α → ℝ≥0∞)
Goal: ∫⁻ (a : α), f a ∂μ = ∑' (a : α), f a * μ {a}
-- MeasureTheory.lintegral_singleton'
theorem lintegral_singleton' {f : α → ℝ≥0∞} (hf : Measurable f) (a : α) :
    ∫⁻ x in {a}, f x ∂μ = f a * μ {a} 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (a : α)
Goal: ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}
-- MeasureTheory.lintegral_singleton
theorem lintegral_singleton [MeasurableSingletonClass α] (f : α → ℝ≥0∞) (a : α) :
    ∫⁻ x in {a}, f x ∂μ = f a * μ {a} 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (f : α → ℝ≥0∞) (a : α)
Goal: ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}
-- MeasureTheory.lintegral_countable
theorem lintegral_countable [MeasurableSingletonClass α] (f : α → ℝ≥0∞) {s : Set α}
    (hs : s.Countable) : ∫⁻ a in s, f a ∂μ = ∑' a : s, f a * μ {(a : α)} 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (f : α → ℝ≥0∞) {s : Set α} (hs : s.Countable)
Goal: ∫⁻ (a : α) in s, f a ∂μ = ∑' (a : ↑s), f ↑a * μ {↑a}
-- MeasureTheory.lintegral_insert
theorem lintegral_insert [MeasurableSingletonClass α] {a : α} {s : Set α} (h : a ∉ s)
    (f : α → ℝ≥0∞) : ∫⁻ x in insert a s, f x ∂μ = f a * μ {a} + ∫⁻ x in s, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] {a : α} {s : Set α} (h : a ∉ s) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α) in insert a s, f x ∂μ = f a * μ {a} + ∫⁻ (x : α) in s, f x ∂μ
-- MeasureTheory.lintegral_finset
theorem lintegral_finset [MeasurableSingletonClass α] (s : Finset α) (f : α → ℝ≥0∞) :
    ∫⁻ x in s, f x ∂μ = ∑ x ∈ s, f x * μ {x} 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (s : Finset α) (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α) in ↑s, f x ∂μ = ∑ x ∈ s, f x * μ {x}
-- MeasureTheory.lintegral_fintype
theorem lintegral_fintype [MeasurableSingletonClass α] [Fintype α] (f : α → ℝ≥0∞) :
    ∫⁻ x, f x ∂μ = ∑ x, f x * μ {x} 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] [Fintype α] (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = ∑ x : α, f x * μ {x}
-- MeasureTheory.lintegral_unique
theorem lintegral_unique [Unique α] (f : α → ℝ≥0∞) : ∫⁻ x, f x ∂μ = f default * μ univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Unique α] (f : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), f x ∂μ = f default * μ Set.univ
-- MeasureTheory.ae_lt_top'
theorem ae_lt_top' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h2f : ∫⁻ x, f x ∂μ ≠ ∞) :
    ∀ᵐ x ∂μ, f x < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: ∀ᵐ (x : α) ∂μ, f x < ⊤
-- MeasureTheory.ae_lt_top
theorem ae_lt_top {f : α → ℝ≥0∞} (hf : Measurable f) (h2f : ∫⁻ x, f x ∂μ ≠ ∞) :
    ∀ᵐ x ∂μ, f x < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: ∀ᵐ (x : α) ∂μ, f x < ⊤
-- MeasureTheory.setLIntegral_lt_top_of_le_nnreal
theorem setLIntegral_lt_top_of_le_nnreal {s : Set α} (hs : μ s ≠ ∞) {f : α → ℝ≥0∞}
    (hbdd : ∃ y : ℝ≥0, ∀ x ∈ s, f x ≤ y) : ∫⁻ x in s, f x ∂μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : μ s ≠ ⊤) {f : α → ℝ≥0∞} (hbdd : ∃ y, ∀ x ∈ s, f x ≤ ↑y)
Goal: ∫⁻ (x : α) in s, f x ∂μ < ⊤
-- MeasureTheory.setLIntegral_lt_top_of_bddAbove
theorem setLIntegral_lt_top_of_bddAbove {s : Set α} (hs : μ s ≠ ∞) {f : α → ℝ≥0}
    (hbdd : BddAbove (f '' s)) : ∫⁻ x in s, f x ∂μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : μ s ≠ ⊤) {f : α → ℝ≥0} (hbdd : BddAbove (f '' s))
Goal: ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤
-- MeasureTheory.setLIntegral_lt_top_of_isCompact
theorem setLIntegral_lt_top_of_isCompact [TopologicalSpace α] {s : Set α}
    (hs : μ s ≠ ∞) (hsc : IsCompact s) {f : α → ℝ≥0} (hf : Continuous f) :
    ∫⁻ x in s, f x ∂μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] {s : Set α} (hs : μ s ≠ ⊤) (hsc : IsCompact s) {f : α → ℝ≥0} (hf : Continuous f)
Goal: ∫⁻ (x : α) in s, ↑(f x) ∂μ < ⊤
-- IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal
theorem _root_.IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal {α : Type*}
    [MeasurableSpace α] (μ : Measure α) [μ_fin : IsFiniteMeasure μ] {f : α → ℝ≥0∞}
    (f_bdd : ∃ c : ℝ≥0, ∀ x, f x ≤ c) : ∫⁻ x, f x ∂μ < ∞ 
Arguments: {α : Type u_5} [MeasurableSpace α] (μ : MeasureTheory.Measure α) [μ_fin : MeasureTheory.IsFiniteMeasure μ] {f : α → ℝ≥0∞} (f_bdd : ∃ c, ∀ (x : α), f x ≤ ↑c)
Goal: ∫⁻ (x : α), f x ∂μ < ⊤
-- MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone_aux
lemma tendsto_of_lintegral_tendsto_of_monotone_aux {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hf_meas : ∀ n, AEMeasurable (f n) μ) (hF_meas : AEMeasurable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) (h_int_finite : ∫⁻ a, F a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
Arguments: {α : Type u_5} {mα : MeasurableSpace α} {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ) (hF_meas : AEMeasurable F μ) (hf_tendsto : Filter.Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), F a ∂μ))) (hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a) (h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤)
Goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))
-- MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone
lemma tendsto_of_lintegral_tendsto_of_monotone {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hF_meas : AEMeasurable F μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) (h_int_finite : ∫⁻ a, F a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
Arguments: {α : Type u_5} {mα : MeasurableSpace α} {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hF_meas : AEMeasurable F μ) (hf_tendsto : Filter.Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), F a ∂μ))) (hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a) (h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤)
Goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))
-- MeasureTheory.tendsto_of_lintegral_tendsto_of_antitone
lemma tendsto_of_lintegral_tendsto_of_antitone {α : Type*} {mα : MeasurableSpace α}
    {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
    (hf_meas : ∀ n, AEMeasurable (f n) μ)
    (hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
    (hf_mono : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a))
    (h_bound : ∀ᵐ a ∂μ, ∀ i, F a ≤ f i a) (h0 : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
    ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a)) 
Arguments: {α : Type u_5} {mα : MeasurableSpace α} {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ) (hf_tendsto : Filter.Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) Filter.atTop (𝓝 (∫⁻ (a : α), F a ∂μ))) (hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a) (h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a) (h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤)
Goal: ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (F a))
-- MeasureTheory.exists_measurable_le_forall_setLIntegral_eq
theorem exists_measurable_le_forall_setLIntegral_eq [SFinite μ] (f : α → ℝ≥0∞) :
    ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∀ s, ∫⁻ a in s, f a ∂μ = ∫⁻ a in s, g a ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → ℝ≥0∞)
Goal: ∃ g, Measurable g ∧ g ≤ f ∧ ∀ (s : Set α), ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
-- MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite
theorem exists_pos_lintegral_lt_of_sigmaFinite (μ : Measure α) [SigmaFinite μ] {ε : ℝ≥0∞}
    (ε0 : ε ≠ 0) : ∃ g : α → ℝ≥0, (∀ x, 0 < g x) ∧ Measurable g ∧ ∫⁻ x, g x ∂μ < ε 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] {ε : ℝ≥0∞} (ε0 : ε ≠ 0)
Goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
-- MeasureTheory.lintegral_trim
theorem lintegral_trim {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
    ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.lintegral_trim_ae
theorem lintegral_trim_ae {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f (μ.trim hm)) : ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.trim hm))
Goal: ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.univ_le_of_forall_fin_meas_le
theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → f s ≤ C) (h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n))
Goal: f Set.univ ≤ C
-- MeasureTheory.lintegral_le_of_forall_fin_meas_trim_le
theorem lintegral_le_of_forall_fin_meas_trim_le {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞}
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C)
Goal: ∫⁻ (x : α), f x ∂μ ≤ C
-- MeasureTheory.lintegral_le_of_forall_fin_meas_le
theorem lintegral_le_of_forall_fin_meas_le [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ]
    (C : ℝ≥0∞) {f : α → ℝ≥0∞}
    (hf : ∀ s, MeasurableSet s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C)
Goal: ∫⁻ (x : α), f x ∂μ ≤ C
-- MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral
theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α}
    {μ : Measure α} [SigmaFinite μ] {f : α →ₛ ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
    ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : MeasureTheory.SimpleFunc α ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ (x : α), ↑(↑f x) ∂μ)
Goal: ∃ g, (∀ (x : α), ↑g x ≤ ↑f x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ
-- MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral
theorem exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α} {μ : Measure α}
    [SigmaFinite μ] {f : α → ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
    ∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {f : α → ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ (x : α), ↑(f x) ∂μ)
Goal: ∃ g, (∀ (x : α), ↑g x ≤ f x) ∧ ∫⁻ (x : α), ↑(↑g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(↑g x) ∂μ
-- MeasureTheory.tendsto_measure_of_ae_tendsto_indicator
lemma tendsto_measure_of_ae_tendsto_indicator {μ : Measure α} (A_mble : MeasurableSet A)
    (As_mble : ∀ i, MeasurableSet (As i)) {B : Set α} (B_mble : MeasurableSet B)
    (B_finmeas : μ B ≠ ∞) (As_le_B : ∀ᶠ i in L, As i ⊆ B)
    (h_lim : ∀ᵐ x ∂μ, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
    Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) 
Arguments: {α : Type u_5} [MeasurableSpace α] {A : Set α} {ι : Type u_6} (L : Filter ι) [L.IsCountablyGenerated] {As : ι → Set α} {μ : MeasureTheory.Measure α} (A_mble : MeasurableSet A) (As_mble : ∀ (i : ι), MeasurableSet (As i)) {B : Set α} (B_mble : MeasurableSet B) (B_finmeas : μ B ≠ ⊤) (As_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B) (h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A)
Goal: Filter.Tendsto (fun i => μ (As i)) L (𝓝 (μ A))
-- MeasureTheory.tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure
lemma tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure [IsCountablyGenerated L]
    {μ : Measure α} [IsFiniteMeasure μ] (A_mble : MeasurableSet A)
    (As_mble : ∀ i, MeasurableSet (As i)) (h_lim : ∀ᵐ x ∂μ, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
    Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) 
Arguments: {α : Type u_5} [MeasurableSpace α] {A : Set α} {ι : Type u_6} (L : Filter ι) {As : ι → Set α} [L.IsCountablyGenerated] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (A_mble : MeasurableSet A) (As_mble : ∀ (i : ι), MeasurableSet (As i)) (h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A)
Goal: Filter.Tendsto (fun i => μ (As i)) L (𝓝 (μ A))
-- MeasureTheory.UniformIntegrable.aeStronglyMeasurable
protected theorem aeStronglyMeasurable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ)
    (i : ι) : AEStronglyMeasurable (f i) μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UniformIntegrable f p μ) (i : ι)
Goal: MeasureTheory.AEStronglyMeasurable (f i) μ
-- MeasureTheory.UniformIntegrable.unifIntegrable
protected theorem unifIntegrable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) :
    UnifIntegrable f p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UniformIntegrable f p μ)
Goal: MeasureTheory.UnifIntegrable f p μ
-- MeasureTheory.UniformIntegrable.memℒp
protected theorem memℒp {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) (i : ι) :
    Memℒp (f i) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UniformIntegrable f p μ) (i : ι)
Goal: MeasureTheory.Memℒp (f i) p μ
-- MeasureTheory.UnifIntegrable.add
protected theorem add (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)
    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
    UnifIntegrable (f + g) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UnifIntegrable f p μ) (hg : MeasureTheory.UnifIntegrable g p μ) (hp : 1 ≤ p) (hf_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (hg_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (g i) μ)
Goal: MeasureTheory.UnifIntegrable (f + g) p μ
-- MeasureTheory.UnifIntegrable.neg
protected theorem neg (hf : UnifIntegrable f p μ) : UnifIntegrable (-f) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UnifIntegrable f p μ)
Goal: MeasureTheory.UnifIntegrable (-f) p μ
-- MeasureTheory.UnifIntegrable.sub
protected theorem sub (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)
    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
    UnifIntegrable (f - g) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UnifIntegrable f p μ) (hg : MeasureTheory.UnifIntegrable g p μ) (hp : 1 ≤ p) (hf_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (hg_meas : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (g i) μ)
Goal: MeasureTheory.UnifIntegrable (f - g) p μ
-- MeasureTheory.UnifIntegrable.ae_eq
protected theorem ae_eq (hf : UnifIntegrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UnifIntegrable g p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f g : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UnifIntegrable f p μ) (hfg : ∀ (n : ι), f n =ᵐ[μ] g n)
Goal: MeasureTheory.UnifIntegrable g p μ
-- MeasureTheory.UnifIntegrable.indicator
protected theorem indicator (hf : UnifIntegrable f p μ) (E : Set α) :
    UnifIntegrable (fun i => E.indicator (f i)) p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UnifIntegrable f p μ) (E : Set α)
Goal: MeasureTheory.UnifIntegrable (fun i => E.indicator (f i)) p μ
-- MeasureTheory.UnifIntegrable.restrict
protected theorem restrict (hf : UnifIntegrable f p μ) (E : Set α) :
    UnifIntegrable f p (μ.restrict E) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : ι → α → β} {p : ℝ≥0∞} (hf : MeasureTheory.UnifIntegrable f p μ) (E : Set α)
Goal: MeasureTheory.UnifIntegrable f p (μ.restrict E)
-- MeasureTheory.unifIntegrable_zero_meas
theorem unifIntegrable_zero_meas [MeasurableSpace α] {p : ℝ≥0∞} {f : ι → α → β} :
    UnifIntegrable f p (0 : Measure α) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} [NormedAddCommGroup β] [MeasurableSpace α] {p : ℝ≥0∞} {f : ι → α → β}
Goal: MeasureTheory.UnifIntegrable f p 0
-- MeasureTheory.unifIntegrable_congr_ae
theorem unifIntegrable_congr_ae {p : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UnifIntegrable f p μ ↔ UnifIntegrable g p μ 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ (n : ι), f n =ᵐ[μ] g n)
Goal: MeasureTheory.UnifIntegrable f p μ ↔ MeasureTheory.UnifIntegrable g p μ
-- MeasureTheory.tendsto_indicator_ge
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} [NormedAddCommGroup β] (f : α → β) (x : α)
Goal: Filter.Tendsto (fun M => {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) Filter.atTop (𝓝 0)
-- MeasureTheory.Memℒp.integral_indicator_norm_ge_le
theorem Memℒp.integral_indicator_norm_ge_le (hf : Memℒp f 1 μ) (hmeas : StronglyMeasurable f)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.Memℒp f 1 μ) (hmeas : MeasureTheory.StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ M, ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε
-- MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le_of_meas
theorem Memℒp.integral_indicator_norm_ge_nonneg_le_of_meas (hf : Memℒp f 1 μ)
    (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.Memℒp f 1 μ) (hmeas : MeasureTheory.StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε
-- MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le
theorem Memℒp.integral_indicator_norm_ge_nonneg_le (hf : Memℒp f 1 μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.Memℒp f 1 μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε
-- MeasureTheory.Memℒp.eLpNormEssSup_indicator_norm_ge_eq_zero
theorem Memℒp.eLpNormEssSup_indicator_norm_ge_eq_zero (hf : Memℒp f ∞ μ)
    (hmeas : StronglyMeasurable f) :
    ∃ M : ℝ, eLpNormEssSup ({ x | M ≤ ‖f x‖₊ }.indicator f) μ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.Memℒp f ⊤ μ) (hmeas : MeasureTheory.StronglyMeasurable f)
Goal: ∃ M, MeasureTheory.eLpNormEssSup ({x | M ≤ ↑‖f x‖₊}.indicator f) μ = 0
-- MeasureTheory.Memℒp.eLpNorm_indicator_norm_ge_le
theorem Memℒp.eLpNorm_indicator_norm_ge_le (hf : Memℒp f p μ) (hmeas : StronglyMeasurable f) {ε : ℝ}
    (hε : 0 < ε) : ∃ M : ℝ, eLpNorm ({ x | M ≤ ‖f x‖₊ }.indicator f) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : α → β} (hf : MeasureTheory.Memℒp f p μ) (hmeas : MeasureTheory.StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ M, MeasureTheory.eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
-- MeasureTheory.Memℒp.eLpNorm_indicator_norm_ge_pos_le
theorem Memℒp.eLpNorm_indicator_norm_ge_pos_le (hf : Memℒp f p μ) (hmeas : StronglyMeasurable f)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, 0 < M ∧ eLpNorm ({ x | M ≤ ‖f x‖₊ }.indicator f) p μ ≤ ENNReal.ofReal ε 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {p : ℝ≥0∞} {f : α → β} (hf : MeasureTheory.Memℒp f p μ) (hmeas : MeasureTheory.StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ M, 0 < M ∧ MeasureTheory.eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε