ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousOn.integrableOn_compact'
theorem ContinuousOn.integrableOn_compact'
    (hK : IsCompact K) (h'K : MeasurableSet K) (hf : ContinuousOn f K) :
    IntegrableOn f K μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] (hK : IsCompact K) (h'K : MeasurableSet K) (hf : ContinuousOn f K)
Goal: MeasureTheory.IntegrableOn f K μ
-- ContinuousOn.integrableOn_compact
theorem ContinuousOn.integrableOn_compact [T2Space X]
    (hK : IsCompact K) (hf : ContinuousOn f K) : IntegrableOn f K μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [T2Space X] (hK : IsCompact K) (hf : ContinuousOn f K)
Goal: MeasureTheory.IntegrableOn f K μ
-- ContinuousOn.integrableOn_Icc
theorem ContinuousOn.integrableOn_Icc [Preorder X] [CompactIccSpace X] [T2Space X]
    (hf : ContinuousOn f (Icc a b)) : IntegrableOn f (Icc a b) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {a b : X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [Preorder X] [CompactIccSpace X] [T2Space X] (hf : ContinuousOn f (Set.Icc a b))
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ
-- Continuous.integrableOn_Icc
theorem Continuous.integrableOn_Icc [Preorder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f (Icc a b) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {a b : X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [Preorder X] [CompactIccSpace X] [T2Space X] (hf : Continuous f)
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ
-- Continuous.integrableOn_Ioc
theorem Continuous.integrableOn_Ioc [Preorder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f (Ioc a b) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {a b : X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [Preorder X] [CompactIccSpace X] [T2Space X] (hf : Continuous f)
Goal: MeasureTheory.IntegrableOn f (Set.Ioc a b) μ
-- ContinuousOn.integrableOn_uIcc
theorem ContinuousOn.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] [T2Space X]
    (hf : ContinuousOn f [[a, b]]) : IntegrableOn f [[a, b]] μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {a b : X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [LinearOrder X] [CompactIccSpace X] [T2Space X] (hf : ContinuousOn f [[a, b]])
Goal: MeasureTheory.IntegrableOn f [[a, b]] μ
-- Continuous.integrableOn_uIcc
theorem Continuous.integrableOn_uIcc [LinearOrder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f [[a, b]] μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {a b : X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [LinearOrder X] [CompactIccSpace X] [T2Space X] (hf : Continuous f)
Goal: MeasureTheory.IntegrableOn f [[a, b]] μ
-- Continuous.integrableOn_uIoc
theorem Continuous.integrableOn_uIoc [LinearOrder X] [CompactIccSpace X] [T2Space X]
    (hf : Continuous f) : IntegrableOn f (Ι a b) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {a b : X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [LinearOrder X] [CompactIccSpace X] [T2Space X] (hf : Continuous f)
Goal: MeasureTheory.IntegrableOn f (Ι a b) μ
-- Continuous.integrable_of_hasCompactSupport
theorem Continuous.integrable_of_hasCompactSupport (hf : Continuous f) (hcf : HasCompactSupport f) :
    Integrable f μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] [MeasureTheory.IsFiniteMeasureOnCompacts μ] (hf : Continuous f) (hcf : HasCompactSupport f)
Goal: MeasureTheory.Integrable f μ
-- MonotoneOn.integrableOn_of_measure_ne_top
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E] [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] (hmono : MonotoneOn f s) {a b : X} (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ⊤) (h's : MeasurableSet s)
Goal: MeasureTheory.IntegrableOn f s μ
-- MonotoneOn.integrableOn_isCompact
theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hmono : MonotoneOn f s) : IntegrableOn f s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E] [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] [MeasureTheory.IsFiniteMeasureOnCompacts μ] (hs : IsCompact s) (hmono : MonotoneOn f s)
Goal: MeasureTheory.IntegrableOn f s μ
-- AntitoneOn.integrableOn_of_measure_ne_top
theorem AntitoneOn.integrableOn_of_measure_ne_top (hanti : AntitoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E] [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] (hanti : AntitoneOn f s) {a b : X} (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ⊤) (h's : MeasurableSet s)
Goal: MeasureTheory.IntegrableOn f s μ
-- AntioneOn.integrableOn_isCompact
theorem AntioneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hanti : AntitoneOn f s) : IntegrableOn f s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E] [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] [MeasureTheory.IsFiniteMeasureOnCompacts μ] (hs : IsCompact s) (hanti : AntitoneOn f s)
Goal: MeasureTheory.IntegrableOn f s μ
-- Monotone.locallyIntegrable
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E] [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] [MeasureTheory.IsLocallyFiniteMeasure μ] (hmono : Monotone f)
Goal: MeasureTheory.LocallyIntegrable f μ
-- Antitone.locallyIntegrable
theorem Antitone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hanti : Antitone f) :
    LocallyIntegrable f μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [BorelSpace X] [ConditionallyCompleteLinearOrder X] [ConditionallyCompleteLinearOrder E] [OrderTopology X] [OrderTopology E] [SecondCountableTopology E] [MeasureTheory.IsLocallyFiniteMeasure μ] (hanti : Antitone f)
Goal: MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.IntegrableOn.mul_continuousOn_of_subset
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ 
Arguments: {X : Type u_1} {R : Type u_5} [MeasurableSpace X] [TopologicalSpace X] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {A K : Set X} [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R} (hg : MeasureTheory.IntegrableOn g A μ) (hg' : ContinuousOn g' K) (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K)
Goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) A μ
-- MeasureTheory.IntegrableOn.mul_continuousOn
theorem IntegrableOn.mul_continuousOn [T2Space X] (hg : IntegrableOn g K μ)
    (hg' : ContinuousOn g' K) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ 
Arguments: {X : Type u_1} {R : Type u_5} [MeasurableSpace X] [TopologicalSpace X] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R} [T2Space X] (hg : MeasureTheory.IntegrableOn g K μ) (hg' : ContinuousOn g' K) (hK : IsCompact K)
Goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) K μ
-- MeasureTheory.IntegrableOn.continuousOn_mul_of_subset
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ 
Arguments: {X : Type u_1} {R : Type u_5} [MeasurableSpace X] [TopologicalSpace X] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {A K : Set X} [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R} (hg : ContinuousOn g K) (hg' : MeasureTheory.IntegrableOn g' A μ) (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K)
Goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) A μ
-- MeasureTheory.IntegrableOn.continuousOn_mul
theorem IntegrableOn.continuousOn_mul [T2Space X] (hg : ContinuousOn g K)
    (hg' : IntegrableOn g' K μ) (hK : IsCompact K) : IntegrableOn (fun x => g x * g' x) K μ 
Arguments: {X : Type u_1} {R : Type u_5} [MeasurableSpace X] [TopologicalSpace X] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} [NormedRing R] [SecondCountableTopologyEither X R] {g g' : X → R} [T2Space X] (hg : ContinuousOn g K) (hg' : MeasureTheory.IntegrableOn g' K μ) (hK : IsCompact K)
Goal: MeasureTheory.IntegrableOn (fun x => g x * g' x) K μ
-- MeasureTheory.IntegrableOn.continuousOn_smul
theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 E] [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E} (hg : MeasureTheory.IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K)
Goal: MeasureTheory.IntegrableOn (fun x => f x • g x) K μ
-- MeasureTheory.IntegrableOn.smul_continuousOn
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} {𝕜 : Type u_6} [NormedField 𝕜] [NormedSpace 𝕜 E] [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜} (hf : MeasureTheory.IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K)
Goal: MeasureTheory.IntegrableOn (fun x => f x • g x) K μ
-- MeasureTheory.LocallyIntegrableOn.continuousOn_mul
theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
    [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ 
Arguments: {X : Type u_1} {R : Type u_5} [MeasurableSpace X] [TopologicalSpace X] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] [LocallyCompactSpace X] [T2Space X] [NormedRing R] [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) (hs : IsOpen s)
Goal: MeasureTheory.LocallyIntegrableOn (fun x => g x * f x) s μ
-- MeasureTheory.LocallyIntegrableOn.mul_continuousOn
theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
    [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ 
Arguments: {X : Type u_1} {R : Type u_5} [MeasurableSpace X] [TopologicalSpace X] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] [LocallyCompactSpace X] [T2Space X] [NormedRing R] [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) (hs : IsOpen s)
Goal: MeasureTheory.LocallyIntegrableOn (fun x => f x * g x) s μ
-- MeasureTheory.LocallyIntegrableOn.continuousOn_smul
theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
    [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
    LocallyIntegrableOn (fun x => g x • f x) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] [LocallyCompactSpace X] [T2Space X] {𝕜 : Type u_6} [NormedField 𝕜] [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X} (hs : IsOpen s) (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hg : ContinuousOn g s)
Goal: MeasureTheory.LocallyIntegrableOn (fun x => g x • f x) s μ
-- MeasureTheory.LocallyIntegrableOn.smul_continuousOn
theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
    [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
    LocallyIntegrableOn (fun x => f x • g x) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] [LocallyCompactSpace X] [T2Space X] {𝕜 : Type u_6} [NormedField 𝕜] [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X} (hs : IsOpen s) (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hg : ContinuousOn g s)
Goal: MeasureTheory.LocallyIntegrableOn (fun x => f x • g x) s μ
-- bergelson'
lemma bergelson' {s : ℕ → Set α} (hs : ∀ n, MeasurableSet (s n)) (hr₀ : r ≠ 0)
    (hr : ∀ n, r ≤ μ (s n)) :
    ∃ t : Set ℕ, t.Infinite ∧ ∀ ⦃u⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n) 
Arguments: {α : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {r : ℝ≥0∞} {s : ℕ → Set α} (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hr₀ : r ≠ 0) (hr : ∀ (n : ℕ), r ≤ μ (s n))
Goal: ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)
-- bergelson
lemma bergelson [Infinite ι] {s : ι → Set α} (hs : ∀ i, MeasurableSet (s i)) (hr₀ : r ≠ 0)
    (hr : ∀ i, r ≤ μ (s i)) :
    ∃ t : Set ι, t.Infinite ∧ ∀ ⦃u⦄, u ⊆ t → u.Finite → 0 < μ (⋂ i ∈ u, s i) 
Arguments: {ι : Type u_1} {α : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {r : ℝ≥0∞} [Infinite ι] {s : ι → Set α} (hs : ∀ (i : ι), MeasurableSet (s i)) (hr₀ : r ≠ 0) (hr : ∀ (i : ι), r ≤ μ (s i))
Goal: ∃ t, t.Infinite ∧ ∀ ⦃u : Set ι⦄, u ⊆ t → u.Finite → 0 < μ (⋂ i ∈ u, s i)
-- MeasureTheory.exists_continuous_eLpNorm_sub_le_of_closed
theorem exists_continuous_eLpNorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
    (s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}
    (hε : ε ≠ 0) :
    ∃ f : α → E,
      Continuous f ∧
        eLpNorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
          (∀ x, ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ Memℒp f p μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedSpace ℝ E] [μ.OuterRegular] (hp : p ≠ ⊤) {s u : Set α} (s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ⊤) (c : E) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ f,
  Continuous f ∧
    MeasureTheory.eLpNorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
      (∀ (x : α), ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ MeasureTheory.Memℒp f p μ
-- MeasureTheory.Memℒp.exists_hasCompactSupport_eLpNorm_sub_le
theorem Memℒp.exists_hasCompactSupport_eLpNorm_sub_le [WeaklyLocallyCompactSpace α] [μ.Regular]
    (hp : p ≠ ∞) {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ g : α → E, HasCompactSupport g ∧ eLpNorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [R1Space α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedSpace ℝ E] [WeaklyLocallyCompactSpace α] [μ.Regular] (hp : p ≠ ⊤) {f : α → E} (hf : MeasureTheory.Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ g, HasCompactSupport g ∧ MeasureTheory.eLpNorm (f - g) p μ ≤ ε ∧ Continuous g ∧ MeasureTheory.Memℒp g p μ
-- MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
theorem Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
    [WeaklyLocallyCompactSpace α] [μ.Regular]
    {p : ℝ} (hp : 0 < p) {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ g : α → E,
      HasCompactSupport g ∧
        (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [R1Space α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [WeaklyLocallyCompactSpace α] [μ.Regular] {p : ℝ} (hp : 0 < p) {f : α → E} (hf : MeasureTheory.Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ MeasureTheory.Memℒp g (ENNReal.ofReal p) μ
-- MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le
theorem Integrable.exists_hasCompactSupport_lintegral_sub_le
    [WeaklyLocallyCompactSpace α] [μ.Regular]
    {f : α → E} (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ g : α → E,
      HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [R1Space α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [WeaklyLocallyCompactSpace α] [μ.Regular] {f : α → E} (hf : MeasureTheory.Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ g, HasCompactSupport g ∧ ∫⁻ (x : α), ↑‖f x - g x‖₊ ∂μ ≤ ε ∧ Continuous g ∧ MeasureTheory.Integrable g μ
-- MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le
theorem Integrable.exists_hasCompactSupport_integral_sub_le
    [WeaklyLocallyCompactSpace α] [μ.Regular]
    {f : α → E} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧
      Continuous g ∧ Integrable g μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [R1Space α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [WeaklyLocallyCompactSpace α] [μ.Regular] {f : α → E} (hf : MeasureTheory.Integrable f μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ Continuous g ∧ MeasureTheory.Integrable g μ
-- MeasureTheory.Memℒp.exists_boundedContinuous_eLpNorm_sub_le
theorem Memℒp.exists_boundedContinuous_eLpNorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}
    (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ g : α →ᵇ E, eLpNorm (f - (g : α → E)) p μ ≤ ε ∧ Memℒp g p μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedSpace ℝ E] [μ.WeaklyRegular] (hp : p ≠ ⊤) {f : α → E} (hf : MeasureTheory.Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ g, MeasureTheory.eLpNorm (f - ⇑g) p μ ≤ ε ∧ MeasureTheory.Memℒp (⇑g) p μ
-- MeasureTheory.Memℒp.exists_boundedContinuous_integral_rpow_sub_le
theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
    {f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p) {f : α → E} (hf : MeasureTheory.Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ g, ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ MeasureTheory.Memℒp (⇑g) (ENNReal.ofReal p) μ
-- MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le
theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
    (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [μ.WeaklyRegular] {f : α → E} (hf : MeasureTheory.Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ g, ∫⁻ (x : α), ↑‖f x - g x‖₊ ∂μ ≤ ε ∧ MeasureTheory.Integrable (⇑g) μ
-- MeasureTheory.Integrable.exists_boundedContinuous_integral_sub_le
theorem Integrable.exists_boundedContinuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
    (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Integrable g μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] {E : Type u_2} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [μ.WeaklyRegular] {f : α → E} (hf : MeasureTheory.Integrable f μ) {ε : ℝ} (hε : 0 < ε)
Goal: ∃ g, ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ MeasureTheory.Integrable (⇑g) μ
-- MeasureTheory.Lp.boundedContinuousFunction_dense
theorem boundedContinuousFunction_dense [SecondCountableTopologyEither α E] [Fact (1 ≤ p)]
    (hp : p ≠ ∞) [μ.WeaklyRegular] :
    Dense (boundedContinuousFunction E p μ : Set (Lp E p μ)) 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] (E : Type u_2) [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) {p : ℝ≥0∞} [NormedSpace ℝ E] [SecondCountableTopologyEither α E] [Fact (1 ≤ p)] (hp : p ≠ ⊤) [μ.WeaklyRegular]
Goal: Dense ↑(MeasureTheory.Lp.boundedContinuousFunction E p μ)
-- MeasureTheory.Lp.boundedContinuousFunction_topologicalClosure
theorem boundedContinuousFunction_topologicalClosure [SecondCountableTopologyEither α E]
    [Fact (1 ≤ p)] (hp : p ≠ ∞) [μ.WeaklyRegular] :
    (boundedContinuousFunction E p μ).topologicalClosure = ⊤ 
Arguments: {α : Type u_1} [TopologicalSpace α] [NormalSpace α] [MeasurableSpace α] [BorelSpace α] (E : Type u_2) [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) {p : ℝ≥0∞} [NormedSpace ℝ E] [SecondCountableTopologyEither α E] [Fact (1 ≤ p)] (hp : p ≠ ⊤) [μ.WeaklyRegular]
Goal: (MeasureTheory.Lp.boundedContinuousFunction E p μ).topologicalClosure = ⊤