ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IntegrableOn.locallyIntegrableOn
theorem IntegrableOn.locallyIntegrableOn (hf : IntegrableOn f s μ) : LocallyIntegrableOn 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} (hf : MeasureTheory.IntegrableOn f s μ)
Goal: MeasureTheory.LocallyIntegrableOn f s μ
-- MeasureTheory.LocallyIntegrableOn.integrableOn_isCompact
theorem LocallyIntegrableOn.integrableOn_isCompact (hf : LocallyIntegrableOn f s μ)
    (hs : IsCompact 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} (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hs : IsCompact s)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.LocallyIntegrableOn.integrableOn_compact_subset
theorem LocallyIntegrableOn.integrableOn_compact_subset (hf : LocallyIntegrableOn f s μ) {t : Set X}
    (hst : t ⊆ s) (ht : IsCompact t) : IntegrableOn f t 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} (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X} (hst : t ⊆ s) (ht : IsCompact t)
Goal: MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.LocallyIntegrableOn.exists_countable_integrableOn
theorem LocallyIntegrableOn.exists_countable_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ T : Set (Set X), T.Countable ∧
    (∀ u ∈ T, IsOpen u) ∧ (s ⊆ ⋃ u ∈ T, u) ∧ (∀ u ∈ T, IntegrableOn f (u ∩ 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} [SecondCountableTopology X] (hf : MeasureTheory.LocallyIntegrableOn f s μ)
Goal: ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, MeasureTheory.IntegrableOn f (u ∩ s) μ
-- MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ 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} [SecondCountableTopology X] (hf : MeasureTheory.LocallyIntegrableOn f s μ)
Goal: ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), MeasureTheory.IntegrableOn f (u n ∩ s) μ
-- MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable
theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
    (hf : LocallyIntegrableOn f s μ) : AEStronglyMeasurable f (μ.restrict s) 
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} [SecondCountableTopology X] (hf : MeasureTheory.LocallyIntegrableOn f s μ)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- MeasureTheory.locallyIntegrableOn_iff
theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s) :
    LocallyIntegrableOn f s μ ↔ ∀ (k : Set X), k ⊆ s → (IsCompact 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} {s : Set X} [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s)
Goal: MeasureTheory.LocallyIntegrableOn f s μ ↔ ∀ k ⊆ s, IsCompact k → MeasureTheory.IntegrableOn f k μ
-- MeasureTheory.LocallyIntegrableOn.add
protected theorem LocallyIntegrableOn.add
    (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
    LocallyIntegrableOn (f + g) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f g : X → E} {μ : MeasureTheory.Measure X} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hg : MeasureTheory.LocallyIntegrableOn g s μ)
Goal: MeasureTheory.LocallyIntegrableOn (f + g) s μ
-- MeasureTheory.LocallyIntegrableOn.sub
protected theorem LocallyIntegrableOn.sub
    (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
    LocallyIntegrableOn (f - g) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f g : X → E} {μ : MeasureTheory.Measure X} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ) (hg : MeasureTheory.LocallyIntegrableOn g s μ)
Goal: MeasureTheory.LocallyIntegrableOn (f - g) s μ
-- MeasureTheory.LocallyIntegrableOn.neg
protected theorem LocallyIntegrableOn.neg (hf : LocallyIntegrableOn f s μ) :
    LocallyIntegrableOn (-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} (hf : MeasureTheory.LocallyIntegrableOn f s μ)
Goal: MeasureTheory.LocallyIntegrableOn (-f) s μ
-- MeasureTheory.locallyIntegrable_comap
theorem locallyIntegrable_comap (hs : MeasurableSet s) :
    LocallyIntegrable (fun x : s ↦ f x) (μ.comap Subtype.val) ↔ LocallyIntegrableOn 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} (hs : MeasurableSet s)
Goal: MeasureTheory.LocallyIntegrable (fun x => f ↑x) (MeasureTheory.Measure.comap Subtype.val μ) ↔
  MeasureTheory.LocallyIntegrableOn f s μ
-- MeasureTheory.locallyIntegrableOn_univ
theorem locallyIntegrableOn_univ : LocallyIntegrableOn f univ μ ↔ 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}
Goal: MeasureTheory.LocallyIntegrableOn f Set.univ μ ↔ MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.LocallyIntegrable.locallyIntegrableOn
theorem LocallyIntegrable.locallyIntegrableOn (hf : LocallyIntegrable f μ) (s : Set X) :
    LocallyIntegrableOn 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} (hf : MeasureTheory.LocallyIntegrable f μ) (s : Set X)
Goal: MeasureTheory.LocallyIntegrableOn f s μ
-- MeasureTheory.Integrable.locallyIntegrable
theorem Integrable.locallyIntegrable (hf : Integrable f μ) : 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} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.LocallyIntegrable.mono
theorem LocallyIntegrable.mono (hf : LocallyIntegrable f μ) {g : X → F}
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
    LocallyIntegrable g μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : X → E} {μ : MeasureTheory.Measure X} (hf : MeasureTheory.LocallyIntegrable f μ) {g : X → F} (hg : MeasureTheory.AEStronglyMeasurable g μ) (h : ∀ᵐ (x : X) ∂μ, ‖g x‖ ≤ ‖f x‖)
Goal: MeasureTheory.LocallyIntegrable g μ
-- MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict
theorem locallyIntegrableOn_of_locallyIntegrable_restrict [OpensMeasurableSpace X]
    (hf : LocallyIntegrable f (μ.restrict s)) : LocallyIntegrableOn 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} [OpensMeasurableSpace X] (hf : MeasureTheory.LocallyIntegrable f (μ.restrict s))
Goal: MeasureTheory.LocallyIntegrableOn f s μ
-- MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict
theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace X]
    (hs : IsClosed s) : LocallyIntegrableOn f s μ ↔ LocallyIntegrable f (μ.restrict s) 
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} [OpensMeasurableSpace X] (hs : IsClosed s)
Goal: MeasureTheory.LocallyIntegrableOn f s μ ↔ MeasureTheory.LocallyIntegrable f (μ.restrict s)
-- MeasureTheory.LocallyIntegrable.integrableOn_isCompact
theorem LocallyIntegrable.integrableOn_isCompact {k : Set X} (hf : LocallyIntegrable f μ)
    (hk : IsCompact 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} {k : Set X} (hf : MeasureTheory.LocallyIntegrable f μ) (hk : IsCompact k)
Goal: MeasureTheory.IntegrableOn f k μ
-- MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact
theorem LocallyIntegrable.integrableOn_nhds_isCompact (hf : LocallyIntegrable f μ) {k : Set X}
    (hk : IsCompact k) : ∃ u, IsOpen u ∧ k ⊆ u ∧ IntegrableOn f u μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} (hf : MeasureTheory.LocallyIntegrable f μ) {k : Set X} (hk : IsCompact k)
Goal: ∃ u, IsOpen u ∧ k ⊆ u ∧ MeasureTheory.IntegrableOn f u μ
-- MeasureTheory.locallyIntegrable_iff
theorem locallyIntegrable_iff [LocallyCompactSpace X] :
    LocallyIntegrable f μ ↔ ∀ k : Set X, IsCompact 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} [LocallyCompactSpace X]
Goal: MeasureTheory.LocallyIntegrable f μ ↔ ∀ (k : Set X), IsCompact k → MeasureTheory.IntegrableOn f k μ
-- MeasureTheory.LocallyIntegrable.aestronglyMeasurable
theorem LocallyIntegrable.aestronglyMeasurable [SecondCountableTopology X]
    (hf : LocallyIntegrable f μ) : AEStronglyMeasurable f μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [SecondCountableTopology X] (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.LocallyIntegrable.exists_nat_integrableOn
theorem LocallyIntegrable.exists_nat_integrableOn [SecondCountableTopology X]
    (hf : LocallyIntegrable f μ) : ∃ u : ℕ → Set X,
    (∀ n, IsOpen (u n)) ∧ ((⋃ n, u n) = univ) ∧ (∀ n, IntegrableOn f (u n) μ) 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [SecondCountableTopology X] (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ ⋃ n, u n = Set.univ ∧ ∀ (n : ℕ), MeasureTheory.IntegrableOn f (u n) μ
-- MeasureTheory.Memℒp.locallyIntegrable
theorem Memℒp.locallyIntegrable [IsLocallyFiniteMeasure μ] {f : X → E} {p : ℝ≥0∞}
    (hf : Memℒp f p μ) (hp : 1 ≤ p) : LocallyIntegrable f μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [MeasureTheory.IsLocallyFiniteMeasure μ] {f : X → E} {p : ℝ≥0∞} (hf : MeasureTheory.Memℒp f p μ) (hp : 1 ≤ p)
Goal: MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.locallyIntegrable_const
theorem locallyIntegrable_const [IsLocallyFiniteMeasure μ] (c : E) :
    LocallyIntegrable (fun _ => c) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [MeasureTheory.IsLocallyFiniteMeasure μ] (c : E)
Goal: MeasureTheory.LocallyIntegrable (fun x => c) μ
-- MeasureTheory.locallyIntegrableOn_const
theorem locallyIntegrableOn_const [IsLocallyFiniteMeasure μ] (c : E) :
    LocallyIntegrableOn (fun _ => c) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} {s : Set X} [MeasureTheory.IsLocallyFiniteMeasure μ] (c : E)
Goal: MeasureTheory.LocallyIntegrableOn (fun x => c) s μ
-- MeasureTheory.locallyIntegrable_zero
theorem locallyIntegrable_zero : LocallyIntegrable (fun _ ↦ (0 : E)) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X}
Goal: MeasureTheory.LocallyIntegrable (fun x => 0) μ
-- MeasureTheory.locallyIntegrableOn_zero
theorem locallyIntegrableOn_zero : LocallyIntegrableOn (fun _ ↦ (0 : E)) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} {s : Set X}
Goal: MeasureTheory.LocallyIntegrableOn (fun x => 0) s μ
-- MeasureTheory.LocallyIntegrable.indicator
theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
    (hs : MeasurableSet s) : LocallyIntegrable (s.indicator f) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} (hf : MeasureTheory.LocallyIntegrable f μ) {s : Set X} (hs : MeasurableSet s)
Goal: MeasureTheory.LocallyIntegrable (s.indicator f) μ
-- MeasureTheory.locallyIntegrable_map_homeomorph
theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E}
    {μ : Measure X} : LocallyIntegrable f (Measure.map e μ) ↔ LocallyIntegrable (f ∘ e) μ 
Arguments: {X : Type u_1} {Y : Type u_2} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [MeasurableSpace Y] [TopologicalSpace Y] [NormedAddCommGroup E] [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E} {μ : MeasureTheory.Measure X}
Goal: MeasureTheory.LocallyIntegrable f (MeasureTheory.Measure.map (⇑e) μ) ↔ MeasureTheory.LocallyIntegrable (f ∘ ⇑e) μ
-- MeasureTheory.LocallyIntegrable.add
protected theorem LocallyIntegrable.add (hf : LocallyIntegrable f μ) (hg : LocallyIntegrable g μ) :
    LocallyIntegrable (f + g) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f g : X → E} {μ : MeasureTheory.Measure X} (hf : MeasureTheory.LocallyIntegrable f μ) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: MeasureTheory.LocallyIntegrable (f + g) μ
-- MeasureTheory.LocallyIntegrable.sub
protected theorem LocallyIntegrable.sub (hf : LocallyIntegrable f μ) (hg : LocallyIntegrable g μ) :
    LocallyIntegrable (f - g) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f g : X → E} {μ : MeasureTheory.Measure X} (hf : MeasureTheory.LocallyIntegrable f μ) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: MeasureTheory.LocallyIntegrable (f - g) μ
-- MeasureTheory.LocallyIntegrable.neg
protected theorem LocallyIntegrable.neg (hf : LocallyIntegrable f μ) :
    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} (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: MeasureTheory.LocallyIntegrable (-f) μ
-- MeasureTheory.LocallyIntegrable.smul
protected theorem LocallyIntegrable.smul {𝕜 : Type*} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E]
    [BoundedSMul 𝕜 E] (hf : LocallyIntegrable f μ) (c : 𝕜) :
    LocallyIntegrable (c • f) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {𝕜 : Type u_6} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E] [BoundedSMul 𝕜 E] (hf : MeasureTheory.LocallyIntegrable f μ) (c : 𝕜)
Goal: MeasureTheory.LocallyIntegrable (c • f) μ
-- MeasureTheory.locallyIntegrable_finset_sum'
theorem locallyIntegrable_finset_sum' {ι} (s : Finset ι) {f : ι → X → E}
    (hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (∑ i ∈ s, f i) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} {ι : Type u_6} (s : Finset ι) {f : ι → X → E} (hf : ∀ i ∈ s, MeasureTheory.LocallyIntegrable (f i) μ)
Goal: MeasureTheory.LocallyIntegrable (∑ i ∈ s, f i) μ
-- MeasureTheory.locallyIntegrable_finset_sum
theorem locallyIntegrable_finset_sum {ι} (s : Finset ι) {f : ι → X → E}
    (hf : ∀ i ∈ s, LocallyIntegrable (f i) μ) : LocallyIntegrable (fun a ↦ ∑ i ∈ s, f i a) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} {ι : Type u_6} (s : Finset ι) {f : ι → X → E} (hf : ∀ i ∈ s, MeasureTheory.LocallyIntegrable (f i) μ)
Goal: MeasureTheory.LocallyIntegrable (fun a => ∑ i ∈ s, f i a) μ
-- MeasureTheory.LocallyIntegrable.integrable_smul_left_of_hasCompactSupport
theorem LocallyIntegrable.integrable_smul_left_of_hasCompactSupport
    [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X]
    (hf : LocallyIntegrable f μ) {g : X → ℝ} (hg : Continuous g) (h'g : HasCompactSupport g) :
    Integrable (fun x ↦ g x • f x) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X] (hf : MeasureTheory.LocallyIntegrable f μ) {g : X → ℝ} (hg : Continuous g) (h'g : HasCompactSupport g)
Goal: MeasureTheory.Integrable (fun x => g x • f x) μ
-- MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
theorem LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
    [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X] {f : X → ℝ} (hf : LocallyIntegrable f μ)
    {g : X → E} (hg : Continuous g) (h'g : HasCompactSupport g) :
    Integrable (fun x ↦ f x • g x) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {μ : MeasureTheory.Measure X} [NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X] {f : X → ℝ} (hf : MeasureTheory.LocallyIntegrable f μ) {g : X → E} (hg : Continuous g) (h'g : HasCompactSupport g)
Goal: MeasureTheory.Integrable (fun x => f x • g x) μ
-- MeasureTheory.integrable_iff_integrableAtFilter_cocompact
theorem integrable_iff_integrableAtFilter_cocompact :
    Integrable f μ ↔ (IntegrableAtFilter f (cocompact X) μ ∧ 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}
Goal: MeasureTheory.Integrable f μ ↔
  MeasureTheory.IntegrableAtFilter f (Filter.cocompact X) μ ∧ MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.integrable_iff_integrableAtFilter_atBot_atTop
theorem integrable_iff_integrableAtFilter_atBot_atTop [LinearOrder X] [CompactIccSpace X] :
    Integrable f μ ↔
    (IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f atTop μ) ∧ 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} [LinearOrder X] [CompactIccSpace X]
Goal: MeasureTheory.Integrable f μ ↔
  (MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.IntegrableAtFilter f Filter.atTop μ) ∧
    MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.integrable_iff_integrableAtFilter_atBot
theorem integrable_iff_integrableAtFilter_atBot [LinearOrder X] [OrderTop X] [CompactIccSpace X] :
    Integrable f μ ↔ IntegrableAtFilter f atBot μ ∧ 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} [LinearOrder X] [OrderTop X] [CompactIccSpace X]
Goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.integrable_iff_integrableAtFilter_atTop
theorem integrable_iff_integrableAtFilter_atTop [LinearOrder X] [OrderBot X] [CompactIccSpace X] :
    Integrable f μ ↔ IntegrableAtFilter f atTop μ ∧ 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} [LinearOrder X] [OrderBot X] [CompactIccSpace X]
Goal: MeasureTheory.Integrable f μ ↔ MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧ MeasureTheory.LocallyIntegrable f μ
-- MeasureTheory.integrableOn_Iic_iff_integrableAtFilter_atBot
theorem integrableOn_Iic_iff_integrableAtFilter_atBot [LinearOrder X] [CompactIccSpace X] :
    IntegrableOn f (Iic a) μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrableOn f (Iic a) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {a : X} [LinearOrder X] [CompactIccSpace X]
Goal: MeasureTheory.IntegrableOn f (Set.Iic a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Iic a) μ
-- MeasureTheory.integrableOn_Ici_iff_integrableAtFilter_atTop
theorem integrableOn_Ici_iff_integrableAtFilter_atTop [LinearOrder X] [CompactIccSpace X] :
    IntegrableOn f (Ici a) μ ↔ IntegrableAtFilter f atTop μ ∧ LocallyIntegrableOn f (Ici a) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {a : X} [LinearOrder X] [CompactIccSpace X]
Goal: MeasureTheory.IntegrableOn f (Set.Ici a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Ici a) μ
-- MeasureTheory.integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin
theorem integrableOn_Iio_iff_integrableAtFilter_atBot_nhdsWithin
    [LinearOrder X] [CompactIccSpace X] [NoMinOrder X] [OrderTopology X] :
    IntegrableOn f (Iio a) μ ↔ IntegrableAtFilter f atBot μ ∧
    IntegrableAtFilter f (𝓝[<] a) μ ∧ LocallyIntegrableOn f (Iio a) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {a : X} [LinearOrder X] [CompactIccSpace X] [NoMinOrder X] [OrderTopology X]
Goal: MeasureTheory.IntegrableOn f (Set.Iio a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atBot μ ∧
    MeasureTheory.IntegrableAtFilter f (𝓝[<] a) μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Iio a) μ
-- MeasureTheory.integrableOn_Ioi_iff_integrableAtFilter_atTop_nhdsWithin
theorem integrableOn_Ioi_iff_integrableAtFilter_atTop_nhdsWithin
    [LinearOrder X] [CompactIccSpace X] [NoMaxOrder X] [OrderTopology X] :
    IntegrableOn f (Ioi a) μ ↔ IntegrableAtFilter f atTop μ ∧
    IntegrableAtFilter f (𝓝[>] a) μ ∧ LocallyIntegrableOn f (Ioi a) μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {a : X} [LinearOrder X] [CompactIccSpace X] [NoMaxOrder X] [OrderTopology X]
Goal: MeasureTheory.IntegrableOn f (Set.Ioi a) μ ↔
  MeasureTheory.IntegrableAtFilter f Filter.atTop μ ∧
    MeasureTheory.IntegrableAtFilter f (𝓝[>] a) μ ∧ MeasureTheory.LocallyIntegrableOn f (Set.Ioi a) μ
-- Continuous.locallyIntegrable
theorem Continuous.locallyIntegrable [IsLocallyFiniteMeasure μ] [SecondCountableTopologyEither X E]
    (hf : Continuous 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} [OpensMeasurableSpace X] [MeasureTheory.IsLocallyFiniteMeasure μ] [SecondCountableTopologyEither X E] (hf : Continuous f)
Goal: MeasureTheory.LocallyIntegrable f μ
-- ContinuousOn.locallyIntegrableOn
theorem ContinuousOn.locallyIntegrableOn [IsLocallyFiniteMeasure μ]
    [SecondCountableTopologyEither X E] (hf : ContinuousOn f K)
    (hK : MeasurableSet K) : LocallyIntegrableOn 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.IsLocallyFiniteMeasure μ] [SecondCountableTopologyEither X E] (hf : ContinuousOn f K) (hK : MeasurableSet K)
Goal: MeasureTheory.LocallyIntegrableOn f K μ