ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.IntegrableOn.add_measure
theorem IntegrableOn.add_measure (hμ : IntegrableOn f s μ) (hν : IntegrableOn f s ν) :
    IntegrableOn f s (μ + ν) 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ ν : MeasureTheory.Measure α} (hμ : MeasureTheory.IntegrableOn f s μ) (hν : MeasureTheory.IntegrableOn f s ν)
Goal: MeasureTheory.IntegrableOn f s (μ + ν)
-- MeasureTheory.integrableOn_add_measure
@[simp]
theorem integrableOn_add_measure :
    IntegrableOn f s (μ + ν) ↔ IntegrableOn f s μ ∧ IntegrableOn f s ν 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ ν : MeasureTheory.Measure α}
Goal: MeasureTheory.IntegrableOn f s (μ + ν) ↔ MeasureTheory.IntegrableOn f s μ ∧ MeasureTheory.IntegrableOn f s ν
-- MeasurableEmbedding.integrableOn_map_iff
theorem _root_.MeasurableEmbedding.integrableOn_map_iff [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} {μ : Measure α} {s : Set β} :
    IntegrableOn f s (μ.map e) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [MeasurableSpace β] {e : α → β} (he : MeasurableEmbedding e) {f : β → E} {μ : MeasureTheory.Measure α} {s : Set β}
Goal: MeasureTheory.IntegrableOn f s (MeasureTheory.Measure.map e μ) ↔ MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) μ
-- MeasurableEmbedding.integrableOn_iff_comap
theorem _root_.MeasurableEmbedding.integrableOn_iff_comap [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} {μ : Measure β} {s : Set β} (hs : s ⊆ range e) :
    IntegrableOn f s μ ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) (μ.comap e) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [MeasurableSpace β] {e : α → β} (he : MeasurableEmbedding e) {f : β → E} {μ : MeasureTheory.Measure β} {s : Set β} (hs : s ⊆ Set.range e)
Goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) (MeasureTheory.Measure.comap e μ)
-- MeasureTheory.integrableOn_map_equiv
theorem integrableOn_map_equiv [MeasurableSpace β] (e : α ≃ᵐ β) {f : β → E} {μ : Measure α}
    {s : Set β} : IntegrableOn f s (μ.map e) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [MeasurableSpace β] (e : α ≃ᵐ β) {f : β → E} {μ : MeasureTheory.Measure α} {s : Set β}
Goal: MeasureTheory.IntegrableOn f s (MeasureTheory.Measure.map (⇑e) μ) ↔ MeasureTheory.IntegrableOn (f ∘ ⇑e) (⇑e ⁻¹' s) μ
-- MeasureTheory.MeasurePreserving.integrableOn_comp_preimage
theorem MeasurePreserving.integrableOn_comp_preimage [MeasurableSpace β] {e : α → β} {ν}
    (h₁ : MeasurePreserving e μ ν) (h₂ : MeasurableEmbedding e) {f : β → E} {s : Set β} :
    IntegrableOn (f ∘ e) (e ⁻¹' s) μ ↔ IntegrableOn f s ν 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [MeasurableSpace β] {e : α → β} {ν : MeasureTheory.Measure β} (h₁ : MeasureTheory.MeasurePreserving e μ ν) (h₂ : MeasurableEmbedding e) {f : β → E} {s : Set β}
Goal: MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) μ ↔ MeasureTheory.IntegrableOn f s ν
-- MeasureTheory.MeasurePreserving.integrableOn_image
theorem MeasurePreserving.integrableOn_image [MeasurableSpace β] {e : α → β} {ν}
    (h₁ : MeasurePreserving e μ ν) (h₂ : MeasurableEmbedding e) {f : β → E} {s : Set α} :
    IntegrableOn f (e '' s) ν ↔ IntegrableOn (f ∘ e) s μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} [MeasurableSpace β] {e : α → β} {ν : MeasureTheory.Measure β} (h₁ : MeasureTheory.MeasurePreserving e μ ν) (h₂ : MeasurableEmbedding e) {f : β → E} {s : Set α}
Goal: MeasureTheory.IntegrableOn f (e '' s) ν ↔ MeasureTheory.IntegrableOn (f ∘ e) s μ
-- MeasureTheory.integrable_indicator_iff
theorem integrable_indicator_iff (hs : MeasurableSet s) :
    Integrable (indicator s f) μ ↔ IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (s.indicator f) μ ↔ MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.integrable_indicator
theorem IntegrableOn.integrable_indicator (h : IntegrableOn f s μ) (hs : MeasurableSet s) :
    Integrable (indicator s f) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s μ) (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (s.indicator f) μ
-- MeasureTheory.Integrable.indicator
theorem Integrable.indicator (h : Integrable f μ) (hs : MeasurableSet s) :
    Integrable (indicator s f) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.Integrable f μ) (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (s.indicator f) μ
-- MeasureTheory.IntegrableOn.indicator
theorem IntegrableOn.indicator (h : IntegrableOn f s μ) (ht : MeasurableSet t) :
    IntegrableOn (indicator t f) s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s μ) (ht : MeasurableSet t)
Goal: MeasureTheory.IntegrableOn (t.indicator f) s μ
-- MeasureTheory.integrable_indicatorConstLp
theorem integrable_indicatorConstLp {E} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α}
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) :
    Integrable (indicatorConstLp p hs hμs c) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : E)
Goal: MeasureTheory.Integrable (↑↑(MeasureTheory.indicatorConstLp p hs hμs c)) μ
-- MeasureTheory.IntegrableOn.restrict_toMeasurable
theorem IntegrableOn.restrict_toMeasurable (hf : IntegrableOn f s μ) (h's : ∀ x ∈ s, f x ≠ 0) :
    μ.restrict (toMeasurable μ s) = μ.restrict s 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.IntegrableOn f s μ) (h's : ∀ x ∈ s, f x ≠ 0)
Goal: μ.restrict (MeasureTheory.toMeasurable μ s) = μ.restrict s
-- MeasureTheory.IntegrableOn.of_ae_diff_eq_zero
theorem IntegrableOn.of_ae_diff_eq_zero (hf : IntegrableOn f s μ) (ht : NullMeasurableSet t μ)
    (h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) : IntegrableOn f t μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.IntegrableOn f s μ) (ht : MeasureTheory.NullMeasurableSet t μ) (h't : ∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0)
Goal: MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.IntegrableOn.of_forall_diff_eq_zero
theorem IntegrableOn.of_forall_diff_eq_zero (hf : IntegrableOn f s μ) (ht : MeasurableSet t)
    (h't : ∀ x ∈ t \ s, f x = 0) : IntegrableOn f t μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.IntegrableOn f s μ) (ht : MeasurableSet t) (h't : ∀ x ∈ t \ s, f x = 0)
Goal: MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.IntegrableOn.integrable_of_ae_not_mem_eq_zero
theorem IntegrableOn.integrable_of_ae_not_mem_eq_zero (hf : IntegrableOn f s μ)
    (h't : ∀ᵐ x ∂μ, x ∉ s → f x = 0) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.IntegrableOn f s μ) (h't : ∀ᵐ (x : α) ∂μ, x ∉ s → f x = 0)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.IntegrableOn.integrable_of_forall_not_mem_eq_zero
theorem IntegrableOn.integrable_of_forall_not_mem_eq_zero (hf : IntegrableOn f s μ)
    (h't : ∀ x, x ∉ s → f x = 0) : Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.IntegrableOn f s μ) (h't : ∀ x ∉ s, f x = 0)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integrableOn_iff_integrable_of_support_subset
theorem integrableOn_iff_integrable_of_support_subset (h1s : support f ⊆ s) :
    IntegrableOn f s μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h1s : Function.support f ⊆ s)
Goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrableOn_Lp_of_measure_ne_top
theorem integrableOn_Lp_of_measure_ne_top {E} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α}
    (f : Lp E p μ) (hp : 1 ≤ p) (hμs : μ s ≠ ∞) : IntegrableOn f s μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α} (f : ↥(MeasureTheory.Lp E p μ)) (hp : 1 ≤ p) (hμs : μ s ≠ ⊤)
Goal: MeasureTheory.IntegrableOn (↑↑f) s μ
-- MeasureTheory.Integrable.lintegral_lt_top
theorem Integrable.lintegral_lt_top {f : α → ℝ} (hf : Integrable f μ) :
    (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ < ⊤
-- MeasureTheory.IntegrableOn.setLIntegral_lt_top
theorem IntegrableOn.setLIntegral_lt_top {f : α → ℝ} {s : Set α} (hf : IntegrableOn f s μ) :
    (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) < ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ℝ} {s : Set α} (hf : MeasureTheory.IntegrableOn f s μ)
Goal: ∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ < ⊤
-- MeasurableEmbedding.integrableAtFilter_map_iff
theorem _root_.MeasurableEmbedding.integrableAtFilter_map_iff [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} :
    IntegrableAtFilter f (l.map e) (μ.map e) ↔ IntegrableAtFilter (f ∘ e) l μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {l : Filter α} [MeasurableSpace β] {e : α → β} (he : MeasurableEmbedding e) {f : β → E}
Goal: MeasureTheory.IntegrableAtFilter f (Filter.map e l) (MeasureTheory.Measure.map e μ) ↔
  MeasureTheory.IntegrableAtFilter (f ∘ e) l μ
-- MeasurableEmbedding.integrableAtFilter_iff_comap
theorem _root_.MeasurableEmbedding.integrableAtFilter_iff_comap [MeasurableSpace β] {e : α → β}
    (he : MeasurableEmbedding e) {f : β → E} {μ : Measure β} :
    IntegrableAtFilter f (l.map e) μ ↔ IntegrableAtFilter (f ∘ e) l (μ.comap e) 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {l : Filter α} [MeasurableSpace β] {e : α → β} (he : MeasurableEmbedding e) {f : β → E} {μ : MeasureTheory.Measure β}
Goal: MeasureTheory.IntegrableAtFilter f (Filter.map e l) μ ↔
  MeasureTheory.IntegrableAtFilter (f ∘ e) l (MeasureTheory.Measure.comap e μ)
-- MeasureTheory.Integrable.integrableAtFilter
theorem Integrable.integrableAtFilter (h : Integrable f μ) (l : Filter α) :
    IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} (h : MeasureTheory.Integrable f μ) (l : Filter α)
Goal: MeasureTheory.IntegrableAtFilter f l μ
-- MeasureTheory.IntegrableAtFilter.eventually
protected theorem IntegrableAtFilter.eventually (h : IntegrableAtFilter f l μ) :
    ∀ᶠ s in l.smallSets, IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α} (h : MeasureTheory.IntegrableAtFilter f l μ)
Goal: ∀ᶠ (s : Set α) in l.smallSets, MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableAtFilter.add
protected theorem IntegrableAtFilter.add {f g : α → E}
    (hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
    IntegrableAtFilter (f + g) l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {l : Filter α} {f g : α → E} (hf : MeasureTheory.IntegrableAtFilter f l μ) (hg : MeasureTheory.IntegrableAtFilter g l μ)
Goal: MeasureTheory.IntegrableAtFilter (f + g) l μ
-- MeasureTheory.IntegrableAtFilter.neg
protected theorem IntegrableAtFilter.neg {f : α → E} (hf : IntegrableAtFilter f l μ) :
    IntegrableAtFilter (-f) l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {l : Filter α} {f : α → E} (hf : MeasureTheory.IntegrableAtFilter f l μ)
Goal: MeasureTheory.IntegrableAtFilter (-f) l μ
-- MeasureTheory.IntegrableAtFilter.sub
protected theorem IntegrableAtFilter.sub {f g : α → E}
    (hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
    IntegrableAtFilter (f - g) l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {l : Filter α} {f g : α → E} (hf : MeasureTheory.IntegrableAtFilter f l μ) (hg : MeasureTheory.IntegrableAtFilter g l μ)
Goal: MeasureTheory.IntegrableAtFilter (f - g) l μ
-- MeasureTheory.IntegrableAtFilter.smul
protected theorem IntegrableAtFilter.smul {𝕜 : Type*} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E]
    [BoundedSMul 𝕜 E] {f : α → E} (hf : IntegrableAtFilter f l μ) (c : 𝕜) :
    IntegrableAtFilter (c • f) l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {l : Filter α} {𝕜 : Type u_5} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E] [BoundedSMul 𝕜 E] {f : α → E} (hf : MeasureTheory.IntegrableAtFilter f l μ) (c : 𝕜)
Goal: MeasureTheory.IntegrableAtFilter (c • f) l μ
-- MeasureTheory.IntegrableAtFilter.norm
protected theorem IntegrableAtFilter.norm (hf : IntegrableAtFilter f l μ) :
    IntegrableAtFilter (fun x => ‖f x‖) l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α} (hf : MeasureTheory.IntegrableAtFilter f l μ)
Goal: MeasureTheory.IntegrableAtFilter (fun x => ‖f x‖) l μ
-- MeasureTheory.IntegrableAtFilter.filter_mono
theorem IntegrableAtFilter.filter_mono (hl : l ≤ l') (hl' : IntegrableAtFilter f l' μ) :
    IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l l' : Filter α} (hl : l ≤ l') (hl' : MeasureTheory.IntegrableAtFilter f l' μ)
Goal: MeasureTheory.IntegrableAtFilter f l μ
-- MeasureTheory.IntegrableAtFilter.inf_of_left
theorem IntegrableAtFilter.inf_of_left (hl : IntegrableAtFilter f l μ) :
    IntegrableAtFilter f (l ⊓ l') μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l l' : Filter α} (hl : MeasureTheory.IntegrableAtFilter f l μ)
Goal: MeasureTheory.IntegrableAtFilter f (l ⊓ l') μ
-- MeasureTheory.IntegrableAtFilter.inf_of_right
theorem IntegrableAtFilter.inf_of_right (hl : IntegrableAtFilter f l μ) :
    IntegrableAtFilter f (l' ⊓ l) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l l' : Filter α} (hl : MeasureTheory.IntegrableAtFilter f l μ)
Goal: MeasureTheory.IntegrableAtFilter f (l' ⊓ l) μ
-- MeasureTheory.IntegrableAtFilter.inf_ae_iff
@[simp]
theorem IntegrableAtFilter.inf_ae_iff {l : Filter α} :
    IntegrableAtFilter f (l ⊓ ae μ) μ ↔ IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α}
Goal: MeasureTheory.IntegrableAtFilter f (l ⊓ MeasureTheory.ae μ) μ ↔ MeasureTheory.IntegrableAtFilter f l μ
-- MeasureTheory.integrableAtFilter_top
@[simp]
theorem integrableAtFilter_top : IntegrableAtFilter f ⊤ μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.IntegrableAtFilter f ⊤ μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.IntegrableAtFilter.sup_iff
theorem IntegrableAtFilter.sup_iff {l l' : Filter α} :
    IntegrableAtFilter f (l ⊔ l') μ ↔ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l l' : Filter α}
Goal: MeasureTheory.IntegrableAtFilter f (l ⊔ l') μ ↔
  MeasureTheory.IntegrableAtFilter f l μ ∧ MeasureTheory.IntegrableAtFilter f l' μ
-- MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter
theorem Measure.FiniteAtFilter.integrableAtFilter {l : Filter α} [IsMeasurablyGenerated l]
    (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l)
    (hf : l.IsBoundedUnder (· ≤ ·) (norm ∘ f)) : IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α} [l.IsMeasurablyGenerated] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f))
Goal: MeasureTheory.IntegrableAtFilter f l μ
-- MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter_of_tendsto_ae
theorem Measure.FiniteAtFilter.integrableAtFilter_of_tendsto_ae {l : Filter α}
    [IsMeasurablyGenerated l] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b}
    (hf : Tendsto f (l ⊓ ae μ) (𝓝 b)) : IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α} [l.IsMeasurablyGenerated] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b : E} (hf : Filter.Tendsto f (l ⊓ MeasureTheory.ae μ) (𝓝 b))
Goal: MeasureTheory.IntegrableAtFilter f l μ
-- MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter_of_tendsto
theorem Measure.FiniteAtFilter.integrableAtFilter_of_tendsto {l : Filter α}
    [IsMeasurablyGenerated l] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b}
    (hf : Tendsto f l (𝓝 b)) : IntegrableAtFilter f l μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α} [l.IsMeasurablyGenerated] (hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l) {b : E} (hf : Filter.Tendsto f l (𝓝 b))
Goal: MeasureTheory.IntegrableAtFilter f l μ
-- MeasureTheory.Measure.integrableOn_of_bounded
lemma Measure.integrableOn_of_bounded (s_finite : μ s ≠ ∞) (f_mble : AEStronglyMeasurable f μ)
    {M : ℝ} (f_bdd : ∀ᵐ a ∂(μ.restrict s), ‖f a‖ ≤ M) :
    IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (s_finite : μ s ≠ ⊤) (f_mble : MeasureTheory.AEStronglyMeasurable f μ) {M : ℝ} (f_bdd : ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ M)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.integrable_add_of_disjoint
theorem integrable_add_of_disjoint {f g : α → E} (h : Disjoint (support f) (support g))
    (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    Integrable (f + g) μ ↔ Integrable f μ ∧ Integrable g μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f g : α → E} (h : Disjoint (Function.support f) (Function.support g)) (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.Integrable (f + g) μ ↔ MeasureTheory.Integrable f μ ∧ MeasureTheory.Integrable g μ
-- MeasureTheory.IntegrableAtFilter.eq_zero_of_tendsto
lemma IntegrableAtFilter.eq_zero_of_tendsto
    (h : IntegrableAtFilter f l μ) (h' : ∀ s ∈ l, μ s = ∞) {a : E}
    (hf : Tendsto f l (𝓝 a)) : a = 0 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {l : Filter α} (h : MeasureTheory.IntegrableAtFilter f l μ) (h' : ∀ s ∈ l, μ s = ⊤) {a : E} (hf : Filter.Tendsto f l (𝓝 a))
Goal: a = 0
-- ContinuousOn.aemeasurable
theorem ContinuousOn.aemeasurable [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [BorelSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : MeasurableSet s) : AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s)
Goal: AEMeasurable f (μ.restrict s)
-- ContinuousOn.aestronglyMeasurable_of_isSeparable
theorem ContinuousOn.aestronglyMeasurable_of_isSeparable [TopologicalSpace α]
    [PseudoMetrizableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β]
    [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s)
    (hs : MeasurableSet s) (h's : TopologicalSpace.IsSeparable s) :
    AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) (h's : TopologicalSpace.IsSeparable s)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- ContinuousOn.aestronglyMeasurable
theorem ContinuousOn.aestronglyMeasurable [TopologicalSpace α] [TopologicalSpace β]
    [h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [PseudoMetrizableSpace β]
    {f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [TopologicalSpace β] [h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [TopologicalSpace.PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- ContinuousOn.aestronglyMeasurable_of_isCompact
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α} (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- ContinuousOn.integrableAt_nhdsWithin_of_isSeparable
theorem ContinuousOn.integrableAt_nhdsWithin_of_isSeparable [TopologicalSpace α]
    [PseudoMetrizableSpace α] [OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ]
    {a : α} {t : Set α} {f : α → E} (hft : ContinuousOn f t) (ht : MeasurableSet t)
    (h't : TopologicalSpace.IsSeparable t) (ha : a ∈ t) : IntegrableAtFilter f (𝓝[t] a) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {a : α} {t : Set α} {f : α → E} (hft : ContinuousOn f t) (ht : MeasurableSet t) (h't : TopologicalSpace.IsSeparable t) (ha : a ∈ t)
Goal: MeasureTheory.IntegrableAtFilter f (𝓝[t] a) μ