ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.aecover_Iic
theorem aecover_Iic : AECover μ l fun i => Iic <| b i 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {b : ι → α} (hb : Filter.Tendsto b l Filter.atTop)
Goal: MeasureTheory.AECover μ l fun i => Set.Iic (b i)
-- MeasureTheory.aecover_Icc
theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop)
Goal: MeasureTheory.AECover μ l fun i => Set.Icc (a i) (b i)
-- MeasureTheory.aecover_Ioi
theorem aecover_Ioi [NoMinOrder α] : AECover μ l fun i => Ioi (a i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a : ι → α} (ha : Filter.Tendsto a l Filter.atBot) [NoMinOrder α]
Goal: MeasureTheory.AECover μ l fun i => Set.Ioi (a i)
-- MeasureTheory.aecover_Iio
theorem aecover_Iio [NoMaxOrder α] : AECover μ l fun i => Iio (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {b : ι → α} (hb : Filter.Tendsto b l Filter.atTop) [NoMaxOrder α]
Goal: MeasureTheory.AECover μ l fun i => Set.Iio (b i)
-- MeasureTheory.aecover_Ioo
theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop) [NoMinOrder α] [NoMaxOrder α]
Goal: MeasureTheory.AECover μ l fun i => Set.Ioo (a i) (b i)
-- MeasureTheory.aecover_Ioc
theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop) [NoMinOrder α]
Goal: MeasureTheory.AECover μ l fun i => Set.Ioc (a i) (b i)
-- MeasureTheory.aecover_Ico
theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop) [NoMaxOrder α]
Goal: MeasureTheory.AECover μ l fun i => Set.Ico (a i) (b i)
-- MeasureTheory.aecover_Ioi_of_Ioi
theorem aecover_Ioi_of_Ioi : AECover (μ.restrict (Ioi A)) l fun i ↦ Ioi (a i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a : ι → α} {A : α} (ha : Filter.Tendsto a l (𝓝 A))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioi A)) l fun i => Set.Ioi (a i)
-- MeasureTheory.aecover_Iio_of_Iio
theorem aecover_Iio_of_Iio : AECover (μ.restrict (Iio B)) l fun i ↦ Iio (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {b : ι → α} {B : α} (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Iio B)) l fun i => Set.Iio (b i)
-- MeasureTheory.aecover_Ioi_of_Ici
theorem aecover_Ioi_of_Ici : AECover (μ.restrict (Ioi A)) l fun i ↦ Ici (a i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a : ι → α} {A : α} (ha : Filter.Tendsto a l (𝓝 A))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioi A)) l fun i => Set.Ici (a i)
-- MeasureTheory.aecover_Iio_of_Iic
theorem aecover_Iio_of_Iic : AECover (μ.restrict (Iio B)) l fun i ↦ Iic (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {b : ι → α} {B : α} (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Iio B)) l fun i => Set.Iic (b i)
-- MeasureTheory.aecover_Ioo_of_Ioo
theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Ioo (a i) (b i)
-- MeasureTheory.aecover_Ioo_of_Icc
theorem aecover_Ioo_of_Icc : AECover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Icc (a i) (b i)
-- MeasureTheory.aecover_Ioo_of_Ico
theorem aecover_Ioo_of_Ico : AECover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Ico (a i) (b i)
-- MeasureTheory.aecover_Ioo_of_Ioc
theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Ioc (a i) (b i)
-- MeasureTheory.aecover_Ioc_of_Icc
theorem aecover_Ioc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ioc A B) l fun i => Icc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Icc (a i) (b i)
-- MeasureTheory.aecover_Ioc_of_Ico
theorem aecover_Ioc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ioc A B) l fun i => Ico (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Ico (a i) (b i)
-- MeasureTheory.aecover_Ioc_of_Ioc
theorem aecover_Ioc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ioc A B) l fun i => Ioc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Ioc (a i) (b i)
-- MeasureTheory.aecover_Ioc_of_Ioo
theorem aecover_Ioc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ioc A B) l fun i => Ioo (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Ioo (a i) (b i)
-- MeasureTheory.aecover_Ico_of_Icc
theorem aecover_Ico_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ico A B) l fun i => Icc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Icc (a i) (b i)
-- MeasureTheory.aecover_Ico_of_Ico
theorem aecover_Ico_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ico A B) l fun i => Ico (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ico (a i) (b i)
-- MeasureTheory.aecover_Ico_of_Ioc
theorem aecover_Ico_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ico A B) l fun i => Ioc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ioc (a i) (b i)
-- MeasureTheory.aecover_Ico_of_Ioo
theorem aecover_Ico_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Ico A B) l fun i => Ioo (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ioo (a i) (b i)
-- MeasureTheory.aecover_Icc_of_Icc
theorem aecover_Icc_of_Icc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Icc A B) l fun i => Icc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Icc (a i) (b i)
-- MeasureTheory.aecover_Icc_of_Ico
theorem aecover_Icc_of_Ico (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Icc A B) l fun i => Ico (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Ico (a i) (b i)
-- MeasureTheory.aecover_Icc_of_Ioc
theorem aecover_Icc_of_Ioc (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Icc A B) l fun i => Ioc (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Ioc (a i) (b i)
-- MeasureTheory.aecover_Icc_of_Ioo
theorem aecover_Icc_of_Ioo (ha : Tendsto a l (𝓝 A)) (hb : Tendsto b l (𝓝 B)) :
    AECover (μ.restrict <| Icc A B) l fun i => Ioo (a i) (b i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a b : ι → α} {A B : α} [MeasureTheory.NoAtoms μ] (ha : Filter.Tendsto a l (𝓝 A)) (hb : Filter.Tendsto b l (𝓝 B))
Goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Ioo (a i) (b i)
-- MeasureTheory.AECover.restrict
protected theorem AECover.restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} :
    AECover (μ.restrict s) l φ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {s : Set α}
Goal: MeasureTheory.AECover (μ.restrict s) l φ
-- MeasureTheory.aecover_restrict_of_ae_imp
theorem aecover_restrict_of_ae_imp {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s)
    (ae_eventually_mem : ∀ᵐ x ∂μ, x ∈ s → ∀ᶠ n in l, x ∈ φ n)
    (measurable : ∀ n, MeasurableSet <| φ n) : AECover (μ.restrict s) l φ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {s : Set α} {φ : ι → Set α} (hs : MeasurableSet s) (ae_eventually_mem : ∀ᵐ (x : α) ∂μ, x ∈ s → ∀ᶠ (n : ι) in l, x ∈ φ n) (measurable : ∀ (n : ι), MeasurableSet (φ n))
Goal: MeasureTheory.AECover (μ.restrict s) l φ
-- MeasureTheory.AECover.inter_restrict
theorem AECover.inter_restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α}
    (hs : MeasurableSet s) : AECover (μ.restrict s) l fun i => φ i ∩ s 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.AECover (μ.restrict s) l fun i => φ i ∩ s
-- MeasureTheory.AECover.ae_tendsto_indicator
theorem AECover.ae_tendsto_indicator {β : Type*} [Zero β] [TopologicalSpace β] (f : α → β)
    {φ : ι → Set α} (hφ : AECover μ l φ) :
    ∀ᵐ x ∂μ, Tendsto (fun i => (φ i).indicator f x) l (𝓝 <| f x) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {β : Type u_3} [Zero β] [TopologicalSpace β] (f : α → β) {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => (φ i).indicator f x) l (𝓝 (f x))
-- MeasureTheory.AECover.aemeasurable
theorem AECover.aemeasurable {β : Type*} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot]
    {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
    (hfm : ∀ i, AEMeasurable f (μ.restrict <| φ i)) : AEMeasurable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {β : Type u_3} [MeasurableSpace β] [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) (hfm : ∀ (i : ι), AEMeasurable f (μ.restrict (φ i)))
Goal: AEMeasurable f μ
-- MeasureTheory.AECover.aestronglyMeasurable
theorem AECover.aestronglyMeasurable {β : Type*} [TopologicalSpace β] [PseudoMetrizableSpace β]
    [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : AECover μ l φ)
    (hfm : ∀ i, AEStronglyMeasurable f (μ.restrict <| φ i)) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {β : Type u_3} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [l.IsCountablyGenerated] [l.NeBot] {f : α → β} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) (hfm : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ.restrict (φ i)))
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.AECover.comp_tendsto
theorem AECover.comp_tendsto {α ι ι' : Type*} [MeasurableSpace α] {μ : Measure α} {l : Filter ι}
    {l' : Filter ι'} {φ : ι → Set α} (hφ : AECover μ l φ) {u : ι' → ι} (hu : Tendsto u l' l) :
    AECover μ l' (φ ∘ u) 
Arguments: {α : Type u_1} {ι : Type u_2} {ι' : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {l' : Filter ι'} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {u : ι' → ι} (hu : Filter.Tendsto u l' l)
Goal: MeasureTheory.AECover μ l' (φ ∘ u)
-- MeasureTheory.AECover.biUnion_Iic_aecover
theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) :
    AECover μ atTop fun n : ι => ⋃ (k) (_h : k ∈ Iic n), φ k 
Arguments: {α : Type u_1} {ι : Type u_2} [Countable ι] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Preorder ι] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ Filter.atTop φ)
Goal: MeasureTheory.AECover μ Filter.atTop fun n => ⋃ k ∈ Set.Iic n, φ k
-- MeasureTheory.AECover.biInter_Ici_aecover
theorem AECover.biInter_Ici_aecover [Preorder ι] {φ : ι → Set α}
    (hφ : AECover μ atTop φ) : AECover μ atTop fun n : ι => ⋂ (k) (_h : k ∈ Ici n), φ k 
Arguments: {α : Type u_1} {ι : Type u_2} [Countable ι] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Preorder ι] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ Filter.atTop φ)
Goal: MeasureTheory.AECover μ Filter.atTop fun n => ⋂ k ∈ Set.Ici n, φ k
-- _private.Mathlib.MeasureTheory.Integral.IntegralEqImproper.0.MeasureTheory.lintegral_tendsto_of_monotone_of_nat
private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ)
    (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
    Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {φ : ℕ → Set α} (hφ : MeasureTheory.AECover μ Filter.atTop φ) (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
Goal: Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, f x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
-- MeasureTheory.AECover.lintegral_tendsto_of_nat
theorem AECover.lintegral_tendsto_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ) {f : α → ℝ≥0∞}
    (hfm : AEMeasurable f μ) : Tendsto (∫⁻ x in φ ·, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {φ : ℕ → Set α} (hφ : MeasureTheory.AECover μ Filter.atTop φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
Goal: Filter.Tendsto (fun x => ∫⁻ (x : α) in φ x, f x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), f x ∂μ))
-- MeasureTheory.AECover.lintegral_tendsto_of_countably_generated
theorem AECover.lintegral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
    (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
    Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 <| ∫⁻ x, f x ∂μ) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
Goal: Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, f x ∂μ) l (𝓝 (∫⁻ (x : α), f x ∂μ))
-- MeasureTheory.AECover.lintegral_eq_of_tendsto
theorem AECover.lintegral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α}
    (hφ : AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ)
    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) l (𝓝 I)) : ∫⁻ x, f x ∂μ = I 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → ℝ≥0∞} (I : ℝ≥0∞) (hfm : AEMeasurable f μ) (htendsto : Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, f x ∂μ) l (𝓝 I))
Goal: ∫⁻ (x : α), f x ∂μ = I
-- MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated
theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.NeBot]
    [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ≥0∞}
    (hfm : AEMeasurable f μ) : ⨆ i : ι, ∫⁻ x in φ i, f x ∂μ = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [Nonempty ι] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
Goal: ⨆ i, ∫⁻ (x : α) in φ i, f x ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded
theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : ℝ) (hfm : MeasureTheory.AEStronglyMeasurable f μ) (hbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ ≤ ENNReal.ofReal I)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto
theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
    Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : ℝ) (hfm : MeasureTheory.AEStronglyMeasurable f μ) (htendsto : Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 (ENNReal.ofReal I)))
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : MeasureTheory.AEStronglyMeasurable f μ) (hbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ ≤ ↑I)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto'
theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : MeasureTheory.AEStronglyMeasurable f μ) (htendsto : Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ) l (𝓝 ↑I))
Goal: MeasureTheory.Integrable f μ