File: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean

full_name: MeasureTheory.AECover.measurableSet
code: /-- A sequence `φ` of subsets of `α` is a `MeasureTheory.AECover` w.r.t. a measure `μ` and a filter
    `l` if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if
    each `φ n` is measurable.  This definition is a technical way to avoid duplicating a lot of
    proofs.  It should be thought of as a sufficient condition for being able to interpret
    `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ n, f x ∂μ` as `n` tends to `l`.

    See for example `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated`,
    `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` and
    `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/
structure AECover (φ : ι → Set α) : Prop where
  ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i
  protected measurableSet : ∀ i, MeasurableSet <| φ i
goal: MeasurableSet (φ i)

full_name: MeasureTheory.AECover
code: /-- A sequence `φ` of subsets of `α` is a `MeasureTheory.AECover` w.r.t. a measure `μ` and a filter
    `l` if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if
    each `φ n` is measurable.  This definition is a technical way to avoid duplicating a lot of
    proofs.  It should be thought of as a sufficient condition for being able to interpret
    `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ n, f x ∂μ` as `n` tends to `l`.

    See for example `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated`,
    `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` and
    `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/
structure AECover (φ : ι → Set α) : Prop where
  ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i
  protected measurableSet : ∀ i, MeasurableSet <| φ i
goal: Prop

full_name: MeasureTheory.AECover.ae_eventually_mem
code: /-- A sequence `φ` of subsets of `α` is a `MeasureTheory.AECover` w.r.t. a measure `μ` and a filter
    `l` if almost every point (w.r.t. `μ`) of `α` eventually belongs to `φ n` (w.r.t. `l`), and if
    each `φ n` is measurable.  This definition is a technical way to avoid duplicating a lot of
    proofs.  It should be thought of as a sufficient condition for being able to interpret
    `∫ x, f x ∂μ` (if it exists) as the limit of `∫ x in φ n, f x ∂μ` as `n` tends to `l`.

    See for example `MeasureTheory.AECover.lintegral_tendsto_of_countably_generated`,
    `MeasureTheory.AECover.integrable_of_integral_norm_tendsto` and
    `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/
structure AECover (φ : ι → Set α) : Prop where
  ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i
  protected measurableSet : ∀ i, MeasurableSet <| φ i
goal: ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in l, x ∈ φ i

full_name: MeasureTheory.AECover.inter
code: /-- Elementwise intersection of two `AECover`s is an `AECover`. -/
theorem inter {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hψ : AECover μ l ψ) :
    AECover μ l (fun i ↦ φ i ∩ ψ i) 
goal: MeasureTheory.AECover μ l fun i => φ i ∩ ψ i

full_name: MeasureTheory.AECover.superset
code: theorem superset {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hsub : ∀ i, φ i ⊆ ψ i)
    (hmeas : ∀ i, MeasurableSet (ψ i)) : AECover μ l ψ 
goal: MeasureTheory.AECover μ l ψ

full_name: MeasureTheory.AECover.mono_ac
code: theorem mono_ac {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≪ μ) :
    AECover ν l φ 
goal: MeasureTheory.AECover ν l φ

full_name: MeasureTheory.AECover.mono
code: theorem mono {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≤ μ) :
    AECover ν l φ 
goal: MeasureTheory.AECover ν l φ

full_name: MeasureTheory.aecover_ball
code: theorem aecover_ball {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
    AECover μ l (fun i ↦ Metric.ball x (r i)) 
goal: MeasureTheory.AECover μ l fun i => Metric.ball x (r i)

full_name: MeasureTheory.aecover_closedBall
code: theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
    AECover μ l (fun i ↦ Metric.closedBall x (r i)) 
goal: MeasureTheory.AECover μ l fun i => Metric.closedBall x (r i)

full_name: MeasureTheory.aecover_Ici
code: theorem aecover_Ici : AECover μ l fun i => Ici (a i) 
goal: MeasureTheory.AECover μ l fun i => Set.Ici (a i)

full_name: MeasureTheory.aecover_Iic
code: theorem aecover_Iic : AECover μ l fun i => Iic <| b i 
goal: MeasureTheory.AECover μ l fun i => Set.Iic (b i)

full_name: MeasureTheory.aecover_Icc
code: theorem aecover_Icc : AECover μ l fun i => Icc (a i) (b i) 
goal: MeasureTheory.AECover μ l fun i => Set.Icc (a i) (b i)

full_name: MeasureTheory.aecover_Ioi
code: theorem aecover_Ioi [NoMinOrder α] : AECover μ l fun i => Ioi (a i) 
goal: MeasureTheory.AECover μ l fun i => Set.Ioi (a i)

full_name: MeasureTheory.aecover_Iio
code: theorem aecover_Iio [NoMaxOrder α] : AECover μ l fun i => Iio (b i) 
goal: MeasureTheory.AECover μ l fun i => Set.Iio (b i)

full_name: MeasureTheory.aecover_Ioo
code: theorem aecover_Ioo [NoMinOrder α] [NoMaxOrder α] : AECover μ l fun i => Ioo (a i) (b i) 
goal: MeasureTheory.AECover μ l fun i => Set.Ioo (a i) (b i)

full_name: MeasureTheory.aecover_Ioc
code: theorem aecover_Ioc [NoMinOrder α] : AECover μ l fun i => Ioc (a i) (b i) 
goal: MeasureTheory.AECover μ l fun i => Set.Ioc (a i) (b i)

full_name: MeasureTheory.aecover_Ico
code: theorem aecover_Ico [NoMaxOrder α] : AECover μ l fun i => Ico (a i) (b i) 
goal: MeasureTheory.AECover μ l fun i => Set.Ico (a i) (b i)

full_name: MeasureTheory.aecover_Ioi_of_Ioi
code: theorem aecover_Ioi_of_Ioi : AECover (μ.restrict (Ioi A)) l fun i ↦ Ioi (a i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioi A)) l fun i => Set.Ioi (a i)

full_name: MeasureTheory.aecover_Iio_of_Iio
code: theorem aecover_Iio_of_Iio : AECover (μ.restrict (Iio B)) l fun i ↦ Iio (b i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Iio B)) l fun i => Set.Iio (b i)

full_name: MeasureTheory.aecover_Ioi_of_Ici
code: theorem aecover_Ioi_of_Ici : AECover (μ.restrict (Ioi A)) l fun i ↦ Ici (a i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioi A)) l fun i => Set.Ici (a i)

full_name: MeasureTheory.aecover_Iio_of_Iic
code: theorem aecover_Iio_of_Iic : AECover (μ.restrict (Iio B)) l fun i ↦ Iic (b i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Iio B)) l fun i => Set.Iic (b i)

full_name: MeasureTheory.aecover_Ioo_of_Ioo
code: theorem aecover_Ioo_of_Ioo : AECover (μ.restrict <| Ioo A B) l fun i => Ioo (a i) (b i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Ioo (a i) (b i)

full_name: MeasureTheory.aecover_Ioo_of_Icc
code: theorem aecover_Ioo_of_Icc : AECover (μ.restrict <| Ioo A B) l fun i => Icc (a i) (b i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Icc (a i) (b i)

full_name: MeasureTheory.aecover_Ioo_of_Ico
code: theorem aecover_Ioo_of_Ico : AECover (μ.restrict <| Ioo A B) l fun i => Ico (a i) (b i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Ico (a i) (b i)

full_name: MeasureTheory.aecover_Ioo_of_Ioc
code: theorem aecover_Ioo_of_Ioc : AECover (μ.restrict <| Ioo A B) l fun i => Ioc (a i) (b i) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioo A B)) l fun i => Set.Ioc (a i) (b i)

full_name: MeasureTheory.aecover_Ioc_of_Icc
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Icc (a i) (b i)

full_name: MeasureTheory.aecover_Ioc_of_Ico
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Ico (a i) (b i)

full_name: MeasureTheory.aecover_Ioc_of_Ioc
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Ioc (a i) (b i)

full_name: MeasureTheory.aecover_Ioc_of_Ioo
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ioc A B)) l fun i => Set.Ioo (a i) (b i)

full_name: MeasureTheory.aecover_Ico_of_Icc
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Icc (a i) (b i)

full_name: MeasureTheory.aecover_Ico_of_Ico
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ico (a i) (b i)

full_name: MeasureTheory.aecover_Ico_of_Ioc
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ioc (a i) (b i)

full_name: MeasureTheory.aecover_Ico_of_Ioo
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ioo (a i) (b i)

full_name: MeasureTheory.aecover_Icc_of_Icc
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Icc (a i) (b i)

full_name: MeasureTheory.aecover_Icc_of_Ico
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Ico (a i) (b i)

full_name: MeasureTheory.aecover_Icc_of_Ioc
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Ioc (a i) (b i)

full_name: MeasureTheory.aecover_Icc_of_Ioo
code: 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) 
goal: MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Ioo (a i) (b i)

full_name: MeasureTheory.AECover.restrict
code: protected theorem AECover.restrict {φ : ι → Set α} (hφ : AECover μ l φ) {s : Set α} :
    AECover (μ.restrict s) l φ 
goal: MeasureTheory.AECover (μ.restrict s) l φ

full_name: MeasureTheory.aecover_restrict_of_ae_imp
code: 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 
goal: MeasureTheory.AECover (μ.restrict s) l φ

full_name: MeasureTheory.AECover.inter_restrict
code: 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 
goal: MeasureTheory.AECover (μ.restrict s) l fun i => φ i ∩ s

full_name: MeasureTheory.AECover.ae_tendsto_indicator
code: 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) 
goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => (φ i).indicator f x) l (𝓝 (f x))

full_name: MeasureTheory.AECover.aemeasurable
code: 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 
goal: AEMeasurable f μ

full_name: MeasureTheory.AECover.aestronglyMeasurable
code: 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 
goal: MeasureTheory.AEStronglyMeasurable f μ

full_name: MeasureTheory.AECover.comp_tendsto
code: 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) 
goal: MeasureTheory.AECover μ l' (φ ∘ u)

full_name: MeasureTheory.AECover.biUnion_Iic_aecover
code: 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 
goal: MeasureTheory.AECover μ Filter.atTop fun n => ⋃ k ∈ Set.Iic n, φ k

full_name: MeasureTheory.AECover.biInter_Ici_aecover
code: 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 
goal: MeasureTheory.AECover μ Filter.atTop fun n => ⋂ k ∈ Set.Ici n, φ k

full_name: _private.Mathlib.MeasureTheory.Integral.IntegralEqImproper.0.MeasureTheory.lintegral_tendsto_of_monotone_of_nat
code: 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) 
goal: Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, f x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), f x ∂μ))

full_name: MeasureTheory.AECover.lintegral_tendsto_of_nat
code: 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) 
goal: Filter.Tendsto (fun x => ∫⁻ (x : α) in φ x, f x ∂μ) Filter.atTop (𝓝 (∫⁻ (x : α), f x ∂μ))

full_name: MeasureTheory.AECover.lintegral_tendsto_of_countably_generated
code: 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) 
goal: Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, f x ∂μ) l (𝓝 (∫⁻ (x : α), f x ∂μ))

full_name: MeasureTheory.AECover.lintegral_eq_of_tendsto
code: 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 
goal: ∫⁻ (x : α), f x ∂μ = I

full_name: MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated
code: 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 
goal: ⨆ i, ∫⁻ (x : α) in φ i, f x ∂μ = ∫⁻ (x : α), f x ∂μ

full_name: MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded
code: 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 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto
code: 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 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
code: 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 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto'
code: 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 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_integral_norm_bounded
code: theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_integral_norm_tendsto
code: theorem AECover.integrable_of_integral_norm_tendsto [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae
code: theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae
code: theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) :
    Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.AECover.integral_tendsto_of_countably_generated
code: theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
    (hφ : AECover μ l φ) {f : α → E} (hfi : Integrable f μ) :
    Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) 
goal: Filter.Tendsto (fun i => ∫ (x : α) in φ i, f x ∂μ) l (𝓝 (∫ (x : α), f x ∂μ))

full_name: MeasureTheory.AECover.integral_eq_of_tendsto
code: /-- Slight reformulation of
    `MeasureTheory.AECover.integral_tendsto_of_countably_generated`. -/
theorem AECover.integral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α}
    (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
    (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I 
goal: ∫ (x : α), f x ∂μ = I

full_name: MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae
code: theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
    (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
    ∫ x, f x ∂μ = I 
goal: ∫ (x : α), f x ∂μ = I

full_name: MeasureTheory.integrable_of_intervalIntegral_norm_bounded
code: theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
    (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.integrable_of_intervalIntegral_norm_tendsto
code: /-- If `f` is integrable on intervals `Ioc (a i) (b i)`,
where `a i` tends to -∞ and `b i` tends to ∞, and
`∫ x in a i .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
then `f` is integrable on the interval (-∞, ∞) -/
theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
    (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a i..b i, ‖f x‖ ∂μ) l (𝓝 I)) :
    Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded
code: theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
    (h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ

full_name: MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto
code: /-- If `f` is integrable on intervals `Ioc (a i) b`,
where `a i` tends to -∞, and
`∫ x in a i .. b, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
then `f` is integrable on the interval (-∞, b) -/
theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
    (h : Tendsto (fun i => ∫ x in a i..b, ‖f x‖ ∂μ) l (𝓝 I)) : IntegrableOn f (Iic b) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ

full_name: MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded
code: theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
    (h : ∀ᶠ i in l, (∫ x in a..b i, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Ioi a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ioi a) μ

full_name: MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
code: /-- If `f` is integrable on intervals `Ioc a (b i)`,
where `b i` tends to ∞, and
`∫ x in a .. b i, ‖f x‖ ∂μ` converges to `I : ℝ` along a filter `l`,
then `f` is integrable on the interval (a, ∞) -/
theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
    (h : Tendsto (fun i => ∫ x in a..b i, ‖f x‖ ∂μ) l (𝓝 <| I)) : IntegrableOn f (Ioi a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ioi a) μ

full_name: MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded
code: theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ}
    (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
    (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) :
    IntegrableOn f (Ioc a₀ b₀) 
goal: MeasureTheory.IntegrableOn f (Set.Ioc a₀ b₀) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left
code: theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ}
    (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
    (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) 
goal: MeasureTheory.IntegrableOn f (Set.Ioc a₀ b) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right
code: theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_right {I a b₀ : ℝ}
    (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
    (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) 
goal: MeasureTheory.IntegrableOn f (Set.Ioc a b₀) MeasureTheory.volume

full_name: MeasureTheory.intervalIntegral_tendsto_integral
code: theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto a l atBot)
    (hb : Tendsto b l atTop) : Tendsto (fun i => ∫ x in a i..b i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) 
goal: Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ), f x ∂μ))

full_name: MeasureTheory.intervalIntegral_tendsto_integral_Iic
code: theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ)
    (ha : Tendsto a l atBot) :
    Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <| ∫ x in Iic b, f x ∂μ) 
goal: Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Set.Iic b, f x ∂μ))

full_name: MeasureTheory.intervalIntegral_tendsto_integral_Ioi
code: theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (Ioi a) μ)
    (hb : Tendsto b l atTop) :
    Tendsto (fun i => ∫ x in a..b i, f x ∂μ) l (𝓝 <| ∫ x in Ioi a, f x ∂μ) 
goal: Filter.Tendsto (fun i => ∫ (x : ℝ) in a..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Set.Ioi a, f x ∂μ))

full_name: MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi
code: /-- If the derivative of a function defined on the real line is integrable close to `+∞`, then
the function has a limit at `+∞`. -/
theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) :
    Tendsto f atTop (𝓝 (limUnder atTop f)) 
goal: Filter.Tendsto f Filter.atTop (𝓝 (limUnder Filter.atTop f))

full_name: MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
code: /-- If a function and its derivative are integrable on `(a, +∞)`, then the function tends to zero
at `+∞`. -/
theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (f'int : IntegrableOn f' (Ioi a)) (fint : IntegrableOn f (Ioi a)) :
    Tendsto f atTop (𝓝 0) 
goal: Filter.Tendsto f Filter.atTop (𝓝 0)

full_name: MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto
code: /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
When a function has a limit at infinity `m`, and its derivative is integrable, then the
integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
on `(a, +∞)` and continuity at `a⁺`.

Note that such a function always has a limit at infinity,
see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/
theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
    (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a 
goal: ∫ (x : ℝ) in Set.Ioi a, f' x = m - f a

full_name: MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto'
code: /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(a, +∞)`.
When a function has a limit at infinity `m`, and its derivative is integrable, then the
integral of the derivative on `(a, +∞)` is `m - f a`. Version assuming differentiability
on `[a, +∞)`.

Note that such a function always has a limit at infinity,
see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/
theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
    (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
    ∫ x in Ioi a, f' x = m - f a 
goal: ∫ (x : ℝ) in Set.Ioi a, f' x = m - f a

full_name: HasCompactSupport.integral_Ioi_deriv_eq
code: /-- A special case of `integral_Ioi_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with
compact support. -/
theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f)
    (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b 
goal: ∫ (x : ℝ) in Set.Ioi b, deriv f x = -f b

full_name: MeasureTheory.integrableOn_Ioi_deriv_of_nonneg
code: /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
is automatically integrable on `(a, +∞)`. Version assuming differentiability
on `(a, +∞)` and continuity at `a⁺`. -/
theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioi_deriv_of_nonneg'
code: /-- When a function has a limit at infinity, and its derivative is nonnegative, then the derivative
is automatically integrable on `(a, +∞)`. Version assuming differentiability
on `[a, +∞)`. -/
theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume

full_name: MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
code: /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
`integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
continuity at `a⁺`. -/
theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a

full_name: MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
code: /-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
`integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a

full_name: MeasureTheory.integrableOn_Ioi_deriv_of_nonpos
code: /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
is automatically integrable on `(a, +∞)`. Version assuming differentiability
on `(a, +∞)` and continuity at `a⁺`. -/
theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioi_deriv_of_nonpos'
code: /-- When a function has a limit at infinity, and its derivative is nonpositive, then the derivative
is automatically integrable on `(a, +∞)`. Version assuming differentiability
on `[a, +∞)`. -/
theorem integrableOn_Ioi_deriv_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume

full_name: MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
code: /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
`integrable_on_Ioi_deriv_of_nonneg`). Version assuming differentiability on `(a, +∞)` and
continuity at `a⁺`. -/
theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a

full_name: MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'
code: /-- When a function has a limit at infinity `l`, and its derivative is nonpositive, then the
integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
`integrable_on_Ioi_deriv_of_nonneg'`). Version assuming differentiability on `[a, +∞)`. -/
theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a

full_name: MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic
code: /-- If the derivative of a function defined on the real line is integrable close to `-∞`, then
the function has a limit at `-∞`. -/
theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic [CompleteSpace E]
    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) :
    Tendsto f atBot (𝓝 (limUnder atBot f)) 
goal: Filter.Tendsto f Filter.atBot (𝓝 (limUnder Filter.atBot f))

full_name: MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
code: /-- If a function and its derivative are integrable on `(-∞, a]`, then the function tends to zero
at `-∞`. -/
theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x)
    (f'int : IntegrableOn f' (Iic a)) (fint : IntegrableOn f (Iic a)) :
    Tendsto f atBot (𝓝 0) 
goal: Filter.Tendsto f Filter.atBot (𝓝 0)

full_name: MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto
code: /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
When a function has a limit `m` at `-∞`, and its derivative is integrable, then the
integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability
on `(-∞, a)` and continuity at `a⁻`.

Note that such a function always has a limit at minus infinity,
see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic`. -/
theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a)
    (hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
    (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m 
goal: ∫ (x : ℝ) in Set.Iic a, f' x = f a - m

full_name: MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto'
code: /-- **Fundamental theorem of calculus-2**, on semi-infinite intervals `(-∞, a)`.
When a function has a limit `m` at `-∞`, and its derivative is integrable, then the
integral of the derivative on `(-∞, a)` is `f a - m`. Version assuming differentiability
on `(-∞, a]`.

Note that such a function always has a limit at minus infinity,
see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic`. -/
theorem integral_Iic_of_hasDerivAt_of_tendsto'
    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
    (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m 
goal: ∫ (x : ℝ) in Set.Iic a, f' x = f a - m

full_name: HasCompactSupport.integral_Iic_deriv_eq
code: /-- A special case of `integral_Iic_of_hasDerivAt_of_tendsto` where we assume that `f` is C^1 with
compact support. -/
theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
    (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Iic b, deriv f x = f b 
goal: ∫ (x : ℝ) in Set.Iic b, deriv f x = f b

full_name: HasCompactSupport.ennnorm_le_lintegral_Ici_deriv
code: lemma _root_.HasCompactSupport.ennnorm_le_lintegral_Ici_deriv
    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
    {f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ) :
    (‖f x‖₊ : ℝ≥0∞) ≤ ∫⁻ y in Iic x, ‖deriv f y‖₊ 
goal: ↑‖f x‖₊ ≤ ∫⁻ (y : ℝ) in Set.Iic x, ↑‖deriv f y‖₊

full_name: MeasureTheory.integral_of_hasDerivAt_of_tendsto
code: /-- **Fundamental theorem of calculus-2**, on the whole real line
When a function has a limit `m` at `-∞` and `n` at `+∞`, and its derivative is integrable, then the
integral of the derivative is `n - m`.

Note that such a function always has a limit at `-∞` and `+∞`,
see `tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic` and
`tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi`. -/
theorem integral_of_hasDerivAt_of_tendsto [CompleteSpace E]
    (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f')
    (hbot : Tendsto f atBot (𝓝 m)) (htop : Tendsto f atTop (𝓝 n)) : ∫ x, f' x = n - m 
goal: ∫ (x : ℝ), f' x = n - m

full_name: MeasureTheory.integral_eq_zero_of_hasDerivAt_of_integrable
code: /-- If a function and its derivative are integrable on the real line, then the integral of the
derivative is zero. -/
theorem integral_eq_zero_of_hasDerivAt_of_integrable
    (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f') (hf : Integrable f) :
    ∫ x, f' x = 0 
goal: ∫ (x : ℝ), f' x = 0

full_name: MeasureTheory.integral_comp_smul_deriv_Ioi
code: /-- Change-of-variables formula for `Ioi` integrals of vector-valued functions, proved by taking
limits from the result for finite intervals. -/
theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : ℝ}
    (hf : ContinuousOn f <| Ici a) (hft : Tendsto f atTop atTop)
    (hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g <| f '' Ioi a) (hg1 : IntegrableOn g <| f '' Ici a)
    (hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) (Ici a)) :
    (∫ x in Ioi a, f' x • (g ∘ f) x) = ∫ u in Ioi (f a), g u 
goal: ∫ (x : ℝ) in Set.Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Set.Ioi (f a), g u

full_name: MeasureTheory.integral_comp_mul_deriv_Ioi
code: /-- Change-of-variables formula for `Ioi` integrals of scalar-valued functions -/
theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a : ℝ}
    (hf : ContinuousOn f <| Ici a) (hft : Tendsto f atTop atTop)
    (hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g <| f '' Ioi a) (hg1 : IntegrableOn g <| f '' Ici a)
    (hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) (Ici a)) :
    (∫ x in Ioi a, (g ∘ f) x * f' x) = ∫ u in Ioi (f a), g u 
goal: ∫ (x : ℝ) in Set.Ioi a, (g ∘ f) x * f' x = ∫ (u : ℝ) in Set.Ioi (f a), g u

full_name: MeasureTheory.integral_comp_rpow_Ioi
code: /-- Substitution `y = x ^ p` in integrals over `Ioi 0` -/
theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
    (∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y 
goal: ∫ (x : ℝ) in Set.Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Set.Ioi 0, g y

full_name: MeasureTheory.integral_comp_rpow_Ioi_of_pos
code: theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
    (∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y 
goal: ∫ (x : ℝ) in Set.Ioi 0, (p * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Set.Ioi 0, g y

full_name: MeasureTheory.integral_comp_mul_left_Ioi
code: theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
    (∫ x in Ioi a, g (b * x)) = b⁻¹ • ∫ x in Ioi (b * a), g x 
goal: ∫ (x : ℝ) in Set.Ioi a, g (b * x) = b⁻¹ • ∫ (x : ℝ) in Set.Ioi (b * a), g x

full_name: MeasureTheory.integral_comp_mul_right_Ioi
code: theorem integral_comp_mul_right_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
    (∫ x in Ioi a, g (x * b)) = b⁻¹ • ∫ x in Ioi (a * b), g x 
goal: ∫ (x : ℝ) in Set.Ioi a, g (x * b) = b⁻¹ • ∫ (x : ℝ) in Set.Ioi (a * b), g x

full_name: MeasureTheory.integrableOn_Ioi_comp_rpow_iff
code: /-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability. -/
theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
    IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) 
goal: MeasureTheory.IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Set.Ioi 0) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi 0) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioi_comp_rpow_iff'
code: /-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability (version
without `|p|` factor) -/
theorem integrableOn_Ioi_comp_rpow_iff' [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
    IntegrableOn (fun x => x ^ (p - 1) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) 
goal: MeasureTheory.IntegrableOn (fun x => x ^ (p - 1) • f (x ^ p)) (Set.Ioi 0) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi 0) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioi_comp_mul_left_iff
code: theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
    IntegrableOn (fun x => f (a * x)) (Ioi c) ↔ IntegrableOn f (Ioi <| a * c) 
goal: MeasureTheory.IntegrableOn (fun x => f (a * x)) (Set.Ioi c) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi (a * c)) MeasureTheory.volume

full_name: MeasureTheory.integrableOn_Ioi_comp_mul_right_iff
code: theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
    IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) 
goal: MeasureTheory.IntegrableOn (fun x => f (x * a)) (Set.Ioi c) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi (c * a)) MeasureTheory.volume

full_name: MeasureTheory.integral_bilinear_hasDerivAt_eq_sub
code: theorem integral_bilinear_hasDerivAt_eq_sub [CompleteSpace G]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv : Integrable (fun x ↦ L (u x) (v' x) + L (u' x) (v x)))
    (h_bot : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 m))
    (h_top : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 n)) :
    ∫ (x : ℝ), L (u x) (v' x) + L (u' x) (v x) = n - m 
goal: ∫ (x : ℝ), (L (u x)) (v' x) + (L (u' x)) (v x) = n - m

full_name: MeasureTheory.integral_bilinear_hasDerivAt_right_eq_sub
code: /-- **Integration by parts on (-∞, ∞).**
With respect to a general bilinear form. For the specific case of multiplication, see
`integral_mul_deriv_eq_deriv_mul`. -/
theorem integral_bilinear_hasDerivAt_right_eq_sub [CompleteSpace G]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (fun x ↦ L (u x) (v' x))) (hu'v : Integrable (fun x ↦ L (u' x) (v x)))
    (h_bot : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 m))
    (h_top : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 n)) :
    ∫ (x : ℝ), L (u x) (v' x) = n - m - ∫ (x : ℝ), L (u' x) (v x) 
goal: ∫ (x : ℝ), (L (u x)) (v' x) = n - m - ∫ (x : ℝ), (L (u' x)) (v x)

full_name: MeasureTheory.integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable
code: /-- **Integration by parts on (-∞, ∞).**
With respect to a general bilinear form, assuming moreover that the total function is integrable.
-/
theorem integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (fun x ↦ L (u x) (v' x))) (hu'v : Integrable (fun x ↦ L (u' x) (v x)))
    (huv : Integrable (fun x ↦ L (u x) (v x))) :
    ∫ (x : ℝ), L (u x) (v' x) = - ∫ (x : ℝ), L (u' x) (v x) 
goal: ∫ (x : ℝ), (L (u x)) (v' x) = -∫ (x : ℝ), (L (u' x)) (v x)

full_name: MeasureTheory.integral_deriv_mul_eq_sub
code: /-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
theorem integral_deriv_mul_eq_sub [CompleteSpace A]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv : Integrable (u' * v + u * v'))
    (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ), u' x * v x + u x * v' x = b' - a' 
goal: ∫ (x : ℝ), u' x * v x + u x * v' x = b' - a'

full_name: MeasureTheory.integral_mul_deriv_eq_deriv_mul
code: /-- **Integration by parts on (-∞, ∞).**
For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
theorem integral_mul_deriv_eq_deriv_mul [CompleteSpace A]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v))
    (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x 
goal: ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x

full_name: MeasureTheory.integral_mul_deriv_eq_deriv_mul_of_integrable
code: /-- **Integration by parts on (-∞, ∞).**
Version assuming that the total function is integrable -/
theorem integral_mul_deriv_eq_deriv_mul_of_integrable
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v)) (huv : Integrable (u * v)) :
    ∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x 
goal: ∫ (x : ℝ), u x * v' x = -∫ (x : ℝ), u' x * v x

full_name: MeasureTheory.integral_Ioi_deriv_mul_eq_sub
code: /-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
theorem integral_Ioi_deriv_mul_eq_sub
    (hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
    (huv : IntegrableOn (u' * v + u * v') (Ioi a))
    (h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a' 
goal: ∫ (x : ℝ) in Set.Ioi a, u' x * v x + u x * v' x = b' - a'

full_name: MeasureTheory.integral_Ioi_mul_deriv_eq_deriv_mul
code: /-- **Integration by parts on (a, ∞).**
For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
theorem integral_Ioi_mul_deriv_eq_deriv_mul
    (hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
    (huv' : IntegrableOn (u * v') (Ioi a)) (hu'v : IntegrableOn (u' * v) (Ioi a))
    (h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ) in Ioi a, u x * v' x = b' - a' - ∫ (x : ℝ) in Ioi a, u' x * v x 
goal: ∫ (x : ℝ) in Set.Ioi a, u x * v' x = b' - a' - ∫ (x : ℝ) in Set.Ioi a, u' x * v x

full_name: MeasureTheory.integral_Iic_deriv_mul_eq_sub
code: /-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
theorem integral_Iic_deriv_mul_eq_sub
    (hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
    (huv : IntegrableOn (u' * v + u * v') (Iic a))
    (h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
    ∫ (x : ℝ) in Iic a, u' x * v x + u x * v' x = a' - b' 
goal: ∫ (x : ℝ) in Set.Iic a, u' x * v x + u x * v' x = a' - b'

full_name: MeasureTheory.integral_Iic_mul_deriv_eq_deriv_mul
code: /-- **Integration by parts on (∞, a].**
For finite intervals, see: `intervalIntegral.integral_mul_deriv_eq_deriv_mul`. -/
theorem integral_Iic_mul_deriv_eq_deriv_mul
    (hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
    (huv' : IntegrableOn (u * v') (Iic a)) (hu'v : IntegrableOn (u' * v) (Iic a))
    (h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
    ∫ (x : ℝ) in Iic a, u x * v' x = a' - b' - ∫ (x : ℝ) in Iic a, u' x * v x 
goal: ∫ (x : ℝ) in Set.Iic a, u x * v' x = a' - b' - ∫ (x : ℝ) in Set.Iic a, u' x * v x