Theorem: VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div
File: Mathlib/MeasureTheory/Covering/Differentiation.lean

tactic: have I : Integrable (hf.1.mk f) μ := hf.congr hf.1.ae_eq_mk
state_before: α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)

tactic: filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div' I hf.1.stronglyMeasurable_mk,
  hf.1.ae_eq_mk] with x hx h'x
state_before: α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)

tactic: apply hx.congr _
state_before: case h
α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
⊢ Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0)

tactic: intro a
state_before: α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
⊢ ∀ (x_1 : Set α),
    (∫⁻ (y : α) in x_1,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ x_1 =
      (∫⁻ (y : α) in x_1, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ x_1

tactic: congr 1
state_before: α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
a : Set α
⊢ (∫⁻ (y : α) in a,
        ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
              AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
      ↑↑μ a =
    (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a

tactic: apply lintegral_congr_ae
state_before: case e_a
α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
a : Set α
⊢ (∫⁻ (y : α) in a,
      ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
            AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) =
    ∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ

tactic: apply ae_restrict_of_ae
state_before: case e_a.h
α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
a : Set α
⊢ (fun a =>
      ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) a -
            AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊) =ᶠ[ae (Measure.restrict μ a)]
    fun a => ↑‖f a - f x‖₊

tactic: filter_upwards [hf.1.ae_eq_mk] with y hy
state_before: case e_a.h.h
α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
a : Set α
⊢ ∀ᵐ (x_1 : α) ∂μ,
    (fun a =>
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) a -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊)
        x_1 =
      (fun a => ↑‖f a - f x‖₊) x_1

tactic: rw [hy, h'x]
state_before: case h
α : Type u_2
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure ρ
f : α → E
hf : Integrable f
I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ))
x : α
hx :
  Tendsto
    (fun a =>
      (∫⁻ (y : α) in a,
          ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
                AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) /
        ↑↑μ a)
    (filterAt v x) (𝓝 0)
h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x
a : Set α
y : α
hy : f y = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y
⊢ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y -
          AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ =
    ↑‖f y - f x‖₊