l3lab/ntp-mathlib huggingface dataset:
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x))

Next tactic:
have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x))

Next tactic:
have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x))

Next tactic:
filter_upwards [A, B, C] with _ Ax Bx Cx
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  filter_upwards [A, B, C] with _ Ax Bx Cx
  

Proof state:
case h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
a✝ : α
Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0)
Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝
Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝))
⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv ρ μ a✝))

Next tactic:
convert Ax.add Cx using 1
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  filter_upwards [A, B, C] with _ Ax Bx Cx
  convert Ax.add Cx using 1
  · 

Proof state:
case h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
a✝ : α
Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0)
Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝
Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝))
⊢ (fun a => ↑↑ρ a / ↑↑μ a) = fun x => ↑↑(singularPart ρ μ) x / ↑↑μ x + ↑↑t x / ↑↑μ x

Next tactic:
ext1 a
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  filter_upwards [A, B, C] with _ Ax Bx Cx
  convert Ax.add Cx using 1
  · ext1 a
    

Proof state:
case h.e'_3.h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
a✝ : α
Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0)
Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝
Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝))
a : Set α
⊢ ↑↑ρ a / ↑↑μ a = ↑↑(singularPart ρ μ) a / ↑↑μ a + ↑↑t a / ↑↑μ a

Next tactic:
conv_lhs => rw [eq_add]
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  filter_upwards [A, B, C] with _ Ax Bx Cx
  convert Ax.add Cx using 1
  · ext1 a
    conv_lhs => 

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
a✝ : α
Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0)
Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝
Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝))
a : Set α
| ↑↑ρ a / ↑↑μ a

Next tactic:
rw [eq_add]
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  filter_upwards [A, B, C] with _ Ax Bx Cx
  convert Ax.add Cx using 1
  · ext1 a
    conv_lhs => rw [eq_add]
    

Proof state:
case h.e'_3.h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
a✝ : α
Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0)
Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝
Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝))
a : Set α
⊢ ↑↑(singularPart ρ μ + t) a / ↑↑μ a = ↑↑(singularPart ρ μ) a / ↑↑μ a + ↑↑t a / ↑↑μ a

Next tactic:
simp only [Pi.add_apply, coe_add, ENNReal.add_div]
/-- Main theorem on differentiation of measures: given a Vitali family `v` for a locally finite
measure `μ`, and another locally finite measure `ρ`, then for `μ`-almost every `x` the
ratio `ρ a / μ a` converges, when `a` shrinks to `x` along the Vitali family, towards the
Radon-Nikodym derivative of `ρ` with respect to `μ`. -/
theorem ae_tendsto_rnDeriv :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by
  let t := μ.withDensity (ρ.rnDeriv μ)
  have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
  have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
    v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
  have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x :=
    rnDeriv_withDensity μ (measurable_rnDeriv ρ μ)
  have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) :=
    v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _)
  filter_upwards [A, B, C] with _ Ax Bx Cx
  convert Ax.add Cx using 1
  · ext1 a
    conv_lhs => rw [eq_add]
    simp only [Pi.add_apply, coe_add, ENNReal.add_div]
  · 

Proof state:
case h.e'_5
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
t : Measure α := withDensity μ (rnDeriv ρ μ)
eq_add : ρ = singularPart ρ μ + t
A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x
C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x))
a✝ : α
Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0)
Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝
Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝))
⊢ 𝓝 (rnDeriv ρ μ a✝) = 𝓝 (0 + rnDeriv t μ a✝)

Next tactic:
simp only [Bx, zero_add]
/-- Given a measurable set `s`, then `μ (s ∩ a) / μ a` converges when `a` shrinks to a typical
point `x` along a Vitali family. The limit is `1` for `x ∈ s` and `0` for `x ∉ s`. This shows that
almost every point of `s` is a Lebesgue density point for `s`. A version for non-measurable sets
holds, but it only gives the first conclusion, see `ae_tendsto_measure_inter_div`. -/
theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x)) := by
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator s 1 x))

Next tactic:
haveI : IsLocallyFiniteMeasure (μ.restrict s) :=
    isLocallyFiniteMeasure_of_le restrict_le_self
/-- Given a measurable set `s`, then `μ (s ∩ a) / μ a` converges when `a` shrinks to a typical
point `x` along a Vitali family. The limit is `1` for `x ∈ s` and `0` for `x ∉ s`. This shows that
almost every point of `s` is a Lebesgue density point for `s`. A version for non-measurable sets
holds, but it only gives the first conclusion, see `ae_tendsto_measure_inter_div`. -/
theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x)) := by
  haveI : IsLocallyFiniteMeasure (μ.restrict s) :=
    isLocallyFiniteMeasure_of_le restrict_le_self
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (Measure.restrict μ s)
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator s 1 x))

Next tactic:
filter_upwards [ae_tendsto_rnDeriv v (μ.restrict s), rnDeriv_restrict_self μ hs]
/-- Given a measurable set `s`, then `μ (s ∩ a) / μ a` converges when `a` shrinks to a typical
point `x` along a Vitali family. The limit is `1` for `x ∈ s` and `0` for `x ∉ s`. This shows that
almost every point of `s` is a Lebesgue density point for `s`. A version for non-measurable sets
holds, but it only gives the first conclusion, see `ae_tendsto_measure_inter_div`. -/
theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x)) := by
  haveI : IsLocallyFiniteMeasure (μ.restrict s) :=
    isLocallyFiniteMeasure_of_le restrict_le_self
  filter_upwards [ae_tendsto_rnDeriv v (μ.restrict s), rnDeriv_restrict_self μ hs]
  

Proof state:
case h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (Measure.restrict μ s)
⊢ ∀ (a : α),
    Tendsto (fun a => ↑↑(Measure.restrict μ s) a / ↑↑μ a) (filterAt v a) (𝓝 (rnDeriv (Measure.restrict μ s) μ a)) →
      rnDeriv (Measure.restrict μ s) μ a = indicator s 1 a →
        Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v a) (𝓝 (indicator s 1 a))

Next tactic:
intro x hx h'x
/-- Given a measurable set `s`, then `μ (s ∩ a) / μ a` converges when `a` shrinks to a typical
point `x` along a Vitali family. The limit is `1` for `x ∈ s` and `0` for `x ∉ s`. This shows that
almost every point of `s` is a Lebesgue density point for `s`. A version for non-measurable sets
holds, but it only gives the first conclusion, see `ae_tendsto_measure_inter_div`. -/
theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x)) := by
  haveI : IsLocallyFiniteMeasure (μ.restrict s) :=
    isLocallyFiniteMeasure_of_le restrict_le_self
  filter_upwards [ae_tendsto_rnDeriv v (μ.restrict s), rnDeriv_restrict_self μ hs]
  intro x hx h'x
  

Proof state:
case h
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
this : IsLocallyFiniteMeasure (Measure.restrict μ s)
x : α
hx : Tendsto (fun a => ↑↑(Measure.restrict μ s) a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv (Measure.restrict μ s) μ x))
h'x : rnDeriv (Measure.restrict μ s) μ x = indicator s 1 x
⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator s 1 x))

Next tactic:
simpa only [h'x, restrict_apply' hs, inter_comm] using hx
/-- Given an arbitrary set `s`, then `μ (s ∩ a) / μ a` converges to `1` when `a` shrinks to a
typical point of `s` along a Vitali family. This shows that almost every point of `s` is a
Lebesgue density point for `s`. A stronger version for measurable sets is given
in `ae_tendsto_measure_inter_div_of_measurableSet`. -/
theorem ae_tendsto_measure_inter_div (s : Set α) :
    ∀ᵐ x ∂μ.restrict s, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 1) := by
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)

Next tactic:
let t := toMeasurable μ s
/-- Given an arbitrary set `s`, then `μ (s ∩ a) / μ a` converges to `1` when `a` shrinks to a
typical point of `s` along a Vitali family. This shows that almost every point of `s` is a
Lebesgue density point for `s`. A stronger version for measurable sets is given
in `ae_tendsto_measure_inter_div_of_measurableSet`. -/
theorem ae_tendsto_measure_inter_div (s : Set α) :
    ∀ᵐ x ∂μ.restrict s, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 1) := by
  let t := toMeasurable μ s
  

Proof state:
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
t : Set α := toMeasurable μ s
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1)

Next tactic:
have A :
    ∀ᵐ x ∂μ.restrict s,
      Tendsto (fun a => μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x)) := by
    apply ae_mono restrict_le_self
    apply ae_tendsto_measure_inter_div_of_measurableSet
    exact measurableSet_toMeasurable _ _