l3lab/ntp-mathlib huggingface dataset:
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        

Proof state:
case bc
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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ↑↑ρ o = 0
μo : ↑↑μ oᶜ = 0
x : α
hx : x ∈ s ∩ o
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a

Next tactic:
rw [hs] at hx
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        

Proof state:
case bc
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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ↑↑ρ o = 0
μo : ↑↑μ oᶜ = 0
x : α
hx : x ∈ {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a

Next tactic:
simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        

Proof state:
case bc
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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ↑↑ρ o = 0
μo : ↑↑μ oᶜ = 0
x : α
hx : (∃ᶠ (x : Set α) in filterAt v x, ↑ε * ↑↑μ x ≤ ↑↑ρ x) ∧ x ∈ o
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a

Next tactic:
exact hx.1
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := 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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ↑↑ρ o = 0
μo : ↑↑μ oᶜ = 0
⊢ (↑ε)⁻¹ * ↑↑ρ (s ∩ o) ≤ (↑ε)⁻¹ * ↑↑ρ o

Next tactic:
gcongr
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := by gcongr; 

Proof state:
case bc.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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ↑↑ρ o = 0
μo : ↑↑μ oᶜ = 0
⊢ s ∩ o ⊆ o

Next tactic:
apply inter_subset_right
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := by gcongr; apply inter_subset_right
      _ = 0 := 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
ε : ℝ≥0
εpos : ε > 0
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a}
o : Set α
left✝ : MeasurableSet o
ρo : ↑↑ρ o = 0
μo : ↑↑μ oᶜ = 0
⊢ (↑ε)⁻¹ * ↑↑ρ o = 0

Next tactic:
rw [ρo, mul_zero]
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := by gcongr; apply inter_subset_right
      _ = 0 := by rw [ρo, mul_zero]
  

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
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0)

Next tactic:
obtain ⟨u, _, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ≥0)
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := by gcongr; apply inter_subset_right
      _ = 0 := by rw [ρo, mul_zero]
  obtain ⟨u, _, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ≥0)
  

Proof state:
case intro.intro.intro
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
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0)

Next tactic:
have B : ∀ᵐ x ∂μ, ∀ n, ∀ᶠ a in v.filterAt x, ρ a < u n * μ a :=
    ae_all_iff.2 fun n => A (u n) (u_pos n)
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := by gcongr; apply inter_subset_right
      _ = 0 := by rw [ρo, mul_zero]
  obtain ⟨u, _, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ≥0)
  have B : ∀ᵐ x ∂μ, ∀ n, ∀ᶠ a in v.filterAt x, ρ a < u n * μ a :=
    ae_all_iff.2 fun n => A (u n) (u_pos n)
  

Proof state:
case intro.intro.intro
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
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0)

Next tactic:
filter_upwards [B, v.ae_eventually_measure_pos]
/-- If a measure `ρ` is singular with respect to `μ`, then for `μ` almost every `x`, the ratio
`ρ a / μ a` tends to zero when `a` shrinks to `x` along the Vitali family. This makes sense
as `μ a` is eventually positive by `ae_eventually_measure_pos`. -/
theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by
  have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
    intro ε εpos
    set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
    change μ s = 0
    obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
    apply le_antisymm _ bot_le
    calc
      μ s ≤ μ (s ∩ o ∪ oᶜ) := by
        conv_lhs => rw [← inter_union_compl s o]
        gcongr
        apply inter_subset_right
      _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _)
      _ = μ (s ∩ o) := by rw [μo, add_zero]
      _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by
        simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)]
        rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul]
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by
        gcongr
        refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _
        intro x hx
        rw [hs] at hx
        simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx
        exact hx.1
      _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := by gcongr; apply inter_subset_right
      _ = 0 := by rw [ρo, mul_zero]
  obtain ⟨u, _, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ≥0)
  have B : ∀ᵐ x ∂μ, ∀ n, ∀ᶠ a in v.filterAt x, ρ a < u n * μ a :=
    ae_all_iff.2 fun n => A (u n) (u_pos n)
  filter_upwards [B, v.ae_eventually_measure_pos]
  

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
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a
⊢ ∀ (a : α),
    (∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v a, ↑↑ρ a < ↑(u n) * ↑↑μ a) →
      (∀ᶠ (a : Set α) in filterAt v a, 0 < ↑↑μ a) → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 0)

Next tactic:
intro x hx h'x