l3lab/ntp-mathlib huggingface dataset:
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := 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
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)
⊢ ↑↑(q • μ) s = ↑↑(q • μ) (s ∩ t ∪ s ∩ tᶜ)

Next tactic:
rw [inter_union_compl]
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := 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
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)
⊢ ↑↑(q • μ) (s ∩ t) + ↑↑(q • μ) (s ∩ tᶜ) ≤ ↑↑ρ (s ∩ t) + ↑↑(q • μ) tᶜ

Next tactic:
gcongr
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; 

Proof state:
case h₂.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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)
⊢ s ∩ tᶜ ⊆ tᶜ

Next tactic:
apply inter_subset_right
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := 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
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)
⊢ ↑↑ρ (s ∩ t) + ↑↑(q • μ) tᶜ = ↑↑ρ (s ∩ t)

Next tactic:
simp [A]
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := 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
hρ : ρ ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)
⊢ ↑↑ρ (s ∩ t) ≤ ↑↑ρ s

Next tactic:
gcongr
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; 

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)
⊢ s ∩ t ⊆ s

Next tactic:
apply inter_subset_left
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; apply inter_subset_left
  

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
⊢ ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t)

Next tactic:
refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; apply inter_subset_left
  refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _
  

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
⊢ ∀ x ∈ s ∩ t, ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a

Next tactic:
intro x hx
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; apply inter_subset_left
  refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _
  intro x hx
  

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
x : α
hx : x ∈ s ∩ t
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a

Next tactic:
have I : ∀ᶠ a in v.filterAt x, (q : ℝ≥0∞) < ρ a / μ a := (tendsto_order.1 hx.2).1 _ (h hx.1)
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; apply inter_subset_left
  refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _
  intro x hx
  have I : ∀ᶠ a in v.filterAt x, (q : ℝ≥0∞) < ρ a / μ a := (tendsto_order.1 hx.2).1 _ (h hx.1)
  

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
x : α
hx : x ∈ s ∩ t
I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a
⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a

Next tactic:
apply I.frequently.mono fun a ha => ?_
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; apply inter_subset_left
  refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _
  intro x hx
  have I : ∀ᶠ a in v.filterAt x, (q : ℝ≥0∞) < ρ a / μ a := (tendsto_order.1 hx.2).1 _ (h hx.1)
  apply I.frequently.mono fun a ha => ?_
  

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
x : α
hx : x ∈ s ∩ t
I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a
a : Set α
ha : ↑q < ↑↑ρ a / ↑↑μ a
⊢ ↑↑(q • μ) a ≤ ↑↑ρ a

Next tactic:
rw [coe_nnreal_smul_apply]
/-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
`v.limRatioMeas hρ x`, the same property holds for sets `s` on which `q < v.limRatioMeas hρ`. -/
theorem mul_measure_le_of_subset_lt_limRatioMeas {q : ℝ≥0} {s : Set α}
    (h : s ⊆ {x | (q : ℝ≥0∞) < v.limRatioMeas hρ x}) : (q : ℝ≥0∞) * μ s ≤ ρ s := by
  let t := {x : α | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}
  have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ
  suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t);
  exact
    calc
      (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl]
      _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _)
      _ ≤ ρ (s ∩ t) + (q • μ) tᶜ := by gcongr; apply inter_subset_right
      _ = ρ (s ∩ t) := by simp [A]
      _ ≤ ρ s := by gcongr; apply inter_subset_left
  refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _
  intro x hx
  have I : ∀ᶠ a in v.filterAt x, (q : ℝ≥0∞) < ρ a / μ a := (tendsto_order.1 hx.2).1 _ (h hx.1)
  apply I.frequently.mono fun a ha => ?_
  rw [coe_nnreal_smul_apply]
  

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 ≪ μ
q : ℝ≥0
s : Set α
h : s ⊆ {x | ↑q < limRatioMeas v hρ x}
t : Set α := {x | Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))}
A : ↑↑μ tᶜ = 0
x : α
hx : x ∈ s ∩ t
I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a
a : Set α
ha : ↑q < ↑↑ρ a / ↑↑μ a
⊢ ↑q * ↑↑μ a ≤ ↑↑ρ a

Next tactic:
exact ENNReal.mul_le_of_le_div ha.le
/-- The points with `v.limRatioMeas hρ x = ∞` have measure `0` for `μ`. -/
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 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
hρ : ρ ≪ μ
⊢ ↑↑μ {x | limRatioMeas v hρ x = ⊤} = 0

Next tactic:
refine' null_of_locally_null _ fun x _ => _
/-- The points with `v.limRatioMeas hρ x = ∞` have measure `0` for `μ`. -/
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by
  refine' null_of_locally_null _ fun x _ => _
  

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 ≪ μ
x : α
x✝ : x ∈ {x | limRatioMeas v hρ x = ⊤}
⊢ ∃ u ∈ 𝓝[{x | limRatioMeas v hρ x = ⊤}] x, ↑↑μ u = 0

Next tactic:
obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top ρ x
/-- The points with `v.limRatioMeas hρ x = ∞` have measure `0` for `μ`. -/
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by
  refine' null_of_locally_null _ fun x _ => _
  obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top ρ x
  

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 ≪ μ
x : α
x✝ : x ∈ {x | limRatioMeas v hρ x = ⊤}
o : Set α
xo : x ∈ o
o_open : IsOpen o
μo : ↑↑ρ o < ⊤
⊢ ∃ u ∈ 𝓝[{x | limRatioMeas v hρ x = ⊤}] x, ↑↑μ u = 0

Next tactic:
let s := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
/-- The points with `v.limRatioMeas hρ x = ∞` have measure `0` for `μ`. -/
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by
  refine' null_of_locally_null _ fun x _ => _
  obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top ρ x
  let s := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
  

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 ≪ μ
x : α
x✝ : x ∈ {x | limRatioMeas v hρ x = ⊤}
o : Set α
xo : x ∈ o
o_open : IsOpen o
μo : ↑↑ρ o < ⊤
s : Set α := {x | limRatioMeas v hρ x = ⊤} ∩ o
⊢ ∃ u ∈ 𝓝[{x | limRatioMeas v hρ x = ⊤}] x, ↑↑μ u = 0

Next tactic:
refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩
/-- The points with `v.limRatioMeas hρ x = ∞` have measure `0` for `μ`. -/
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by
  refine' null_of_locally_null _ fun x _ => _
  obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top ρ x
  let s := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
  refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩
  

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 ≪ μ
x : α
x✝ : x ∈ {x | limRatioMeas v hρ x = ⊤}
o : Set α
xo : x ∈ o
o_open : IsOpen o
μo : ↑↑ρ o < ⊤
s : Set α := {x | limRatioMeas v hρ x = ⊤} ∩ o
⊢ ↑↑μ s ≤ 0

Next tactic:
have ρs : ρ s ≠ ∞ := ((measure_mono (inter_subset_right _ _)).trans_lt μo).ne
/-- The points with `v.limRatioMeas hρ x = ∞` have measure `0` for `μ`. -/
theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by
  refine' null_of_locally_null _ fun x _ => _
  obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top ρ x
  let s := {x : α | v.limRatioMeas hρ x = ∞} ∩ o
  refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩
  have ρs : ρ s ≠ ∞ := ((measure_mono (inter_subset_right _ _)).trans_lt μo).ne
  

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 ≪ μ
x : α
x✝ : x ∈ {x | limRatioMeas v hρ x = ⊤}
o : Set α
xo : x ∈ o
o_open : IsOpen o
μo : ↑↑ρ o < ⊤
s : Set α := {x | limRatioMeas v hρ x = ⊤} ∩ o
ρs : ↑↑ρ s ≠ ⊤
⊢ ↑↑μ s ≤ 0

Next tactic:
have A : ∀ q : ℝ≥0, 1 ≤ q → μ s ≤ (q : ℝ≥0∞)⁻¹ * ρ s := by
    intro q hq
    rw [mul_comm, ← div_eq_mul_inv, ENNReal.le_div_iff_mul_le _ (Or.inr ρs), mul_comm]
    · apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ
      intro y hy
      have : v.limRatioMeas hρ y = ∞ := hy.1
      simp only [this, ENNReal.coe_lt_top, mem_setOf_eq]
    · simp only [(zero_lt_one.trans_le hq).ne', true_or_iff, ENNReal.coe_eq_zero, Ne.def,
        not_false_iff]