-- Mathlib/MeasureTheory/Covering/Differentiation.lean
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
    ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
      {x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0

Proof state:
case neg
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
x : α
hx : x ∈ {x | ↑q < v.limRatio ρ x}
h : x ∉ s
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)

Tactic:
exact Or.inl (subset_toMeasurable μ sᶜ h)

Result:
no goals

Proof state:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
⊢ ∃ a b,
    MeasurableSet a ∧ MeasurableSet b ∧ {x | v.limRatio ρ x < ↑p} ⊆ a ∧ {x | ↑q < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0

Tactic:
let w n := s ∩ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n

Result:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
⊢ ∃ a b,
    MeasurableSet a ∧ MeasurableSet b ∧ {x | v.limRatio ρ x < ↑p} ⊆ a ∧ {x | ↑q < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0

Proof state:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
⊢ (p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) = ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))

Tactic:
simp only [coe_nnreal_smul_apply,
  measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I]

Result:
no goals

Proof state:
case refine_3
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) (u i) ∩ toMeasurable (ρ + μ) (w i_1) ⊆
    toMeasurable μ sᶜ ∪ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) (u i) ∩ toMeasurable (ρ + μ) (w i_1)

Tactic:
exact subset_union_right

Result:
no goals

Proof state:
case neg
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
x : α
hx : x ∈ {x | v.limRatio ρ x < ↑p}
h : x ∉ s
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)

Tactic:
exact Or.inl (subset_toMeasurable μ sᶜ h)

Result:
no goals

Proof state:
case pos
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
x : α
hx : x ∈ {x | v.limRatio ρ x < ↑p}
h : x ∈ s
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)

Tactic:
refine Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, ?_⟩)

Result:
case pos
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
x : α
hx : x ∈ {x | v.limRatio ρ x < ↑p}
h : x ∈ s
⊢ x ∈ toMeasurable (ρ + μ) (u (spanningSetsIndex (ρ + μ) x))