-- Mathlib/MeasureTheory/Covering/Differentiation.lean
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a

Proof state:
case intro.intro.intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
x : α
ε : ℝ
εpos : ε > 0
hx : ∀ (ε : ℝ), 0 < ε → ∃ t ∈ v.setsAt x, t ⊆ closedBall x ε ∧ μ t = 0
a : Set α
a_sets : a ∈ v.setsAt x
ax : a ⊆ closedBall x ε
μa : μ a = 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Tactic:
exact ⟨a, ⟨a_sets, μa⟩, ax⟩

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
x : α
hx : x ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Tactic:
rw [hs] at hx

Result:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
x : α
hx : x ∈ {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
ε : ℝ
εpos : ε > 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Proof state:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
x : α
ε : ℝ
εpos : ε > 0
hx : ∀ (ε : ℝ), 0 < ε → ∃ t ∈ v.setsAt x, t ⊆ closedBall x ε ∧ μ t = 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Tactic:
rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩

Result:
case intro.intro.intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
x : α
ε : ℝ
εpos : ε > 0
hx : ∀ (ε : ℝ), 0 < ε → ∃ t ∈ v.setsAt x, t ⊆ closedBall x ε ∧ μ t = 0
a : Set α
a_sets : a ∈ v.setsAt x
ax : a ⊆ closedBall x ε
μa : μ a = 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Proof state:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
h : v.FineSubfamilyOn f s
⊢ μ s ≤ 0

Tactic:
calc
  μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum
  _ = ∑' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2
  _ = 0 := by simp only [tsum_zero, add_zero]

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
⊢ v.FineSubfamilyOn f s

Tactic:
intro x hx ε εpos

Result:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}
hs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}
f : α → Set (Set α) := fun x => {a | μ a = 0}
x : α
hx : x ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε