-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
theorem exists_null_frontier_thickening (μ : Measure Ω) [SFinite μ] (s : Set Ω) {a b : ℝ}
    (hab : a < b) : ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Proof state:
case intro.intro
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
len_pos : 0 < volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))})
r : ℝ
r_in_Ioo : r ∈ Ioo a b
hr : r ∉ {i | 0 < μ (frontier (Metric.thickening i s))}
⊢ μ (frontier (Metric.thickening r s)) = 0

Tactic:
simpa only [mem_setOf_eq, not_lt, le_zero_iff] using hr

Result:
no goals

Proof state:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
len_pos : 0 < ENNReal.ofReal (b - a)
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Tactic:
rw [← Real.volume_Ioo, ← aux] at len_pos

Result:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
len_pos : 0 < volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))})
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Proof state:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Tactic:
have aux := measure_diff_null (s := Ioo a b) (Set.Countable.measure_zero key volume)

Result:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Proof state:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Tactic:
have len_pos : 0 < ENNReal.ofReal (b - a) := by simp only [hab, ENNReal.ofReal_pos, sub_pos]

Result:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
len_pos : 0 < ENNReal.ofReal (b - a)
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Proof state:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
len_pos : 0 < volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))})
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Tactic:
rcases nonempty_of_measure_ne_zero len_pos.ne.symm with ⟨r, ⟨r_in_Ioo, hr⟩⟩

Result:
case intro.intro
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
key : {i | 0 < μ (frontier (Metric.thickening i s))}.Countable
aux : volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))}) = volume (Ioo a b)
len_pos : 0 < volume (Ioo a b \ {i | 0 < μ (frontier (Metric.thickening i s))})
r : ℝ
r_in_Ioo : r ∈ Ioo a b
hr : r ∉ {i | 0 < μ (frontier (Metric.thickening i s))}
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Proof state:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0

Tactic:
have disjs := Metric.frontier_thickening_disjoint s

Result:
Ω : Type u_1
inst✝³ : PseudoEMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : SFinite μ
s : Set Ω
a b : ℝ
hab : a < b
mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))
disjs : Pairwise (Function.onFun Disjoint fun r => frontier (Metric.thickening r s))
⊢ ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0