Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ μ = ν

Tactic:
refine
    Measure.ext_of_generateFrom_of_cover_subset
      (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico α)) (isPiSystem_Ico id id) ?_ this
      ?_ ?_ ?_

Goal state after:
case refine_1
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2} ⊆ {S | ∃ l u, l < u ∧ Ico l u = S}

case refine_2
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ ⋃₀ {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2} = univ

case refine_3
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ ∀ s_1 ∈ {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}, μ s_1 ≠ ∞

case refine_4
α : Type u_5
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
inst✝ : NoMaxOrder α
μ ν : Measure α
hμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞
h : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) = ν (Ico a b)
s : Set α
hsc : s.Countable
hsd : Dense s
hsb : ∀ (x : α), IsBot x → x ∈ s
right✝ : ∀ (x : α), IsTop x → x ∈ s
this : {x | ∃ i, ∃ (_ : i ∈ s), ∃ i_2, ∃ (_ : i_2 ∈ s) (_ : i < i_2), x = Ico i i_2}.Countable
⊢ ∀ s ∈ {S | ∃ l u, l < u ∧ Ico l u = S}, μ s = ν s