Theorem: MeasureTheory.Measure.ext_of_Ioc_finite
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: refine' @ext_of_Ico_finite αᵒᵈ _ _ _ _ _ ‹_› μ ν _ hμν fun a b hab => _
state_before: α✝ : Type ?u.679146
β : Type ?u.679149
γ : Type ?u.679152
γ₂ : Type ?u.679155
δ : Type ?u.679158
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.679251
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a b x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
⊢ μ = ν

tactic: erw [dual_Ico (α := α)]
state_before: α✝ : Type ?u.679146
β : Type ?u.679149
γ : Type ?u.679152
γ₂ : Type ?u.679155
δ : Type ?u.679158
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.679251
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (Ico a b) = ↑↑ν (Ico a b)

tactic: exact h hab
state_before: α✝ : Type ?u.679146
β : Type ?u.679149
γ : Type ?u.679152
γ₂ : Type ?u.679155
δ : Type ?u.679158
ι : Sort y
s t u : Set α✝
inst✝²² : TopologicalSpace α✝
inst✝²¹ : MeasurableSpace α✝
inst✝²⁰ : OpensMeasurableSpace α✝
inst✝¹⁹ : TopologicalSpace β
inst✝¹⁸ : MeasurableSpace β
inst✝¹⁷ : OpensMeasurableSpace β
inst✝¹⁶ : TopologicalSpace γ
inst✝¹⁵ : MeasurableSpace γ
inst✝¹⁴ : BorelSpace γ
inst✝¹³ : TopologicalSpace γ₂
inst✝¹² : MeasurableSpace γ₂
inst✝¹¹ : BorelSpace γ₂
inst✝¹⁰ : MeasurableSpace δ
α' : Type ?u.679251
inst✝⁹ : TopologicalSpace α'
inst✝⁸ : MeasurableSpace α'
inst✝⁷ : LinearOrder α✝
inst✝⁶ : OrderClosedTopology α✝
a✝ b✝ x : α✝
α : Type u_1
inst✝⁵ : TopologicalSpace α
m : MeasurableSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : BorelSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure μ
hμν : ↑↑μ univ = ↑↑ν univ
h : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ioc a b) = ↑↑ν (Ioc a b)
a b : αᵒᵈ
hab : a < b
⊢ ↑↑μ (↑OrderDual.ofDual ⁻¹' Ioc b a) = ↑↑ν (↑OrderDual.ofDual ⁻¹' Ioc b a)