Theorem: measurable_of_isClosed
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: apply measurable_of_isOpen
state_before: α : Type ?u.149654
β : Type ?u.149657
γ : Type u_1
γ₂ : Type ?u.149663
δ : Type u_2
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f

tactic: intro s hs
state_before: case hf
α : Type ?u.149654
β : Type ?u.149657
γ : Type u_1
γ₂ : Type ?u.149663
δ : Type u_2
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
⊢ ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)

tactic: rw [← MeasurableSet.compl_iff, ← preimage_compl]
state_before: case hf
α : Type ?u.149654
β : Type ?u.149657
γ : Type u_1
γ₂ : Type ?u.149663
δ : Type u_2
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ MeasurableSet (f ⁻¹' s)

tactic: apply hf
state_before: case hf
α : Type ?u.149654
β : Type ?u.149657
γ : Type u_1
γ₂ : Type ?u.149663
δ : Type u_2
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ MeasurableSet (f ⁻¹' sᶜ)

tactic: rw [isClosed_compl_iff]
state_before: case hf.a
α : Type ?u.149654
β : Type ?u.149657
γ : Type u_1
γ₂ : Type ?u.149663
δ : Type u_2
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ IsClosed (sᶜ)

tactic: exact hs
state_before: case hf.a
α : Type ?u.149654
β : Type ?u.149657
γ : Type u_1
γ₂ : Type ?u.149663
δ : Type u_2
ι : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen s
⊢ IsOpen s