Theorem: MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace
File: Mathlib/MeasureTheory/Measure/Regular.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
⊢ μ.InnerRegularWRT IsClosed IsOpen
state_after: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
⊢ μ.InnerRegularWRT IsClosed IsOpen
tactic: let A : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
⊢ μ.InnerRegularWRT IsClosed IsOpen
state_after: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U✝ s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
U : Set X
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ U
⊢ ∃ K ⊆ U, IsClosed K ∧ r < μ K
tactic: intro U hU r hr
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U✝ s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
U : Set X
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ U
⊢ ∃ K ⊆ U, IsClosed K ∧ r < μ K
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
r : ℝ≥0∞
F : ℕ → Set X
F_closed : ∀ (n : ℕ), IsClosed (F n)
F_mono : Monotone F
hU : IsOpen (⋃ n, F n)
hr : r < μ (⋃ n, F n)
⊢ ∃ K ⊆ ⋃ n, F n, IsClosed K ∧ r < μ K
tactic: rcases hU.exists_iUnion_isClosed with ⟨F, F_closed, -, rfl, F_mono⟩
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
r : ℝ≥0∞
F : ℕ → Set X
F_closed : ∀ (n : ℕ), IsClosed (F n)
F_mono : Monotone F
hU : IsOpen (⋃ n, F n)
hr : r < μ (⋃ n, F n)
⊢ ∃ K ⊆ ⋃ n, F n, IsClosed K ∧ r < μ K
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
r : ℝ≥0∞
F : ℕ → Set X
F_closed : ∀ (n : ℕ), IsClosed (F n)
F_mono : Monotone F
hU : IsOpen (⋃ n, F n)
hr : r < ⨆ i, μ (F i)
⊢ ∃ K ⊆ ⋃ n, F n, IsClosed K ∧ r < μ K
tactic: rw [measure_iUnion_eq_iSup F_mono.directed_le] at hr
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
r : ℝ≥0∞
F : ℕ → Set X
F_closed : ∀ (n : ℕ), IsClosed (F n)
F_mono : Monotone F
hU : IsOpen (⋃ n, F n)
hr : r < ⨆ i, μ (F i)
⊢ ∃ K ⊆ ⋃ n, F n, IsClosed K ∧ r < μ K
state_after: case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
r : ℝ≥0∞
F : ℕ → Set X
F_closed : ∀ (n : ℕ), IsClosed (F n)
F_mono : Monotone F
hU : IsOpen (⋃ n, F n)
hr : r < ⨆ i, μ (F i)
n : ℕ
hn : r < μ (F n)
⊢ ∃ K ⊆ ⋃ n, F n, IsClosed K ∧ r < μ K
tactic: rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
state_before: case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s : Set α
ε r✝ : ℝ≥0∞
X : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : PseudoMetrizableSpace X
inst✝ : MeasurableSpace X
μ : Measure X
A : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
r : ℝ≥0∞
F : ℕ → Set X
F_closed : ∀ (n : ℕ), IsClosed (F n)
F_mono : Monotone F
hU : IsOpen (⋃ n, F n)
hr : r < ⨆ i, μ (F i)
n : ℕ
hn : r < μ (F n)
⊢ ∃ K ⊆ ⋃ n, F n, IsClosed K ∧ r < μ K
state_after: no goals
tactic: exact ⟨F n, subset_iUnion _ _, F_closed n, hn⟩