Theorem: MeasureTheory.Measure.Regular.restrict_of_measure_ne_top
File: Mathlib/MeasureTheory/Measure/Regular.lean

tactic: have : WeaklyRegular (μ.restrict A) := WeaklyRegular.restrict_of_measure_ne_top h'A
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
⊢ (μ.restrict A).Regular
state_after: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).Regular

tactic: constructor
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).Regular
state_after: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).InnerRegularWRT IsCompact IsOpen

tactic: intro V hV r hr
state_before: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).InnerRegularWRT IsCompact IsOpen
state_after: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ ∃ K ⊆ V, IsCompact K ∧ r < (μ.restrict A) K

tactic: have R : restrict μ A V ≠ ∞ := by
  rw [restrict_apply hV.measurableSet]
  exact ((measure_mono inter_subset_right).trans_lt h'A.lt_top).ne
state_before: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ ∃ K ⊆ V, IsCompact K ∧ r < (μ.restrict A) K
state_after: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
R : (μ.restrict A) V ≠ ⊤
⊢ ∃ K ⊆ V, IsCompact K ∧ r < (μ.restrict A) K

tactic: exact MeasurableSet.exists_lt_isCompact_of_ne_top hV.measurableSet R hr
state_before: case innerRegular
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
R : (μ.restrict A) V ≠ ⊤
⊢ ∃ K ⊆ V, IsCompact K ∧ r < (μ.restrict A) K
state_after: no goals

tactic: rw [restrict_apply hV.measurableSet]
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ (μ.restrict A) V ≠ ⊤
state_after: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ μ (V ∩ A) ≠ ⊤

tactic: exact ((measure_mono inter_subset_right).trans_lt h'A.lt_top).ne
state_before: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
μ : Measure α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ μ (V ∩ A) ≠ ⊤
state_after: no goals