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

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
⊢ μ.InnerRegularWRT p MeasurableSet
state_after: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: intro F hF r hr
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: have hBU : ⋃ n, F ∩ s n = F := by  rw [← inter_iUnion, univ_subset_iff.mp hs, inter_univ]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: have : μ F = ⨆ n, μ (F ∩ s n) := by
    rw [← measure_iUnion_eq_iSup, hBU]
    exact Monotone.directed_le fun m n h ↦ inter_subset_inter_right _ (hmono h)
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < μ F
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: rw [this] at hr
state_before: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: case intro
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < μ (F ∩ s n)
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: case intro
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < (μ.restrict (s n)) F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: rw [← restrict_apply hF] at hn
state_before: case intro
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < (μ.restrict (s n)) F
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < (μ.restrict (s n)) F
K : Set α
KF : K ⊆ F
hKp : p K
hK : r < (μ.restrict (s n)) K
⊢ ∃ K ⊆ F, p K ∧ r < μ K
tactic: rcases h n hF _ hn with ⟨K, KF, hKp, hK⟩
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : TopologicalSpace α
μ✝ : Measure α
p q : Set α → Prop
U s✝ : Set α
ε r✝ : ℝ≥0∞
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet
hs : univ ⊆ ⋃ n, s n
hmono : Monotone s
F : Set α
hF : MeasurableSet F
r : ℝ≥0∞
hr : r < ⨆ n, μ (F ∩ s n)
hBU : ⋃ n, F ∩ s n = F
this : μ F = ⨆ n, μ (F ∩ s n)
n : ℕ
hn : r < (μ.restrict (s n)) F
K : Set α
KF : K ⊆ F
hKp : p K
hK : r < (μ.restrict (s n)) K
⊢ ∃ K ⊆ F, p K ∧ r < μ K
state_after: no goals
tactic: exact ⟨K, KF, hKp, hK.trans_le (restrict_apply_le _ _)⟩