Theorem: MeasureTheory.inducedOuterMeasure_exists_set
File: Mathlib/MeasureTheory/Measure/OuterMeasure.lean

tactic: have h := ENNReal.lt_add_right hs hε
state_before: α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))
msU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε

tactic: conv at h =>
  lhs
  rw [inducedOuterMeasure_eq_iInf _ msU m_mono]
state_before: α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))
msU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
h : ↑(inducedOuterMeasure m P0 m0) s < ↑(inducedOuterMeasure m P0 m0) s + ε
⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε

tactic: simp only [iInf_lt_iff] at h
state_before: α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))
msU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
h : (⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht) < ↑(inducedOuterMeasure m P0 m0) s + ε
⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε

tactic: rcases h with ⟨t, h1t, h2t, h3t⟩
state_before: α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))
msU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
h : ∃ i h i_1, m i (_ : P i) < ↑(inducedOuterMeasure m P0 m0) s + ε
⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε

tactic: exact
  ⟨t, h1t, h2t, le_trans (le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono h1t) (le_of_lt h3t)⟩
state_before: case intro.intro.intro
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)
mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))
msU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),
    m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s : Set α
hs : ↑(inducedOuterMeasure m P0 m0) s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
t : Set α
h1t : P t
h2t : s ⊆ t
h3t : m t (_ : P t) < ↑(inducedOuterMeasure m P0 m0) s + ε
⊢ ∃ t _ht, s ⊆ t ∧ ↑(inducedOuterMeasure m P0 m0) t ≤ ↑(inducedOuterMeasure m P0 m0) s + ε