-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
theorem inducedOuterMeasure_caratheodory (s : Set α) :
    MeasurableSet[(inducedOuterMeasure m P0 m0).caratheodory] s ↔
      ∀ t : Set α,
        P t →
          inducedOuterMeasure m P0 m0 (t ∩ s) + inducedOuterMeasure m P0 m0 (t \ s) ≤
            inducedOuterMeasure m P0 m0 t

Proof state:
case mpr
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, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (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
h :
  ∀ (t : Set α),
    P t →
      (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t
u t : Set α
ht : P t
h2t : u ⊆ t
⊢ (inducedOuterMeasure m P0 m0) (u ∩ s) + (inducedOuterMeasure m P0 m0) (u \ s) ≤ m t ht

Tactic:
refine le_trans ?_ ((h t ht).trans_eq <| inducedOuterMeasure_eq' _ msU m_mono ht)

Result:
case mpr
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, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (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
h :
  ∀ (t : Set α),
    P t →
      (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t
u t : Set α
ht : P t
h2t : u ⊆ t
⊢ (inducedOuterMeasure m P0 m0) (u ∩ s) + (inducedOuterMeasure m P0 m0) (u \ s) ≤
    (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s)

Proof state:
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, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (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
⊢ MeasurableSet s ↔
    ∀ (t : Set α),
      P t →
        (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t

Tactic:
rw [isCaratheodory_iff_le]

Result:
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, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (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
⊢ (∀ (t : Set α),
      (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t) ↔
    ∀ (t : Set α),
      P t →
        (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t

Proof state:
case mpr
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, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (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
h :
  ∀ (t : Set α),
    P t →
      (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t
u : Set α
⊢ (inducedOuterMeasure m P0 m0) (u ∩ s) + (inducedOuterMeasure m P0 m0) (u \ s) ≤
    ⨅ t, ⨅ (ht : P t), ⨅ (_ : u ⊆ t), m t ht

Tactic:
refine le_iInf ?_

Result:
case mpr
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, f i)
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (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
h :
  ∀ (t : Set α),
    P t →
      (inducedOuterMeasure m P0 m0) (t ∩ s) + (inducedOuterMeasure m P0 m0) (t \ s) ≤ (inducedOuterMeasure m P0 m0) t
u : Set α
⊢ ∀ (i : Set α),
    (inducedOuterMeasure m P0 m0) (u ∩ s) + (inducedOuterMeasure m P0 m0) (u \ s) ≤ ⨅ (ht : P i), ⨅ (_ : u ⊆ i), m i ht