File: Mathlib/MeasureTheory/OuterMeasure/Defs.lean

full_name: MeasureTheory.OuterMeasure
code: /-- An outer measure is a countably subadditive monotone function that sends `∅` to `0`. -/
structure OuterMeasure (α : Type*) where
  /-- Outer measure function. Use automatic coercion instead. -/
  protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
goal: Type u_2

full_name: MeasureTheory.OuterMeasure.mono
code: /-- An outer measure is a countably subadditive monotone function that sends `∅` to `0`. -/
structure OuterMeasure (α : Type*) where
  /-- Outer measure function. Use automatic coercion instead. -/
  protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
goal: s₁ ⊆ s₂ → self.measureOf s₁ ≤ self.measureOf s₂

full_name: MeasureTheory.OuterMeasure.iUnion_nat
code: /-- An outer measure is a countably subadditive monotone function that sends `∅` to `0`. -/
structure OuterMeasure (α : Type*) where
  /-- Outer measure function. Use automatic coercion instead. -/
  protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
goal: Pairwise (Disjoint on s) → self.measureOf (⋃ i, s i) ≤ ∑' (i : ℕ), self.measureOf (s i)

full_name: MeasureTheory.OuterMeasure.empty
code: /-- An outer measure is a countably subadditive monotone function that sends `∅` to `0`. -/
structure OuterMeasure (α : Type*) where
  /-- Outer measure function. Use automatic coercion instead. -/
  protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
goal: self.measureOf ∅ = 0

full_name: MeasureTheory.OuterMeasure.measureOf
code: /-- An outer measure is a countably subadditive monotone function that sends `∅` to `0`. -/
structure OuterMeasure (α : Type*) where
  /-- Outer measure function. Use automatic coercion instead. -/
  protected measureOf : Set α → ℝ≥0∞
  protected empty : measureOf ∅ = 0
  protected mono : ∀ {s₁ s₂}, s₁ ⊆ s₂ → measureOf s₁ ≤ measureOf s₂
  protected iUnion_nat : ∀ s : ℕ → Set α, Pairwise (Disjoint on s) →
    measureOf (⋃ i, s i) ≤ ∑' i, measureOf (s i)
goal: Set α → ℝ≥0∞

full_name: MeasureTheory.OuterMeasureClass.measure_empty
code: /-- A mixin class saying that elements `μ : F` are outer measures on `α`.

This typeclass is used to unify some API for outer measures and measures. -/
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
goal: f ∅ = 0

full_name: MeasureTheory.OuterMeasureClass.measure_mono
code: /-- A mixin class saying that elements `μ : F` are outer measures on `α`.

This typeclass is used to unify some API for outer measures and measures. -/
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
goal: s ⊆ t → f s ≤ f t

full_name: MeasureTheory.OuterMeasureClass.measure_iUnion_nat_le
code: /-- A mixin class saying that elements `μ : F` are outer measures on `α`.

This typeclass is used to unify some API for outer measures and measures. -/
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
goal: Pairwise (Disjoint on s) → f (⋃ i, s i) ≤ ∑' (i : ℕ), f (s i)

full_name: MeasureTheory.OuterMeasureClass
code: /-- A mixin class saying that elements `μ : F` are outer measures on `α`.

This typeclass is used to unify some API for outer measures and measures. -/
class OuterMeasureClass (F : Type*) (α : outParam Type*) [FunLike F (Set α) ℝ≥0∞] : Prop where
  protected measure_empty (f : F) : f ∅ = 0
  protected measure_mono (f : F) {s t} : s ⊆ t → f s ≤ f t
  protected measure_iUnion_nat_le (f : F) (s : ℕ → Set α) : Pairwise (Disjoint on s) →
    f (⋃ i, s i) ≤ ∑' i, f (s i)
goal: Prop

full_name: MeasureTheory.OuterMeasure.instFunLikeSetENNReal
code: instance : FunLike (OuterMeasure α) (Set α) ℝ≥0∞ 
goal: FunLike (MeasureTheory.OuterMeasure α) (Set α) ℝ≥0∞

full_name: MeasureTheory.OuterMeasure.instCoeFun
code: instance instCoeFun : CoeFun (OuterMeasure α) (fun _ => Set α → ℝ≥0∞) 
goal: CoeFun (MeasureTheory.OuterMeasure α) fun x => Set α → ℝ≥0∞

full_name: MeasureTheory.OuterMeasure.measureOf_eq_coe
code: @[simp] theorem measureOf_eq_coe (m : OuterMeasure α) : m.measureOf = m 
goal: m.measureOf = ⇑m

full_name: MeasureTheory.OuterMeasure.instOuterMeasureClass
code: instance : OuterMeasureClass (OuterMeasure α) α 
goal: MeasureTheory.OuterMeasureClass (MeasureTheory.OuterMeasure α) α