File: Mathlib/MeasureTheory/Measure/AddContent.lean

full_name: MeasureTheory.AddContent
code: /-- An additive content is a set function with value 0 at the empty set which is finitely additive
on a given set of sets. -/
structure AddContent (C : Set (Set α)) where
  /-- The value of the content on a set. -/
  toFun : Set α → ℝ≥0∞
  empty' : toFun ∅ = 0
  sUnion' (I : Finset (Set α)) (_h_ss : ↑I ⊆ C)
      (_h_dis : PairwiseDisjoint (I : Set (Set α)) id) (_h_mem : ⋃₀ ↑I ∈ C) :
    toFun (⋃₀ I) = ∑ u ∈ I, toFun u
goal: Type u_1

full_name: MeasureTheory.AddContent.sUnion'
code: /-- An additive content is a set function with value 0 at the empty set which is finitely additive
on a given set of sets. -/
structure AddContent (C : Set (Set α)) where
  /-- The value of the content on a set. -/
  toFun : Set α → ℝ≥0∞
  empty' : toFun ∅ = 0
  sUnion' (I : Finset (Set α)) (_h_ss : ↑I ⊆ C)
      (_h_dis : PairwiseDisjoint (I : Set (Set α)) id) (_h_mem : ⋃₀ ↑I ∈ C) :
    toFun (⋃₀ I) = ∑ u ∈ I, toFun u
goal: self.toFun (⋃₀ ↑I) = ∑ u ∈ I, self.toFun u

full_name: MeasureTheory.AddContent.empty'
code: /-- An additive content is a set function with value 0 at the empty set which is finitely additive
on a given set of sets. -/
structure AddContent (C : Set (Set α)) where
  /-- The value of the content on a set. -/
  toFun : Set α → ℝ≥0∞
  empty' : toFun ∅ = 0
  sUnion' (I : Finset (Set α)) (_h_ss : ↑I ⊆ C)
      (_h_dis : PairwiseDisjoint (I : Set (Set α)) id) (_h_mem : ⋃₀ ↑I ∈ C) :
    toFun (⋃₀ I) = ∑ u ∈ I, toFun u
goal: self.toFun ∅ = 0

full_name: MeasureTheory.AddContent.toFun
code: /-- An additive content is a set function with value 0 at the empty set which is finitely additive
on a given set of sets. -/
structure AddContent (C : Set (Set α)) where
  /-- The value of the content on a set. -/
  toFun : Set α → ℝ≥0∞
  empty' : toFun ∅ = 0
  sUnion' (I : Finset (Set α)) (_h_ss : ↑I ⊆ C)
      (_h_dis : PairwiseDisjoint (I : Set (Set α)) id) (_h_mem : ⋃₀ ↑I ∈ C) :
    toFun (⋃₀ I) = ∑ u ∈ I, toFun u
goal: Set α → ℝ≥0∞

full_name: MeasureTheory.instInhabitedAddContent
code: instance : Inhabited (AddContent C) 
goal: Inhabited (MeasureTheory.AddContent C)

full_name: MeasureTheory.instDFunLikeAddContentSetENNReal
code: instance : DFunLike (AddContent C) (Set α) (fun _ ↦ ℝ≥0∞) 
goal: DFunLike (MeasureTheory.AddContent C) (Set α) fun x => ℝ≥0∞

full_name: MeasureTheory.AddContent.ext
code: @[ext] protected lemma AddContent.ext (h : ∀ s, m s = m' s) : m = m' 
goal: m = m'

full_name: MeasureTheory.AddContent.ext_iff
code: protected lemma AddContent.ext_iff (m m' : AddContent C) : m = m' ↔ ∀ s, m s = m' s 
goal: m = m' ↔ ∀ (s : Set α), m s = m' s

full_name: MeasureTheory.addContent_empty
code: @[simp] lemma addContent_empty : m ∅ = 0 
goal: m ∅ = 0

full_name: MeasureTheory.addContent_sUnion
code: lemma addContent_sUnion (h_ss : ↑I ⊆ C)
    (h_dis : PairwiseDisjoint (I : Set (Set α)) id) (h_mem : ⋃₀ ↑I ∈ C) :
    m (⋃₀ I) = ∑ u ∈ I, m u 
goal: m (⋃₀ ↑I) = ∑ u ∈ I, m u

full_name: MeasureTheory.addContent_union'
code: lemma addContent_union' (hs : s ∈ C) (ht : t ∈ C) (hst : s ∪ t ∈ C) (h_dis : Disjoint s t) :
    m (s ∪ t) = m s + m t 
goal: m (s ∪ t) = m s + m t

full_name: MeasureTheory.addContent_eq_add_diffFinset₀_of_subset
code: lemma addContent_eq_add_diffFinset₀_of_subset (hC : IsSetSemiring C)
    (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s)
    (h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
    m s = ∑ i ∈ I, m i + ∑ i ∈ hC.diffFinset₀ hs hI, m i 
goal: m s = ∑ i ∈ I, m i + ∑ i ∈ hC.diffFinset₀ hs hI, m i

full_name: MeasureTheory.sum_addContent_le_of_subset
code: lemma sum_addContent_le_of_subset (hC : IsSetSemiring C)
    (h_ss : ↑I ⊆ C) (h_dis : PairwiseDisjoint (I : Set (Set α)) id)
    (ht : t ∈ C) (hJt : ∀ s ∈ I, s ⊆ t) :
    ∑ u ∈ I, m u ≤ m t 
goal: ∑ u ∈ I, m u ≤ m t

full_name: MeasureTheory.addContent_mono
code: lemma addContent_mono (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
    (hst : s ⊆ t) :
    m s ≤ m t 
goal: m s ≤ m t

full_name: MeasureTheory.addContent_union
code: lemma addContent_union (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C)
    (h_dis : Disjoint s t) :
    m (s ∪ t) = m s + m t 
goal: m (s ∪ t) = m s + m t

full_name: MeasureTheory.addContent_union_le
code: lemma addContent_union_le (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) :
    m (s ∪ t) ≤ m s + m t 
goal: m (s ∪ t) ≤ m s + m t

full_name: MeasureTheory.addContent_biUnion_le
code: lemma addContent_biUnion_le {ι : Type*} (hC : IsSetRing C) {s : ι → Set α}
    {S : Finset ι} (hs : ∀ n ∈ S, s n ∈ C) :
    m (⋃ i ∈ S, s i) ≤ ∑ i ∈ S, m (s i) 
goal: m (⋃ i ∈ S, s i) ≤ ∑ i ∈ S, m (s i)

full_name: MeasureTheory.le_addContent_diff
code: lemma le_addContent_diff (m : AddContent C) (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) :
    m s - m t ≤ m (s \ t) 
goal: m s - m t ≤ m (s \ t)