File: Mathlib/MeasureTheory/Measure/AEDisjoint.lean

full_name: MeasureTheory.AEDisjoint
code: /-- Two sets are said to be `μ`-a.e. disjoint if their intersection has measure zero. -/
def AEDisjoint (s t : Set α) 
goal: Prop

full_name: MeasureTheory.exists_null_pairwise_disjoint_diff
code: /-- If `s : ι → Set α` is a countable family of pairwise a.e. disjoint sets, then there exists a
family of measurable null sets `t i` such that `s i \ t i` are pairwise disjoint. -/
theorem exists_null_pairwise_disjoint_diff [Countable ι] {s : ι → Set α}
    (hd : Pairwise (AEDisjoint μ on s)) : ∃ t : ι → Set α, (∀ i, MeasurableSet (t i)) ∧
    (∀ i, μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i) 
goal: ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i)

full_name: MeasureTheory.AEDisjoint.eq
code: protected theorem eq (h : AEDisjoint μ s t) : μ (s ∩ t) = 0 
goal: μ (s ∩ t) = 0

full_name: MeasureTheory.AEDisjoint.symm
code: @[symm]
protected theorem symm (h : AEDisjoint μ s t) : AEDisjoint μ t s 
goal: MeasureTheory.AEDisjoint μ t s

full_name: MeasureTheory.AEDisjoint.symmetric
code: protected theorem symmetric : Symmetric (AEDisjoint μ) 
goal: Symmetric (MeasureTheory.AEDisjoint μ)

full_name: MeasureTheory.AEDisjoint.comm
code: protected theorem comm : AEDisjoint μ s t ↔ AEDisjoint μ t s 
goal: MeasureTheory.AEDisjoint μ s t ↔ MeasureTheory.AEDisjoint μ t s

full_name: Disjoint.aedisjoint
code: protected theorem _root_.Disjoint.aedisjoint (h : Disjoint s t) : AEDisjoint μ s t 
goal: MeasureTheory.AEDisjoint μ s t

full_name: Pairwise.aedisjoint
code: protected theorem _root_.Pairwise.aedisjoint {f : ι → Set α} (hf : Pairwise (Disjoint on f)) :
    Pairwise (AEDisjoint μ on f) 
goal: Pairwise (MeasureTheory.AEDisjoint μ on f)

full_name: Set.PairwiseDisjoint.aedisjoint
code: protected theorem _root_.Set.PairwiseDisjoint.aedisjoint {f : ι → Set α} {s : Set ι}
    (hf : s.PairwiseDisjoint f) : s.Pairwise (AEDisjoint μ on f) 
goal: s.Pairwise (MeasureTheory.AEDisjoint μ on f)

full_name: MeasureTheory.AEDisjoint.mono_ae
code: theorem mono_ae (h : AEDisjoint μ s t) (hu : u ≤ᵐ[μ] s) (hv : v ≤ᵐ[μ] t) : AEDisjoint μ u v 
goal: MeasureTheory.AEDisjoint μ u v

full_name: MeasureTheory.AEDisjoint.mono
code: protected theorem mono (h : AEDisjoint μ s t) (hu : u ⊆ s) (hv : v ⊆ t) : AEDisjoint μ u v 
goal: MeasureTheory.AEDisjoint μ u v

full_name: MeasureTheory.AEDisjoint.congr
code: protected theorem congr (h : AEDisjoint μ s t) (hu : u =ᵐ[μ] s) (hv : v =ᵐ[μ] t) :
    AEDisjoint μ u v 
goal: MeasureTheory.AEDisjoint μ u v

full_name: MeasureTheory.AEDisjoint.iUnion_left_iff
code: @[simp]
theorem iUnion_left_iff [Countable ι] {s : ι → Set α} :
    AEDisjoint μ (⋃ i, s i) t ↔ ∀ i, AEDisjoint μ (s i) t 
goal: MeasureTheory.AEDisjoint μ (⋃ i, s i) t ↔ ∀ (i : ι), MeasureTheory.AEDisjoint μ (s i) t

full_name: MeasureTheory.AEDisjoint.iUnion_right_iff
code: @[simp]
theorem iUnion_right_iff [Countable ι] {t : ι → Set α} :
    AEDisjoint μ s (⋃ i, t i) ↔ ∀ i, AEDisjoint μ s (t i) 
goal: MeasureTheory.AEDisjoint μ s (⋃ i, t i) ↔ ∀ (i : ι), MeasureTheory.AEDisjoint μ s (t i)

full_name: MeasureTheory.AEDisjoint.union_left_iff
code: @[simp]
theorem union_left_iff : AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u 
goal: MeasureTheory.AEDisjoint μ (s ∪ t) u ↔ MeasureTheory.AEDisjoint μ s u ∧ MeasureTheory.AEDisjoint μ t u

full_name: MeasureTheory.AEDisjoint.union_right_iff
code: @[simp]
theorem union_right_iff : AEDisjoint μ s (t ∪ u) ↔ AEDisjoint μ s t ∧ AEDisjoint μ s u 
goal: MeasureTheory.AEDisjoint μ s (t ∪ u) ↔ MeasureTheory.AEDisjoint μ s t ∧ MeasureTheory.AEDisjoint μ s u

full_name: MeasureTheory.AEDisjoint.union_left
code: theorem union_left (hs : AEDisjoint μ s u) (ht : AEDisjoint μ t u) : AEDisjoint μ (s ∪ t) u 
goal: MeasureTheory.AEDisjoint μ (s ∪ t) u

full_name: MeasureTheory.AEDisjoint.union_right
code: theorem union_right (ht : AEDisjoint μ s t) (hu : AEDisjoint μ s u) : AEDisjoint μ s (t ∪ u) 
goal: MeasureTheory.AEDisjoint μ s (t ∪ u)

full_name: MeasureTheory.AEDisjoint.diff_ae_eq_left
code: theorem diff_ae_eq_left (h : AEDisjoint μ s t) : (s \ t : Set α) =ᵐ[μ] s 
goal: s \ t =ᵐ[μ] s

full_name: MeasureTheory.AEDisjoint.diff_ae_eq_right
code: theorem diff_ae_eq_right (h : AEDisjoint μ s t) : (t \ s : Set α) =ᵐ[μ] t 
goal: t \ s =ᵐ[μ] t

full_name: MeasureTheory.AEDisjoint.measure_diff_left
code: theorem measure_diff_left (h : AEDisjoint μ s t) : μ (s \ t) = μ s 
goal: μ (s \ t) = μ s

full_name: MeasureTheory.AEDisjoint.measure_diff_right
code: theorem measure_diff_right (h : AEDisjoint μ s t) : μ (t \ s) = μ t 
goal: μ (t \ s) = μ t

full_name: MeasureTheory.AEDisjoint.exists_disjoint_diff
code: /-- If `s` and `t` are `μ`-a.e. disjoint, then `s \ u` and `t` are disjoint for some measurable null
set `u`. -/
theorem exists_disjoint_diff (h : AEDisjoint μ s t) :
    ∃ u, MeasurableSet u ∧ μ u = 0 ∧ Disjoint (s \ u) t 
goal: ∃ u, MeasurableSet u ∧ μ u = 0 ∧ Disjoint (s \ u) t

full_name: MeasureTheory.AEDisjoint.of_null_right
code: theorem of_null_right (h : μ t = 0) : AEDisjoint μ s t 
goal: MeasureTheory.AEDisjoint μ s t

full_name: MeasureTheory.AEDisjoint.of_null_left
code: theorem of_null_left (h : μ s = 0) : AEDisjoint μ s t 
goal: MeasureTheory.AEDisjoint μ s t

full_name: MeasureTheory.aedisjoint_compl_left
code: theorem aedisjoint_compl_left : AEDisjoint μ sᶜ s 
goal: MeasureTheory.AEDisjoint μ sᶜ s

full_name: MeasureTheory.aedisjoint_compl_right
code: theorem aedisjoint_compl_right : AEDisjoint μ s sᶜ 
goal: MeasureTheory.AEDisjoint μ s sᶜ