File: Mathlib/MeasureTheory/Measure/NullMeasurable.lean

full_name: MeasureTheory.NullMeasurableSpace
code: /-- A type tag for `α` with `MeasurableSet` given by `NullMeasurableSet`. -/
@[nolint unusedArguments]
def NullMeasurableSpace (α : Type*) [MeasurableSpace α]
    (_μ : Measure α := by volume_tac) : Type _ 
goal: Type u_5

full_name: MeasureTheory.NullMeasurableSpace.instInhabited
code: instance NullMeasurableSpace.instInhabited [h : Inhabited α] :
    Inhabited (NullMeasurableSpace α μ) 
goal: Inhabited (MeasureTheory.NullMeasurableSpace α μ)

full_name: MeasureTheory.NullMeasurableSpace.instSubsingleton
code: instance NullMeasurableSpace.instSubsingleton [h : Subsingleton α] :
    Subsingleton (NullMeasurableSpace α μ) 
goal: Subsingleton (MeasureTheory.NullMeasurableSpace α μ)

full_name: MeasureTheory.NullMeasurableSpace.instMeasurableSpace
code: instance NullMeasurableSpace.instMeasurableSpace : MeasurableSpace (NullMeasurableSpace α μ) 
goal: MeasurableSpace (MeasureTheory.NullMeasurableSpace α μ)

full_name: MeasureTheory.NullMeasurableSet
code: /-- A set is called `NullMeasurableSet` if it can be approximated by a measurable set up to
a set of null measure. -/
def NullMeasurableSet [MeasurableSpace α] (s : Set α)
    (μ : Measure α := by volume_tac) : Prop 
goal: Prop

full_name: MeasurableSet.nullMeasurableSet
code: @[simp]
theorem _root_.MeasurableSet.nullMeasurableSet (h : MeasurableSet s) : NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet s μ

full_name: MeasureTheory.nullMeasurableSet_empty
code: theorem nullMeasurableSet_empty : NullMeasurableSet ∅ μ 
goal: MeasureTheory.NullMeasurableSet ∅ μ

full_name: MeasureTheory.nullMeasurableSet_univ
code: theorem nullMeasurableSet_univ : NullMeasurableSet univ μ 
goal: MeasureTheory.NullMeasurableSet Set.univ μ

full_name: MeasureTheory.NullMeasurableSet.of_null
code: theorem of_null (h : μ s = 0) : NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet s μ

full_name: MeasureTheory.NullMeasurableSet.compl
code: theorem compl (h : NullMeasurableSet s μ) : NullMeasurableSet sᶜ μ 
goal: MeasureTheory.NullMeasurableSet sᶜ μ

full_name: MeasureTheory.NullMeasurableSet.of_compl
code: theorem of_compl (h : NullMeasurableSet sᶜ μ) : NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet s μ

full_name: MeasureTheory.NullMeasurableSet.compl_iff
code: @[simp]
theorem compl_iff : NullMeasurableSet sᶜ μ ↔ NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet sᶜ μ ↔ MeasureTheory.NullMeasurableSet s μ

full_name: MeasureTheory.NullMeasurableSet.of_subsingleton
code: @[nontriviality]
theorem of_subsingleton [Subsingleton α] : NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet s μ

full_name: MeasureTheory.NullMeasurableSet.congr
code: protected theorem congr (hs : NullMeasurableSet s μ) (h : s =ᵐ[μ] t) : NullMeasurableSet t μ 
goal: MeasureTheory.NullMeasurableSet t μ

full_name: MeasureTheory.NullMeasurableSet.iUnion
code: protected theorem iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
    (h : ∀ i, NullMeasurableSet (s i) μ) : NullMeasurableSet (⋃ i, s i) μ 
goal: MeasureTheory.NullMeasurableSet (⋃ i, s i) μ

full_name: MeasureTheory.NullMeasurableSet.biUnion
code: protected theorem biUnion {f : ι → Set α} {s : Set ι} (hs : s.Countable)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ 
goal: MeasureTheory.NullMeasurableSet (⋃ b ∈ s, f b) μ

full_name: MeasureTheory.NullMeasurableSet.sUnion
code: protected theorem sUnion {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, NullMeasurableSet t μ) :
    NullMeasurableSet (⋃₀ s) μ 
goal: MeasureTheory.NullMeasurableSet (⋃₀ s) μ

full_name: MeasureTheory.NullMeasurableSet.iInter
code: protected theorem iInter {ι : Sort*} [Countable ι] {f : ι → Set α}
    (h : ∀ i, NullMeasurableSet (f i) μ) : NullMeasurableSet (⋂ i, f i) μ 
goal: MeasureTheory.NullMeasurableSet (⋂ i, f i) μ

full_name: MeasureTheory.NullMeasurableSet.biInter
code: protected theorem biInter {f : β → Set α} {s : Set β} (hs : s.Countable)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋂ b ∈ s, f b) μ 
goal: MeasureTheory.NullMeasurableSet (⋂ b ∈ s, f b) μ

full_name: MeasureTheory.NullMeasurableSet.sInter
code: protected theorem sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, NullMeasurableSet t μ) :
    NullMeasurableSet (⋂₀ s) μ 
goal: MeasureTheory.NullMeasurableSet (⋂₀ s) μ

full_name: MeasureTheory.NullMeasurableSet.union
code: @[simp]
protected theorem union (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
    NullMeasurableSet (s ∪ t) μ 
goal: MeasureTheory.NullMeasurableSet (s ∪ t) μ

full_name: MeasureTheory.NullMeasurableSet.union_null
code: protected theorem union_null (hs : NullMeasurableSet s μ) (ht : μ t = 0) :
    NullMeasurableSet (s ∪ t) μ 
goal: MeasureTheory.NullMeasurableSet (s ∪ t) μ

full_name: MeasureTheory.NullMeasurableSet.inter
code: @[simp]
protected theorem inter (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
    NullMeasurableSet (s ∩ t) μ 
goal: MeasureTheory.NullMeasurableSet (s ∩ t) μ

full_name: MeasureTheory.NullMeasurableSet.diff
code: @[simp]
protected theorem diff (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
    NullMeasurableSet (s \ t) μ 
goal: MeasureTheory.NullMeasurableSet (s \ t) μ

full_name: MeasureTheory.NullMeasurableSet.disjointed
code: @[simp]
protected theorem disjointed {f : ℕ → Set α} (h : ∀ i, NullMeasurableSet (f i) μ) (n) :
    NullMeasurableSet (disjointed f n) μ 
goal: MeasureTheory.NullMeasurableSet (disjointed f n) μ

full_name: MeasureTheory.NullMeasurableSet.const
code: protected theorem const (p : Prop) : NullMeasurableSet { _a : α | p } μ 
goal: MeasureTheory.NullMeasurableSet {_a | p} μ

full_name: MeasureTheory.NullMeasurableSet.instMeasurableSingletonClass
code: instance instMeasurableSingletonClass [MeasurableSingletonClass α] :
    MeasurableSingletonClass (NullMeasurableSpace α μ) 
goal: MeasurableSingletonClass (MeasureTheory.NullMeasurableSpace α μ)

full_name: MeasureTheory.NullMeasurableSet.insert
code: protected theorem insert [MeasurableSingletonClass (NullMeasurableSpace α μ)]
    (hs : NullMeasurableSet s μ) (a : α) : NullMeasurableSet (insert a s) μ 
goal: MeasureTheory.NullMeasurableSet (insert a s) μ

full_name: MeasureTheory.NullMeasurableSet.exists_measurable_superset_ae_eq
code: theorem exists_measurable_superset_ae_eq (h : NullMeasurableSet s μ) :
    ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s 
goal: ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s

full_name: MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq
code: theorem toMeasurable_ae_eq (h : NullMeasurableSet s μ) : toMeasurable μ s =ᵐ[μ] s 
goal: MeasureTheory.toMeasurable μ s =ᵐ[μ] s

full_name: MeasureTheory.NullMeasurableSet.compl_toMeasurable_compl_ae_eq
code: theorem compl_toMeasurable_compl_ae_eq (h : NullMeasurableSet s μ) : (toMeasurable μ sᶜ)ᶜ =ᵐ[μ] s 
goal: (MeasureTheory.toMeasurable μ sᶜ)ᶜ =ᵐ[μ] s

full_name: MeasureTheory.NullMeasurableSet.exists_measurable_subset_ae_eq
code: theorem exists_measurable_subset_ae_eq (h : NullMeasurableSet s μ) :
    ∃ t ⊆ s, MeasurableSet t ∧ t =ᵐ[μ] s 
goal: ∃ t ⊆ s, MeasurableSet t ∧ t =ᵐ[μ] s

full_name: MeasureTheory.exists_subordinate_pairwise_disjoint
code: /-- If `sᵢ` is a countable family of (null) measurable pairwise `μ`-a.e. disjoint sets, then there
exists a subordinate family `tᵢ ⊆ sᵢ` of measurable pairwise disjoint sets such that
`tᵢ =ᵐ[μ] sᵢ`. -/
theorem exists_subordinate_pairwise_disjoint [Countable ι] {s : ι → Set α}
    (h : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s)) :
    ∃ t : ι → Set α,
      (∀ i, t i ⊆ s i) ∧
        (∀ i, s i =ᵐ[μ] t i) ∧ (∀ i, MeasurableSet (t i)) ∧ Pairwise (Disjoint on t) 
goal: ∃ t, (∀ (i : ι), t i ⊆ s i) ∧ (∀ (i : ι), s i =ᵐ[μ] t i) ∧ (∀ (i : ι), MeasurableSet (t i)) ∧ Pairwise (Disjoint on t)

full_name: MeasureTheory.measure_iUnion
code: theorem measure_iUnion {m0 : MeasurableSpace α} {μ : Measure α} [Countable ι] {f : ι → Set α}
    (hn : Pairwise (Disjoint on f)) (h : ∀ i, MeasurableSet (f i)) :
    μ (⋃ i, f i) = ∑' i, μ (f i) 
goal: μ (⋃ i, f i) = ∑' (i : ι), μ (f i)

full_name: MeasureTheory.measure_iUnion₀
code: theorem measure_iUnion₀ [Countable ι] {f : ι → Set α} (hd : Pairwise (AEDisjoint μ on f))
    (h : ∀ i, NullMeasurableSet (f i) μ) : μ (⋃ i, f i) = ∑' i, μ (f i) 
goal: μ (⋃ i, f i) = ∑' (i : ι), μ (f i)

full_name: MeasureTheory.measure_union₀_aux
code: theorem measure_union₀_aux (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ)
    (hd : AEDisjoint μ s t) : μ (s ∪ t) = μ s + μ t 
goal: μ (s ∪ t) = μ s + μ t

full_name: MeasureTheory.measure_inter_add_diff₀
code: /-- A null measurable set `t` is Carathéodory measurable: for any `s`, we have
`μ (s ∩ t) + μ (s \ t) = μ s`. -/
theorem measure_inter_add_diff₀ (s : Set α) (ht : NullMeasurableSet t μ) :
    μ (s ∩ t) + μ (s \ t) = μ s 
goal: μ (s ∩ t) + μ (s \ t) = μ s

full_name: MeasureTheory.measure_union_add_inter₀
code: theorem measure_union_add_inter₀ (s : Set α) (ht : NullMeasurableSet t μ) :
    μ (s ∪ t) + μ (s ∩ t) = μ s + μ t 
goal: μ (s ∪ t) + μ (s ∩ t) = μ s + μ t

full_name: MeasureTheory.measure_union_add_inter₀'
code: theorem measure_union_add_inter₀' (hs : NullMeasurableSet s μ) (t : Set α) :
    μ (s ∪ t) + μ (s ∩ t) = μ s + μ t 
goal: μ (s ∪ t) + μ (s ∩ t) = μ s + μ t

full_name: MeasureTheory.measure_union₀
code: theorem measure_union₀ (ht : NullMeasurableSet t μ) (hd : AEDisjoint μ s t) :
    μ (s ∪ t) = μ s + μ t 
goal: μ (s ∪ t) = μ s + μ t

full_name: MeasureTheory.measure_union₀'
code: theorem measure_union₀' (hs : NullMeasurableSet s μ) (hd : AEDisjoint μ s t) :
    μ (s ∪ t) = μ s + μ t 
goal: μ (s ∪ t) = μ s + μ t

full_name: MeasureTheory.measure_add_measure_compl₀
code: theorem measure_add_measure_compl₀ {s : Set α} (hs : NullMeasurableSet s μ) :
    μ s + μ sᶜ = μ univ 
goal: μ s + μ sᶜ = μ Set.univ

full_name: MeasureTheory.nullMeasurableSet_singleton
code: theorem nullMeasurableSet_singleton (x : α) : NullMeasurableSet {x} μ 
goal: MeasureTheory.NullMeasurableSet {x} μ

full_name: MeasureTheory.nullMeasurableSet_insert
code: @[simp]
theorem nullMeasurableSet_insert {a : α} {s : Set α} :
    NullMeasurableSet (insert a s) μ ↔ NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet (insert a s) μ ↔ MeasureTheory.NullMeasurableSet s μ

full_name: MeasureTheory.nullMeasurableSet_eq
code: theorem nullMeasurableSet_eq {a : α} : NullMeasurableSet { x | x = a } μ 
goal: MeasureTheory.NullMeasurableSet {x | x = a} μ

full_name: Set.Finite.nullMeasurableSet
code: protected theorem _root_.Set.Finite.nullMeasurableSet (hs : s.Finite) : NullMeasurableSet s μ 
goal: MeasureTheory.NullMeasurableSet s μ

full_name: Finset.nullMeasurableSet
code: protected theorem _root_.Finset.nullMeasurableSet (s : Finset α) : NullMeasurableSet (↑s) μ 
goal: MeasureTheory.NullMeasurableSet (↑s) μ

full_name: Set.Finite.nullMeasurableSet_biUnion
code: theorem _root_.Set.Finite.nullMeasurableSet_biUnion {f : ι → Set α} {s : Set ι} (hs : s.Finite)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ 
goal: MeasureTheory.NullMeasurableSet (⋃ b ∈ s, f b) μ

full_name: Finset.nullMeasurableSet_biUnion
code: theorem _root_.Finset.nullMeasurableSet_biUnion {f : ι → Set α} (s : Finset ι)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋃ b ∈ s, f b) μ 
goal: MeasureTheory.NullMeasurableSet (⋃ b ∈ s, f b) μ

full_name: Set.Finite.nullMeasurableSet_sUnion
code: theorem _root_.Set.Finite.nullMeasurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)
    (h : ∀ t ∈ s, NullMeasurableSet t μ) : NullMeasurableSet (⋃₀ s) μ 
goal: MeasureTheory.NullMeasurableSet (⋃₀ s) μ

full_name: Set.Finite.nullMeasurableSet_biInter
code: theorem _root_.Set.Finite.nullMeasurableSet_biInter {f : ι → Set α} {s : Set ι} (hs : s.Finite)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋂ b ∈ s, f b) μ 
goal: MeasureTheory.NullMeasurableSet (⋂ b ∈ s, f b) μ

full_name: Finset.nullMeasurableSet_biInter
code: theorem _root_.Finset.nullMeasurableSet_biInter {f : ι → Set α} (s : Finset ι)
    (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) : NullMeasurableSet (⋂ b ∈ s, f b) μ 
goal: MeasureTheory.NullMeasurableSet (⋂ b ∈ s, f b) μ

full_name: Set.Finite.nullMeasurableSet_sInter
code: theorem _root_.Set.Finite.nullMeasurableSet_sInter {s : Set (Set α)} (hs : s.Finite)
    (h : ∀ t ∈ s, NullMeasurableSet t μ) : NullMeasurableSet (⋂₀ s) μ 
goal: MeasureTheory.NullMeasurableSet (⋂₀ s) μ

full_name: MeasureTheory.nullMeasurableSet_toMeasurable
code: theorem nullMeasurableSet_toMeasurable : NullMeasurableSet (toMeasurable μ s) μ 
goal: MeasureTheory.NullMeasurableSet (MeasureTheory.toMeasurable μ s) μ

full_name: MeasureTheory.NullMeasurable
code: /-- A function `f : α → β` is null measurable if the preimage of a measurable set is a null
measurable set. -/
def NullMeasurable (f : α → β) (μ : Measure α := by volume_tac) : Prop 
goal: Prop

full_name: Measurable.nullMeasurable
code: protected theorem _root_.Measurable.nullMeasurable (h : Measurable f) : NullMeasurable f μ 
goal: MeasureTheory.NullMeasurable f μ

full_name: MeasureTheory.NullMeasurable.measurable'
code: protected theorem NullMeasurable.measurable' (h : NullMeasurable f μ) :
    @Measurable (NullMeasurableSpace α μ) β _ _ f 
goal: Measurable f

full_name: MeasureTheory.Measurable.comp_nullMeasurable
code: theorem Measurable.comp_nullMeasurable {g : β → γ} (hg : Measurable g) (hf : NullMeasurable f μ) :
    NullMeasurable (g ∘ f) μ 
goal: MeasureTheory.NullMeasurable (g ∘ f) μ

full_name: MeasureTheory.NullMeasurable.congr
code: theorem NullMeasurable.congr {g : α → β} (hf : NullMeasurable f μ) (hg : f =ᵐ[μ] g) :
    NullMeasurable g μ 
goal: MeasureTheory.NullMeasurable g μ

full_name: MeasureTheory.Measure.IsComplete.out'
code: /-- A measure is complete if every null set is also measurable.
  A null set is a subset of a measurable set with measure `0`.
  Since every measure is defined as a special case of an outer measure, we can more simply state
  that a set `s` is null if `μ s = 0`. -/
class Measure.IsComplete {_ : MeasurableSpace α} (μ : Measure α) : Prop where
  out' : ∀ s, μ s = 0 → MeasurableSet s
goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : μ.IsComplete] (s : Set α), μ s = 0 → MeasurableSet s

full_name: MeasureTheory.Measure.IsComplete
code: /-- A measure is complete if every null set is also measurable.
  A null set is a subset of a measurable set with measure `0`.
  Since every measure is defined as a special case of an outer measure, we can more simply state
  that a set `s` is null if `μ s = 0`. -/
class Measure.IsComplete {_ : MeasurableSpace α} (μ : Measure α) : Prop where
  out' : ∀ s, μ s = 0 → MeasurableSet s
goal: {x : MeasurableSpace α} → MeasureTheory.Measure α → Prop

full_name: MeasureTheory.Measure.isComplete_iff
code: theorem Measure.isComplete_iff : μ.IsComplete ↔ ∀ s, μ s = 0 → MeasurableSet s 
goal: μ.IsComplete ↔ ∀ (s : Set α), μ s = 0 → MeasurableSet s

full_name: MeasureTheory.Measure.IsComplete.out
code: theorem Measure.IsComplete.out (h : μ.IsComplete) : ∀ s, μ s = 0 → MeasurableSet s 
goal: μ s = 0 → MeasurableSet s

full_name: MeasureTheory.measurableSet_of_null
code: theorem measurableSet_of_null [μ.IsComplete] (hs : μ s = 0) : MeasurableSet s 
goal: MeasurableSet s

full_name: MeasureTheory.NullMeasurableSet.measurable_of_complete
code: theorem NullMeasurableSet.measurable_of_complete (hs : NullMeasurableSet s μ) [μ.IsComplete] :
    MeasurableSet s 
goal: MeasurableSet s

full_name: MeasureTheory.NullMeasurable.measurable_of_complete
code: theorem NullMeasurable.measurable_of_complete [μ.IsComplete] {_m1 : MeasurableSpace β} {f : α → β}
    (hf : NullMeasurable f μ) : Measurable f 
goal: Measurable f

full_name: Measurable.congr_ae
code: theorem _root_.Measurable.congr_ae {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
    [_hμ : μ.IsComplete] {f g : α → β} (hf : Measurable f) (hfg : f =ᵐ[μ] g) : Measurable g 
goal: Measurable g

full_name: MeasureTheory.Measure.completion
code: /-- Given a measure we can complete it to a (complete) measure on all null measurable sets. -/
def completion {_ : MeasurableSpace α} (μ : Measure α) :
    MeasureTheory.Measure (NullMeasurableSpace α μ) 
goal: {x : MeasurableSpace α} → (μ : MeasureTheory.Measure α) → MeasureTheory.Measure (MeasureTheory.NullMeasurableSpace α μ)

full_name: MeasureTheory.Measure.completion.isComplete
code: instance completion.isComplete {_m : MeasurableSpace α} (μ : Measure α) : μ.completion.IsComplete 
goal: μ.completion.IsComplete

full_name: MeasureTheory.Measure.coe_completion
code: @[simp]
theorem coe_completion {_ : MeasurableSpace α} (μ : Measure α) : ⇑μ.completion = μ 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), ⇑μ.completion = ⇑μ

full_name: MeasureTheory.Measure.completion_apply
code: theorem completion_apply {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) :
    μ.completion s = μ s 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α), μ.completion s = μ s

full_name: MeasureTheory.Measure.ae_completion
code: @[simp]
theorem ae_completion {_ : MeasurableSpace α} (μ : Measure α) : ae μ.completion = ae μ 
goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α), MeasureTheory.ae μ.completion = MeasureTheory.ae μ