File: Mathlib/MeasureTheory/OuterMeasure/AE.lean

full_name: MeasureTheory.ae
code: /-- The “almost everywhere” filter of co-null sets. -/
def ae (μ : F) : Filter α 
goal: Filter α

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_OuterMeasure_AE___macroRules_MeasureTheory_term∀ᵐ_∂_,__1»
code: /-- `∀ᵐ a ∂μ, p a` means that `p a` for a.e. `a`, i.e. `p` holds true away from a null set.

This is notation for `Filter.Eventually p (MeasureTheory.ae μ)`. -/
notation3 "∀ᵐ "(...)" ∂"μ", "r:(scoped p => Filter.Eventually p <| MeasureTheory.ae μ) => r
goal: Lean.Macro

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_OuterMeasure_AE___macroRules_MeasureTheory_term∃ᵐ_∂_,__1»
code: /-- `∃ᵐ a ∂μ, p a` means that `p` holds `∂μ`-frequently,
i.e. `p` holds on a set of positive measure.

This is notation for `Filter.Frequently p (MeasureTheory.ae μ)`. -/
notation3 "∃ᵐ "(...)" ∂"μ", "r:(scoped P => Filter.Frequently P <| MeasureTheory.ae μ) => r
goal: Lean.Macro

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_OuterMeasure_AE___unexpand_Filter_EventuallyEq_1
code: /-- `f =ᵐ[μ] g` means `f` and `g` are eventually equal along the a.e. filter,
i.e. `f=g` away from a null set.

This is notation for `Filter.EventuallyEq (MeasureTheory.ae μ) f g`. -/
notation:50 f " =ᵐ[" μ:50 "] " g:50 => Filter.EventuallyEq (MeasureTheory.ae μ) f g
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_OuterMeasure_AE___macroRules_MeasureTheory_term_=ᵐ[_]__1»
code: /-- `f =ᵐ[μ] g` means `f` and `g` are eventually equal along the a.e. filter,
i.e. `f=g` away from a null set.

This is notation for `Filter.EventuallyEq (MeasureTheory.ae μ) f g`. -/
notation:50 f " =ᵐ[" μ:50 "] " g:50 => Filter.EventuallyEq (MeasureTheory.ae μ) f g
goal: Lean.Macro

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_OuterMeasure_AE___unexpand_Filter_EventuallyLE_1
code: /-- `f ≤ᵐ[μ] g` means `f` is eventually less than `g` along the a.e. filter,
i.e. `f ≤ g` away from a null set.

This is notation for `Filter.EventuallyLE (MeasureTheory.ae μ) f g`. -/
notation:50 f " ≤ᵐ[" μ:50 "] " g:50 => Filter.EventuallyLE (MeasureTheory.ae μ) f g
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_OuterMeasure_AE___macroRules_MeasureTheory_term_≤ᵐ[_]__1»
code: /-- `f ≤ᵐ[μ] g` means `f` is eventually less than `g` along the a.e. filter,
i.e. `f ≤ g` away from a null set.

This is notation for `Filter.EventuallyLE (MeasureTheory.ae μ) f g`. -/
notation:50 f " ≤ᵐ[" μ:50 "] " g:50 => Filter.EventuallyLE (MeasureTheory.ae μ) f g
goal: Lean.Macro

full_name: MeasureTheory.mem_ae_iff
code: theorem mem_ae_iff {s : Set α} : s ∈ ae μ ↔ μ sᶜ = 0 
goal: s ∈ MeasureTheory.ae μ ↔ μ sᶜ = 0

full_name: MeasureTheory.ae_iff
code: theorem ae_iff {p : α → Prop} : (∀ᵐ a ∂μ, p a) ↔ μ { a | ¬p a } = 0 
goal: (∀ᵐ (a : α) ∂μ, p a) ↔ μ {a | ¬p a} = 0

full_name: MeasureTheory.compl_mem_ae_iff
code: theorem compl_mem_ae_iff {s : Set α} : sᶜ ∈ ae μ ↔ μ s = 0 
goal: sᶜ ∈ MeasureTheory.ae μ ↔ μ s = 0

full_name: MeasureTheory.frequently_ae_iff
code: theorem frequently_ae_iff {p : α → Prop} : (∃ᵐ a ∂μ, p a) ↔ μ { a | p a } ≠ 0 
goal: (∃ᵐ (a : α) ∂μ, p a) ↔ μ {a | p a} ≠ 0

full_name: MeasureTheory.frequently_ae_mem_iff
code: theorem frequently_ae_mem_iff {s : Set α} : (∃ᵐ a ∂μ, a ∈ s) ↔ μ s ≠ 0 
goal: (∃ᵐ (a : α) ∂μ, a ∈ s) ↔ μ s ≠ 0

full_name: MeasureTheory.measure_zero_iff_ae_nmem
code: theorem measure_zero_iff_ae_nmem {s : Set α} : μ s = 0 ↔ ∀ᵐ a ∂μ, a ∉ s 
goal: μ s = 0 ↔ ∀ᵐ (a : α) ∂μ, a ∉ s

full_name: MeasureTheory.ae_of_all
code: theorem ae_of_all {p : α → Prop} (μ : F) : (∀ a, p a) → ∀ᵐ a ∂μ, p a 
goal: (∀ (a : α), p a) → ∀ᵐ (a : α) ∂μ, p a

full_name: MeasureTheory.instCountableInterFilter
code: instance instCountableInterFilter : CountableInterFilter (ae μ) 
goal: CountableInterFilter (MeasureTheory.ae μ)

full_name: MeasureTheory.ae_all_iff
code: theorem ae_all_iff {ι : Sort*} [Countable ι] {p : α → ι → Prop} :
    (∀ᵐ a ∂μ, ∀ i, p a i) ↔ ∀ i, ∀ᵐ a ∂μ, p a i 
goal: (∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i) ↔ ∀ (i : ι), ∀ᵐ (a : α) ∂μ, p a i

full_name: MeasureTheory.all_ae_of
code: theorem all_ae_of {ι : Sort*} {p : α → ι → Prop} (hp : ∀ᵐ a ∂μ, ∀ i, p a i) (i : ι) :
    ∀ᵐ a ∂μ, p a i 
goal: ∀ᵐ (a : α) ∂μ, p a i

full_name: MeasureTheory.ae_iff_of_countable
code: lemma ae_iff_of_countable [Countable α] {p : α → Prop} : (∀ᵐ x ∂μ, p x) ↔ ∀ x, μ {x} ≠ 0 → p x 
goal: (∀ᵐ (x : α) ∂μ, p x) ↔ ∀ (x : α), μ {x} ≠ 0 → p x

full_name: MeasureTheory.ae_ball_iff
code: theorem ae_ball_iff {ι : Type*} {S : Set ι} (hS : S.Countable) {p : α → ∀ i ∈ S, Prop} :
    (∀ᵐ x ∂μ, ∀ i (hi : i ∈ S), p x i hi) ↔ ∀ i (hi : i ∈ S), ∀ᵐ x ∂μ, p x i hi 
goal: (∀ᵐ (x : α) ∂μ, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᵐ (x : α) ∂μ, p x i hi

full_name: MeasureTheory.ae_eq_refl
code: theorem ae_eq_refl (f : α → β) : f =ᵐ[μ] f 
goal: f =ᵐ[μ] f

full_name: MeasureTheory.ae_eq_symm
code: theorem ae_eq_symm {f g : α → β} (h : f =ᵐ[μ] g) : g =ᵐ[μ] f 
goal: g =ᵐ[μ] f

full_name: MeasureTheory.ae_eq_trans
code: theorem ae_eq_trans {f g h : α → β} (h₁ : f =ᵐ[μ] g) (h₂ : g =ᵐ[μ] h) : f =ᵐ[μ] h 
goal: f =ᵐ[μ] h

full_name: MeasureTheory.ae_le_of_ae_lt
code: theorem ae_le_of_ae_lt {β : Type*} [Preorder β] {f g : α → β} (h : ∀ᵐ x ∂μ, f x < g x) :
    f ≤ᵐ[μ] g 
goal: f ≤ᵐ[μ] g

full_name: MeasureTheory.ae_eq_empty
code: @[simp]
theorem ae_eq_empty : s =ᵐ[μ] (∅ : Set α) ↔ μ s = 0 
goal: s =ᵐ[μ] ∅ ↔ μ s = 0

full_name: MeasureTheory.ae_eq_univ
code: @[simp high]
theorem ae_eq_univ : s =ᵐ[μ] (univ : Set α) ↔ μ sᶜ = 0 
goal: s =ᵐ[μ] Set.univ ↔ μ sᶜ = 0

full_name: MeasureTheory.ae_le_set
code: theorem ae_le_set : s ≤ᵐ[μ] t ↔ μ (s \ t) = 0 
goal: s ≤ᵐ[μ] t ↔ μ (s \ t) = 0

full_name: MeasureTheory.ae_le_set_inter
code: theorem ae_le_set_inter {s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t') :
    (s ∩ s' : Set α) ≤ᵐ[μ] (t ∩ t' : Set α) 
goal: s ∩ s' ≤ᵐ[μ] t ∩ t'

full_name: MeasureTheory.ae_le_set_union
code: theorem ae_le_set_union {s' t' : Set α} (h : s ≤ᵐ[μ] t) (h' : s' ≤ᵐ[μ] t') :
    (s ∪ s' : Set α) ≤ᵐ[μ] (t ∪ t' : Set α) 
goal: s ∪ s' ≤ᵐ[μ] t ∪ t'

full_name: MeasureTheory.union_ae_eq_right
code: theorem union_ae_eq_right : (s ∪ t : Set α) =ᵐ[μ] t ↔ μ (s \ t) = 0 
goal: s ∪ t =ᵐ[μ] t ↔ μ (s \ t) = 0

full_name: MeasureTheory.diff_ae_eq_self
code: theorem diff_ae_eq_self : (s \ t : Set α) =ᵐ[μ] s ↔ μ (s ∩ t) = 0 
goal: s \ t =ᵐ[μ] s ↔ μ (s ∩ t) = 0

full_name: MeasureTheory.diff_null_ae_eq_self
code: theorem diff_null_ae_eq_self (ht : μ t = 0) : (s \ t : Set α) =ᵐ[μ] s 
goal: s \ t =ᵐ[μ] s

full_name: MeasureTheory.ae_eq_set
code: theorem ae_eq_set {s t : Set α} : s =ᵐ[μ] t ↔ μ (s \ t) = 0 ∧ μ (t \ s) = 0 
goal: s =ᵐ[μ] t ↔ μ (s \ t) = 0 ∧ μ (t \ s) = 0

full_name: MeasureTheory.measure_symmDiff_eq_zero_iff
code: @[simp]
theorem measure_symmDiff_eq_zero_iff {s t : Set α} : μ (s ∆ t) = 0 ↔ s =ᵐ[μ] t 
goal: μ (symmDiff s t) = 0 ↔ s =ᵐ[μ] t

full_name: MeasureTheory.ae_eq_set_compl_compl
code: @[simp]
theorem ae_eq_set_compl_compl {s t : Set α} : sᶜ =ᵐ[μ] tᶜ ↔ s =ᵐ[μ] t 
goal: sᶜ =ᵐ[μ] tᶜ ↔ s =ᵐ[μ] t

full_name: MeasureTheory.ae_eq_set_compl
code: theorem ae_eq_set_compl {s t : Set α} : sᶜ =ᵐ[μ] t ↔ s =ᵐ[μ] tᶜ 
goal: sᶜ =ᵐ[μ] t ↔ s =ᵐ[μ] tᶜ

full_name: MeasureTheory.ae_eq_set_inter
code: theorem ae_eq_set_inter {s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t') :
    (s ∩ s' : Set α) =ᵐ[μ] (t ∩ t' : Set α) 
goal: s ∩ s' =ᵐ[μ] t ∩ t'

full_name: MeasureTheory.ae_eq_set_union
code: theorem ae_eq_set_union {s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t') :
    (s ∪ s' : Set α) =ᵐ[μ] (t ∪ t' : Set α) 
goal: s ∪ s' =ᵐ[μ] t ∪ t'

full_name: MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left
code: theorem union_ae_eq_univ_of_ae_eq_univ_left (h : s =ᵐ[μ] univ) : (s ∪ t : Set α) =ᵐ[μ] univ 
goal: s ∪ t =ᵐ[μ] Set.univ

full_name: MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right
code: theorem union_ae_eq_univ_of_ae_eq_univ_right (h : t =ᵐ[μ] univ) : (s ∪ t : Set α) =ᵐ[μ] univ 
goal: s ∪ t =ᵐ[μ] Set.univ

full_name: MeasureTheory.union_ae_eq_right_of_ae_eq_empty
code: theorem union_ae_eq_right_of_ae_eq_empty (h : s =ᵐ[μ] (∅ : Set α)) : (s ∪ t : Set α) =ᵐ[μ] t 
goal: s ∪ t =ᵐ[μ] t

full_name: MeasureTheory.union_ae_eq_left_of_ae_eq_empty
code: theorem union_ae_eq_left_of_ae_eq_empty (h : t =ᵐ[μ] (∅ : Set α)) : (s ∪ t : Set α) =ᵐ[μ] s 
goal: s ∪ t =ᵐ[μ] s

full_name: MeasureTheory.inter_ae_eq_right_of_ae_eq_univ
code: theorem inter_ae_eq_right_of_ae_eq_univ (h : s =ᵐ[μ] univ) : (s ∩ t : Set α) =ᵐ[μ] t 
goal: s ∩ t =ᵐ[μ] t

full_name: MeasureTheory.inter_ae_eq_left_of_ae_eq_univ
code: theorem inter_ae_eq_left_of_ae_eq_univ (h : t =ᵐ[μ] univ) : (s ∩ t : Set α) =ᵐ[μ] s 
goal: s ∩ t =ᵐ[μ] s

full_name: MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_left
code: theorem inter_ae_eq_empty_of_ae_eq_empty_left (h : s =ᵐ[μ] (∅ : Set α)) :
    (s ∩ t : Set α) =ᵐ[μ] (∅ : Set α) 
goal: s ∩ t =ᵐ[μ] ∅

full_name: MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_right
code: theorem inter_ae_eq_empty_of_ae_eq_empty_right (h : t =ᵐ[μ] (∅ : Set α)) :
    (s ∩ t : Set α) =ᵐ[μ] (∅ : Set α) 
goal: s ∩ t =ᵐ[μ] ∅

full_name: Set.mulIndicator_ae_eq_one
code: @[to_additive]
theorem _root_.Set.mulIndicator_ae_eq_one {M : Type*} [One M] {f : α → M} {s : Set α} :
    s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ f.mulSupport) = 0 
goal: s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ Function.mulSupport f) = 0

full_name: Set.indicator_ae_eq_zero
code: @[to_additive]
theorem _root_.Set.mulIndicator_ae_eq_one {M : Type*} [One M] {f : α → M} {s : Set α} :
    s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ f.mulSupport) = 0 
goal: s.indicator f =ᵐ[μ] 0 ↔ μ (s ∩ Function.support f) = 0

full_name: MeasureTheory.measure_mono_ae
code: /-- If `s ⊆ t` modulo a set of measure `0`, then `μ s ≤ μ t`. -/
@[mono]
theorem measure_mono_ae (H : s ≤ᵐ[μ] t) : μ s ≤ μ t 
goal: μ s ≤ μ t

full_name: MeasureTheory.measure_congr
code: /-- If two sets are equal modulo a set of measure zero, then `μ s = μ t`. -/
theorem measure_congr (H : s =ᵐ[μ] t) : μ s = μ t 
goal: μ s = μ t

full_name: MeasureTheory.measure_mono_null_ae
code: theorem measure_mono_null_ae (H : s ≤ᵐ[μ] t) (ht : μ t = 0) : μ s = 0 
goal: μ s = 0