File: Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean

full_name: MeasureTheory.Measure.trim_le
code: /-- A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure. -/
structure Measure (α : Type*) [MeasurableSpace α] extends OuterMeasure α where
  m_iUnion ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    toOuterMeasure (⋃ i, f i) = ∑' i, toOuterMeasure (f i)
  trim_le : toOuterMeasure.trim ≤ toOuterMeasure
goal: self.trim ≤ self.toOuterMeasure

full_name: MeasureTheory.Measure
code: /-- A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure. -/
structure Measure (α : Type*) [MeasurableSpace α] extends OuterMeasure α where
  m_iUnion ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    toOuterMeasure (⋃ i, f i) = ∑' i, toOuterMeasure (f i)
  trim_le : toOuterMeasure.trim ≤ toOuterMeasure
goal: Type u_6

full_name: MeasureTheory.Measure.m_iUnion
code: /-- A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure. -/
structure Measure (α : Type*) [MeasurableSpace α] extends OuterMeasure α where
  m_iUnion ⦃f : ℕ → Set α⦄ : (∀ i, MeasurableSet (f i)) → Pairwise (Disjoint on f) →
    toOuterMeasure (⋃ i, f i) = ∑' i, toOuterMeasure (f i)
  trim_le : toOuterMeasure.trim ≤ toOuterMeasure
goal: (∀ (i : ℕ), MeasurableSet (f i)) →
  Pairwise (Disjoint on f) → self.toOuterMeasure (⋃ i, f i) = ∑' (i : ℕ), self.toOuterMeasure (f i)

full_name: MeasureTheory.Measure.toOuterMeasure_injective
code: theorem Measure.toOuterMeasure_injective [MeasurableSpace α] :
    Injective (toOuterMeasure : Measure α → OuterMeasure α)
  
goal: Function.Injective MeasureTheory.Measure.toOuterMeasure

full_name: MeasureTheory.Measure.instFunLike
code: instance Measure.instFunLike [MeasurableSpace α] : FunLike (Measure α) (Set α) ℝ≥0∞ 
goal: FunLike (MeasureTheory.Measure α) (Set α) ℝ≥0∞

full_name: MeasureTheory.Measure.instOuterMeasureClass
code: instance Measure.instOuterMeasureClass [MeasurableSpace α] : OuterMeasureClass (Measure α) α 
goal: MeasureTheory.OuterMeasureClass (MeasureTheory.Measure α) α

full_name: MeasureTheory.Measure.trimmed
code: theorem trimmed (μ : Measure α) : μ.toOuterMeasure.trim = μ.toOuterMeasure 
goal: μ.trim = μ.toOuterMeasure

full_name: MeasureTheory.Measure.ofMeasurable
code: /-- Obtain a measure by giving a countably additive function that sends `∅` to `0`. -/
def ofMeasurable (m : ∀ s : Set α, MeasurableSet s → ℝ≥0∞) (m0 : m ∅ MeasurableSet.empty = 0)
    (mU :
      ∀ ⦃f : ℕ → Set α⦄ (h : ∀ i, MeasurableSet (f i)),
        Pairwise (Disjoint on f) → m (⋃ i, f i) (MeasurableSet.iUnion h) = ∑' i, m (f i) (h i)) :
    Measure α 
goal: MeasureTheory.Measure α

full_name: MeasureTheory.Measure.ofMeasurable_apply
code: theorem ofMeasurable_apply {m : ∀ s : Set α, MeasurableSet s → ℝ≥0∞}
    {m0 : m ∅ MeasurableSet.empty = 0}
    {mU :
      ∀ ⦃f : ℕ → Set α⦄ (h : ∀ i, MeasurableSet (f i)),
        Pairwise (Disjoint on f) → m (⋃ i, f i) (MeasurableSet.iUnion h) = ∑' i, m (f i) (h i)}
    (s : Set α) (hs : MeasurableSet s) : ofMeasurable m m0 mU s = m s hs 
goal: (MeasureTheory.Measure.ofMeasurable m m0 mU) s = m s hs

full_name: MeasureTheory.Measure.ext
code: @[ext]
theorem ext (h : ∀ s, MeasurableSet s → μ₁ s = μ₂ s) : μ₁ = μ₂ 
goal: μ₁ = μ₂

full_name: MeasureTheory.Measure.ext_iff
code: theorem ext_iff : μ₁ = μ₂ ↔ ∀ s, MeasurableSet s → μ₁ s = μ₂ s 
goal: μ₁ = μ₂ ↔ ∀ (s : Set α), MeasurableSet s → μ₁ s = μ₂ s

full_name: MeasureTheory.Measure.ext_iff'
code: theorem ext_iff' : μ₁ = μ₂ ↔ ∀ s, μ₁ s = μ₂ s 
goal: μ₁ = μ₂ ↔ ∀ (s : Set α), μ₁ s = μ₂ s

full_name: MeasureTheory.Measure.outerMeasure_le_iff
code: theorem outerMeasure_le_iff {m : OuterMeasure α} : m ≤ μ.1 ↔ ∀ s, MeasurableSet s → m s ≤ μ s 
goal: m ≤ μ.toOuterMeasure ↔ ∀ (s : Set α), MeasurableSet s → m s ≤ μ s

full_name: MeasureTheory.Measure.coe_toOuterMeasure
code: @[simp] theorem Measure.coe_toOuterMeasure (μ : Measure α) : ⇑μ.toOuterMeasure = μ 
goal: ⇑μ.toOuterMeasure = ⇑μ

full_name: MeasureTheory.Measure.toOuterMeasure_apply
code: theorem Measure.toOuterMeasure_apply (μ : Measure α) (s : Set α) :
    μ.toOuterMeasure s = μ s 
goal: μ.toOuterMeasure s = μ s

full_name: MeasureTheory.measure_eq_trim
code: theorem measure_eq_trim (s : Set α) : μ s = μ.toOuterMeasure.trim s 
goal: μ s = μ.trim s

full_name: MeasureTheory.measure_eq_iInf
code: theorem measure_eq_iInf (s : Set α) : μ s = ⨅ (t) (_ : s ⊆ t) (_ : MeasurableSet t), μ t 
goal: μ s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), μ t

full_name: MeasureTheory.measure_eq_iInf'
code: /-- A variant of `measure_eq_iInf` which has a single `iInf`. This is useful when applying a
  lemma next that only works for non-empty infima, in which case you can use
  `nonempty_measurable_superset`. -/
theorem measure_eq_iInf' (μ : Measure α) (s : Set α) :
    μ s = ⨅ t : { t // s ⊆ t ∧ MeasurableSet t }, μ t 
goal: μ s = ⨅ t, μ ↑t

full_name: MeasureTheory.measure_eq_inducedOuterMeasure
code: theorem measure_eq_inducedOuterMeasure :
    μ s = inducedOuterMeasure (fun s _ => μ s) MeasurableSet.empty μ.empty s 
goal: μ s = (MeasureTheory.inducedOuterMeasure (fun s x => μ s) ⋯ ⋯) s

full_name: MeasureTheory.toOuterMeasure_eq_inducedOuterMeasure
code: theorem toOuterMeasure_eq_inducedOuterMeasure :
    μ.toOuterMeasure = inducedOuterMeasure (fun s _ => μ s) MeasurableSet.empty μ.empty 
goal: μ.toOuterMeasure = MeasureTheory.inducedOuterMeasure (fun s x => μ s) ⋯ ⋯

full_name: MeasureTheory.measure_eq_extend
code: theorem measure_eq_extend (hs : MeasurableSet s) :
    μ s = extend (fun t (_ht : MeasurableSet t) => μ t) s 
goal: μ s = MeasureTheory.extend (fun t _ht => μ t) s

full_name: MeasureTheory.nonempty_of_measure_ne_zero
code: theorem nonempty_of_measure_ne_zero (h : μ s ≠ 0) : s.Nonempty 
goal: s.Nonempty

full_name: MeasureTheory.measure_mono_top
code: theorem measure_mono_top (h : s₁ ⊆ s₂) (h₁ : μ s₁ = ∞) : μ s₂ = ∞ 
goal: μ s₂ = ⊤

full_name: MeasureTheory.measure_le_measure_union_left
code: @[simp, mono]
theorem measure_le_measure_union_left : μ s ≤ μ (s ∪ t) 
goal: μ s ≤ μ (s ∪ t)

full_name: MeasureTheory.measure_le_measure_union_right
code: @[simp, mono]
theorem measure_le_measure_union_right : μ t ≤ μ (s ∪ t) 
goal: μ t ≤ μ (s ∪ t)

full_name: MeasureTheory.exists_measurable_superset
code: /-- For every set there exists a measurable superset of the same measure. -/
theorem exists_measurable_superset (μ : Measure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s 
goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s

full_name: MeasureTheory.exists_measurable_superset_forall_eq
code: /-- For every set `s` and a countable collection of measures `μ i` there exists a measurable
superset `t ⊇ s` such that each measure `μ i` takes the same value on `s` and `t`. -/
theorem exists_measurable_superset_forall_eq [Countable ι] (μ : ι → Measure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = μ i s 
goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), (μ i) t = (μ i) s

full_name: MeasureTheory.exists_measurable_superset₂
code: theorem exists_measurable_superset₂ (μ ν : Measure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s 
goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s

full_name: MeasureTheory.exists_measurable_superset_of_null
code: theorem exists_measurable_superset_of_null (h : μ s = 0) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0 
goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0

full_name: MeasureTheory.exists_measurable_superset_iff_measure_eq_zero
code: theorem exists_measurable_superset_iff_measure_eq_zero :
    (∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0) ↔ μ s = 0 
goal: (∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = 0) ↔ μ s = 0

full_name: MeasureTheory.measure_biUnion_lt_top
code: theorem measure_biUnion_lt_top {s : Set β} {f : β → Set α} (hs : s.Finite)
    (hfin : ∀ i ∈ s, μ (f i) ≠ ∞) : μ (⋃ i ∈ s, f i) < ∞ 
goal: μ (⋃ i ∈ s, f i) < ⊤

full_name: MeasureTheory.measure_iUnion_null_iff'
code: @[deprecated measure_iUnion_null_iff (since := "2024-01-14")]
theorem measure_iUnion_null_iff' {ι : Prop} {s : ι → Set α} : μ (⋃ i, s i) = 0 ↔ ∀ i, μ (s i) = 0 
goal: μ (⋃ (i : ι), s i) = 0 ↔ ∀ (i : ι), μ (s i) = 0

full_name: MeasureTheory.measure_union_lt_top
code: theorem measure_union_lt_top (hs : μ s < ∞) (ht : μ t < ∞) : μ (s ∪ t) < ∞ 
goal: μ (s ∪ t) < ⊤

full_name: MeasureTheory.measure_union_lt_top_iff
code: @[simp]
theorem measure_union_lt_top_iff : μ (s ∪ t) < ∞ ↔ μ s < ∞ ∧ μ t < ∞ 
goal: μ (s ∪ t) < ⊤ ↔ μ s < ⊤ ∧ μ t < ⊤

full_name: MeasureTheory.measure_union_ne_top
code: theorem measure_union_ne_top (hs : μ s ≠ ∞) (ht : μ t ≠ ∞) : μ (s ∪ t) ≠ ∞ 
goal: μ (s ∪ t) ≠ ⊤

full_name: MeasureTheory.measure_symmDiff_ne_top
code: theorem measure_symmDiff_ne_top (hs : μ s ≠ ∞) (ht : μ t ≠ ∞) : μ (s ∆ t) ≠ ∞ 
goal: μ (symmDiff s t) ≠ ⊤

full_name: MeasureTheory.measure_union_eq_top_iff
code: @[simp]
theorem measure_union_eq_top_iff : μ (s ∪ t) = ∞ ↔ μ s = ∞ ∨ μ t = ∞ 
goal: μ (s ∪ t) = ⊤ ↔ μ s = ⊤ ∨ μ t = ⊤

full_name: MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null
code: theorem exists_measure_pos_of_not_measure_iUnion_null [Countable ι] {s : ι → Set α}
    (hs : μ (⋃ n, s n) ≠ 0) : ∃ n, 0 < μ (s n) 
goal: ∃ n, 0 < μ (s n)

full_name: MeasureTheory.measure_lt_top_of_subset
code: theorem measure_lt_top_of_subset (hst : t ⊆ s) (hs : μ s ≠ ∞) : μ t < ∞ 
goal: μ t < ⊤

full_name: MeasureTheory.measure_inter_lt_top_of_left_ne_top
code: theorem measure_inter_lt_top_of_left_ne_top (hs_finite : μ s ≠ ∞) : μ (s ∩ t) < ∞ 
goal: μ (s ∩ t) < ⊤

full_name: MeasureTheory.measure_inter_lt_top_of_right_ne_top
code: theorem measure_inter_lt_top_of_right_ne_top (ht_finite : μ t ≠ ∞) : μ (s ∩ t) < ∞ 
goal: μ (s ∩ t) < ⊤

full_name: MeasureTheory.measure_inter_null_of_null_right
code: theorem measure_inter_null_of_null_right (S : Set α) {T : Set α} (h : μ T = 0) : μ (S ∩ T) = 0 
goal: μ (S ∩ T) = 0

full_name: MeasureTheory.measure_inter_null_of_null_left
code: theorem measure_inter_null_of_null_left {S : Set α} (T : Set α) (h : μ S = 0) : μ (S ∩ T) = 0 
goal: μ (S ∩ T) = 0

full_name: MeasurableSpace.ae_induction_on_inter
code: /-- Given a predicate on `β` and `Set α` where both `α` and `β` are measurable spaces, if the
predicate holds for almost every `x : β` and
- `∅ : Set α`
- a family of sets generating the σ-algebra of `α`
Moreover, if for almost every `x : β`, the predicate is closed under complements and countable
disjoint unions, then the predicate holds for almost every `x : β` and all measurable sets of `α`.

This is an AE version of `MeasurableSpace.induction_on_inter` where the condition is dependent
on a measurable space `β`. -/
theorem _root_.MeasurableSpace.ae_induction_on_inter {β} [MeasurableSpace β] {μ : Measure β}
    {C : β → Set α → Prop} {s : Set (Set α)} [m : MeasurableSpace α]
    (h_eq : m = MeasurableSpace.generateFrom s)
    (h_inter : IsPiSystem s) (h_empty : ∀ᵐ x ∂μ, C x ∅) (h_basic : ∀ᵐ x ∂μ, ∀ t ∈ s, C x t)
    (h_compl : ∀ᵐ x ∂μ, ∀ t, MeasurableSet t → C x t → C x tᶜ)
    (h_union : ∀ᵐ x ∂μ, ∀ f : ℕ → Set α,
        Pairwise (Disjoint on f) → (∀ i, MeasurableSet (f i)) → (∀ i, C x (f i)) → C x (⋃ i, f i)) :
    ∀ᵐ x ∂μ, ∀ ⦃t⦄, MeasurableSet t → C x t 
goal: ∀ᵐ (x : β) ∂μ, ∀ ⦃t : Set α⦄, MeasurableSet t → C x t

full_name: MeasureTheory.toMeasurable_def
code: /-- A measurable set `t ⊇ s` such that `μ t = μ s`. It even satisfies `μ (t ∩ u) = μ (s ∩ u)` for
any measurable set `u` if `μ s ≠ ∞`, see `measure_toMeasurable_inter`.
(This property holds without the assumption `μ s ≠ ∞` when the space is s-finite -- for example
σ-finite), see `measure_toMeasurable_inter_of_sFinite`).
If `s` is a null measurable set, then
we also have `t =ᵐ[μ] s`, see `NullMeasurableSet.toMeasurable_ae_eq`.
This notion is sometimes called a "measurable hull" in the literature. -/
irreducible_def toMeasurable (μ : Measure α) (s : Set α) : Set α 
goal: MeasureTheory.toMeasurable μ s =
  if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose
  else
    if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose
    else ⋯.choose

full_name: MeasureTheory.definition._@.Mathlib.MeasureTheory.Measure.MeasureSpaceDef._hyg.3711
code: /-- A measurable set `t ⊇ s` such that `μ t = μ s`. It even satisfies `μ (t ∩ u) = μ (s ∩ u)` for
any measurable set `u` if `μ s ≠ ∞`, see `measure_toMeasurable_inter`.
(This property holds without the assumption `μ s ≠ ∞` when the space is s-finite -- for example
σ-finite), see `measure_toMeasurable_inter_of_sFinite`).
If `s` is a null measurable set, then
we also have `t =ᵐ[μ] s`, see `NullMeasurableSet.toMeasurable_ae_eq`.
This notion is sometimes called a "measurable hull" in the literature. -/
irreducible_def toMeasurable (μ : Measure α) (s : Set α) : Set α 
goal: Set α

full_name: MeasureTheory.toMeasurable
code: /-- A measurable set `t ⊇ s` such that `μ t = μ s`. It even satisfies `μ (t ∩ u) = μ (s ∩ u)` for
any measurable set `u` if `μ s ≠ ∞`, see `measure_toMeasurable_inter`.
(This property holds without the assumption `μ s ≠ ∞` when the space is s-finite -- for example
σ-finite), see `measure_toMeasurable_inter_of_sFinite`).
If `s` is a null measurable set, then
we also have `t =ᵐ[μ] s`, see `NullMeasurableSet.toMeasurable_ae_eq`.
This notion is sometimes called a "measurable hull" in the literature. -/
irreducible_def toMeasurable (μ : Measure α) (s : Set α) : Set α 
goal: Set α

full_name: MeasureTheory.wrapped._@.Mathlib.MeasureTheory.Measure.MeasureSpaceDef._hyg.3711
code: /-- A measurable set `t ⊇ s` such that `μ t = μ s`. It even satisfies `μ (t ∩ u) = μ (s ∩ u)` for
any measurable set `u` if `μ s ≠ ∞`, see `measure_toMeasurable_inter`.
(This property holds without the assumption `μ s ≠ ∞` when the space is s-finite -- for example
σ-finite), see `measure_toMeasurable_inter_of_sFinite`).
If `s` is a null measurable set, then
we also have `t =ᵐ[μ] s`, see `NullMeasurableSet.toMeasurable_ae_eq`.
This notion is sometimes called a "measurable hull" in the literature. -/
irreducible_def toMeasurable (μ : Measure α) (s : Set α) : Set α 
goal: Subtype (Eq @MeasureTheory.definition✝)

full_name: MeasureTheory.subset_toMeasurable
code: theorem subset_toMeasurable (μ : Measure α) (s : Set α) : s ⊆ toMeasurable μ s 
goal: s ⊆ MeasureTheory.toMeasurable μ s

full_name: MeasureTheory.ae_le_toMeasurable
code: theorem ae_le_toMeasurable : s ≤ᵐ[μ] toMeasurable μ s 
goal: s ≤ᵐ[μ] MeasureTheory.toMeasurable μ s

full_name: MeasureTheory.measurableSet_toMeasurable
code: @[simp]
theorem measurableSet_toMeasurable (μ : Measure α) (s : Set α) :
    MeasurableSet (toMeasurable μ s) 
goal: MeasurableSet (MeasureTheory.toMeasurable μ s)

full_name: MeasureTheory.measure_toMeasurable
code: @[simp]
theorem measure_toMeasurable (s : Set α) : μ (toMeasurable μ s) = μ s 
goal: μ (MeasureTheory.toMeasurable μ s) = μ s

full_name: MeasureTheory.MeasureSpace
code: /-- A measure space is a measurable space equipped with a
  measure, referred to as `volume`. -/
class MeasureSpace (α : Type*) extends MeasurableSpace α where
  volume : Measure α
goal: Type u_6

full_name: MeasureTheory.MeasureSpace.volume
code: /-- A measure space is a measurable space equipped with a
  measure, referred to as `volume`. -/
class MeasureSpace (α : Type*) extends MeasurableSpace α where
  volume : Measure α
goal: MeasureTheory.Measure α

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Measure_MeasureSpaceDef___macroRules_MeasureTheory_term∀ᵐ_,__1»
code: /-- `∀ᵐ a, 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 MeasureSpace.volume)`. -/
notation3 "∀ᵐ "(...)", "r:(scoped P =>
  Filter.Eventually P <| MeasureTheory.ae MeasureTheory.MeasureSpace.volume) => r
goal: Lean.Macro

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Measure_MeasureSpaceDef___macroRules_MeasureTheory_term∃ᵐ_,__1»
code: /-- `∃ᵐ a, p a` means that `p` holds frequently, i.e. on a set of positive measure,
w.r.t. the volume measure.

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

full_name: MeasureTheory.tacticVolume_tac
code: macro "volume_tac" : tactic =>
  `(tactic| (first | exact MeasureTheory.MeasureSpace.volume))
goal: Lean.ParserDescr

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_Measure_MeasureSpaceDef___macroRules_MeasureTheory_tacticVolume_tac_1
code: macro "volume_tac" : tactic =>
  `(tactic| (first | exact MeasureTheory.MeasureSpace.volume))
goal: Lean.Macro

full_name: AEMeasurable
code: /-- A function is almost everywhere measurable if it coincides almost everywhere with a measurable
function. -/
@[fun_prop]
def AEMeasurable {_m : MeasurableSpace α} (f : α → β) (μ : Measure α := by volume_tac) : Prop 
goal: Prop

full_name: Measurable.aemeasurable
code: @[fun_prop, aesop unsafe 30% apply (rule_sets := [Measurable])]
theorem Measurable.aemeasurable (h : Measurable f) : AEMeasurable f μ 
goal: AEMeasurable f μ

full_name: AEMeasurable.mk
code: /-- Given an almost everywhere measurable function `f`, associate to it a measurable function
that coincides with it almost everywhere. `f` is explicit in the definition to make sure that
it shows in pretty-printing. -/
def mk (f : α → β) (h : AEMeasurable f μ) : α → β 
goal: α → β

full_name: AEMeasurable.measurable_mk
code: @[measurability]
theorem measurable_mk (h : AEMeasurable f μ) : Measurable (h.mk f) 
goal: Measurable (AEMeasurable.mk f h)

full_name: AEMeasurable.ae_eq_mk
code: theorem ae_eq_mk (h : AEMeasurable f μ) : f =ᵐ[μ] h.mk f 
goal: f =ᵐ[μ] AEMeasurable.mk f h

full_name: AEMeasurable.congr
code: theorem congr (hf : AEMeasurable f μ) (h : f =ᵐ[μ] g) : AEMeasurable g μ 
goal: AEMeasurable g μ

full_name: aemeasurable_congr
code: theorem aemeasurable_congr (h : f =ᵐ[μ] g) : AEMeasurable f μ ↔ AEMeasurable g μ 
goal: AEMeasurable f μ ↔ AEMeasurable g μ

full_name: aemeasurable_const
code: @[simp, fun_prop, measurability]
theorem aemeasurable_const {b : β} : AEMeasurable (fun _a : α => b) μ 
goal: AEMeasurable (fun _a => b) μ

full_name: aemeasurable_id
code: @[measurability]
theorem aemeasurable_id : AEMeasurable id μ 
goal: AEMeasurable id μ

full_name: aemeasurable_id'
code: @[measurability]
theorem aemeasurable_id' : AEMeasurable (fun x => x) μ 
goal: AEMeasurable (fun x => x) μ

full_name: Measurable.comp_aemeasurable
code: theorem Measurable.comp_aemeasurable [MeasurableSpace δ] {f : α → δ} {g : δ → β} (hg : Measurable g)
    (hf : AEMeasurable f μ) : AEMeasurable (g ∘ f) μ 
goal: AEMeasurable (g ∘ f) μ

full_name: Measurable.comp_aemeasurable'
code: @[fun_prop, measurability]
theorem Measurable.comp_aemeasurable' [MeasurableSpace δ] {f : α → δ} {g : δ → β}
    (hg : Measurable g) (hf : AEMeasurable f μ) : AEMeasurable (fun x ↦ g (f x)) μ 
goal: AEMeasurable (fun x => g (f x)) μ