ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.AEStronglyMeasurable.edist
@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem edist {β : Type*} [SeminormedAddCommGroup β] {f g : α → β}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEMeasurable (fun a => edist (f a) (g a)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [SeminormedAddCommGroup β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: AEMeasurable (fun a => edist (f a) (g a)) μ
-- MeasureTheory.AEStronglyMeasurable.real_toNNReal
@[measurability]
protected theorem real_toNNReal {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (f x).toNNReal) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (f x).toNNReal) μ
-- aestronglyMeasurable_indicator_iff
theorem _root_.aestronglyMeasurable_indicator_iff [Zero β] {s : Set α} (hs : MeasurableSet s) :
    AEStronglyMeasurable (indicator s f) μ ↔ AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.AEStronglyMeasurable (s.indicator f) μ ↔ MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- MeasureTheory.AEStronglyMeasurable.indicator
@[measurability]
protected theorem indicator [Zero β] (hfm : AEStronglyMeasurable f μ) {s : Set α}
    (hs : MeasurableSet s) : AEStronglyMeasurable (s.indicator f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] (hfm : MeasureTheory.AEStronglyMeasurable f μ) {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.AEStronglyMeasurable (s.indicator f) μ
-- MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun
theorem nullMeasurableSet_eq_fun {E} [TopologicalSpace E] [MetrizableSpace E] {f g : α → E}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    NullMeasurableSet { x | f x = g x } μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [TopologicalSpace E] [TopologicalSpace.MetrizableSpace E] {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.NullMeasurableSet {x | f x = g x} μ
-- MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_support
@[to_additive]
lemma nullMeasurableSet_mulSupport {E} [TopologicalSpace E] [MetrizableSpace E] [One E] {f : α → E}
    (hf : AEStronglyMeasurable f μ) : NullMeasurableSet (mulSupport f) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [TopologicalSpace E] [TopologicalSpace.MetrizableSpace E] [Zero E] {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.NullMeasurableSet (Function.support f) μ
-- MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_mulSupport
@[to_additive]
lemma nullMeasurableSet_mulSupport {E} [TopologicalSpace E] [MetrizableSpace E] [One E] {f : α → E}
    (hf : AEStronglyMeasurable f μ) : NullMeasurableSet (mulSupport f) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [TopologicalSpace E] [TopologicalSpace.MetrizableSpace E] [One E] {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.NullMeasurableSet (Function.mulSupport f) μ
-- MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt
theorem nullMeasurableSet_lt [LinearOrder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
    {f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    NullMeasurableSet { a | f a < g a } μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [LinearOrder β] [OrderClosedTopology β] [TopologicalSpace.PseudoMetrizableSpace β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.NullMeasurableSet {a | f a < g a} μ
-- MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le
theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
    {f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    NullMeasurableSet { a | f a ≤ g a } μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [Preorder β] [OrderClosedTopology β] [TopologicalSpace.PseudoMetrizableSpace β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.NullMeasurableSet {a | f a ≤ g a} μ
-- aestronglyMeasurable_of_aestronglyMeasurable_trim
theorem _root_.aestronglyMeasurable_of_aestronglyMeasurable_trim {α} {m m0 : MeasurableSpace α}
    {μ : Measure α} (hm : m ≤ m0) {f : α → β} (hf : AEStronglyMeasurable f (μ.trim hm)) :
    AEStronglyMeasurable f μ 
Arguments: {β : Type u_2} [TopologicalSpace β] {α : Type u_5} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f (μ.trim hm))
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.AEStronglyMeasurable.comp_aemeasurable
theorem comp_aemeasurable {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
    {μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
    AEStronglyMeasurable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {g : α → β} {γ : Type u_5}
Goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ},
  MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) →
    AEMeasurable f μ → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ
-- MeasureTheory.AEStronglyMeasurable.comp_measurable
theorem comp_measurable {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α}
    {μ : Measure γ} (hg : AEStronglyMeasurable g (Measure.map f μ)) (hf : Measurable f) :
    AEStronglyMeasurable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {g : α → β} {γ : Type u_5}
Goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ},
  MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) →
    Measurable f → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ
-- MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving
theorem comp_quasiMeasurePreserving {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α}
    {f : γ → α} {μ : Measure γ} {ν : Measure α} (hg : AEStronglyMeasurable g ν)
    (hf : QuasiMeasurePreserving f μ ν) : AEStronglyMeasurable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {g : α → β} {γ : Type u_5}
Goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ}
  {ν : MeasureTheory.Measure α},
  MeasureTheory.AEStronglyMeasurable g ν →
    MeasureTheory.Measure.QuasiMeasurePreserving f μ ν → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ
-- MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range
theorem isSeparable_ae_range (hf : AEStronglyMeasurable f μ) :
    ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t
-- aestronglyMeasurable_iff_aemeasurable_separable
theorem _root_.aestronglyMeasurable_iff_aemeasurable_separable [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] :
    AEStronglyMeasurable f μ ↔
      AEMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
Goal: MeasureTheory.AEStronglyMeasurable f μ ↔ AEMeasurable f μ ∧ ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t
-- aestronglyMeasurable_iff_nullMeasurable_separable
theorem _root_.aestronglyMeasurable_iff_nullMeasurable_separable [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] :
    AEStronglyMeasurable f μ ↔
      NullMeasurable f μ ∧ ∃ t : Set β, IsSeparable t ∧ ∀ᵐ x ∂μ, f x ∈ t 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
Goal: MeasureTheory.AEStronglyMeasurable f μ ↔
  MeasureTheory.NullMeasurable f μ ∧ ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t
-- MeasurableEmbedding.aestronglyMeasurable_map_iff
theorem _root_.MeasurableEmbedding.aestronglyMeasurable_map_iff {γ : Type*}
    {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α} {μ : Measure γ}
    (hf : MeasurableEmbedding f) {g : α → β} :
    AEStronglyMeasurable g (Measure.map f μ) ↔ AEStronglyMeasurable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {γ : Type u_5} {mγ : MeasurableSpace γ} {mα : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ} (hf : MeasurableEmbedding f) {g : α → β}
Goal: MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.AEStronglyMeasurable (g ∘ f) μ
-- Embedding.aestronglyMeasurable_comp_iff
theorem _root_.Embedding.aestronglyMeasurable_comp_iff [PseudoMetrizableSpace β]
    [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β} (hg : Embedding g) :
    AEStronglyMeasurable (fun x => g (f x)) μ ↔ AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace β] [TopologicalSpace.PseudoMetrizableSpace γ] {g : β → γ} {f : α → β} (hg : Embedding g)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => g (f x)) μ ↔ MeasureTheory.AEStronglyMeasurable f μ
-- aestronglyMeasurable_of_tendsto_ae
theorem _root_.aestronglyMeasurable_of_tendsto_ae {ι : Type*} [PseudoMetrizableSpace β]
    (u : Filter ι) [NeBot u] [IsCountablyGenerated u] {f : ι → α → β} {g : α → β}
    (hf : ∀ i, AEStronglyMeasurable (f i) μ) (lim : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) :
    AEStronglyMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {ι : Type u_5} [TopologicalSpace.PseudoMetrizableSpace β] (u : Filter ι) [u.NeBot] [u.IsCountablyGenerated] {f : ι → α → β} {g : α → β} (hf : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) (lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) u (𝓝 (g x)))
Goal: MeasureTheory.AEStronglyMeasurable g μ
-- exists_stronglyMeasurable_limit_of_tendsto_ae
theorem _root_.exists_stronglyMeasurable_limit_of_tendsto_ae [PseudoMetrizableSpace β]
    {f : ℕ → α → β} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) atTop (𝓝 l)) :
    ∃ f_lim : α → β, StronglyMeasurable f_lim ∧
      ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x)) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] {f : ℕ → α → β} (hf : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) (h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 l))
Goal: ∃ f_lim,
  MeasureTheory.StronglyMeasurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x))
-- MeasureTheory.AEStronglyMeasurable.piecewise
theorem piecewise {s : Set α} [DecidablePred (· ∈ s)]
    (hs : MeasurableSet s) (hf : AEStronglyMeasurable f (μ.restrict s))
    (hg : AEStronglyMeasurable g (μ.restrict sᶜ)) :
    AEStronglyMeasurable (s.piecewise f g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} {s : Set α} [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s) (hf : MeasureTheory.AEStronglyMeasurable f (μ.restrict s)) (hg : MeasureTheory.AEStronglyMeasurable g (μ.restrict sᶜ))
Goal: MeasureTheory.AEStronglyMeasurable (s.piecewise f g) μ
-- MeasureTheory.AEStronglyMeasurable.sum_measure
theorem sum_measure [PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → Measure α}
    (h : ∀ i, AEStronglyMeasurable f (μ i)) : AEStronglyMeasurable f (Measure.sum μ) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [Countable ι] [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] {m : MeasurableSpace α} {μ : ι → MeasureTheory.Measure α} (h : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ i))
Goal: MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.sum μ)
-- aestronglyMeasurable_sum_measure_iff
@[simp]
theorem _root_.aestronglyMeasurable_sum_measure_iff [PseudoMetrizableSpace β]
    {_m : MeasurableSpace α} {μ : ι → Measure α} :
    AEStronglyMeasurable f (sum μ) ↔ ∀ i, AEStronglyMeasurable f (μ i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [Countable ι] [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] {_m : MeasurableSpace α} {μ : ι → MeasureTheory.Measure α}
Goal: MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.sum μ) ↔
  ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ i)
-- aestronglyMeasurable_add_measure_iff
@[simp]
theorem _root_.aestronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Measure α} :
    AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] {ν : MeasureTheory.Measure α}
Goal: MeasureTheory.AEStronglyMeasurable f (μ + ν) ↔
  MeasureTheory.AEStronglyMeasurable f μ ∧ MeasureTheory.AEStronglyMeasurable f ν
-- MeasureTheory.AEStronglyMeasurable.add_measure
@[measurability]
theorem add_measure [PseudoMetrizableSpace β] {ν : Measure α} {f : α → β}
    (hμ : AEStronglyMeasurable f μ) (hν : AEStronglyMeasurable f ν) :
    AEStronglyMeasurable f (μ + ν) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] {ν : MeasureTheory.Measure α} {f : α → β} (hμ : MeasureTheory.AEStronglyMeasurable f μ) (hν : MeasureTheory.AEStronglyMeasurable f ν)
Goal: MeasureTheory.AEStronglyMeasurable f (μ + ν)
-- MeasureTheory.AEStronglyMeasurable.iUnion
@[measurability]
protected theorem iUnion [PseudoMetrizableSpace β] {s : ι → Set α}
    (h : ∀ i, AEStronglyMeasurable f (μ.restrict (s i))) :
    AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [Countable ι] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] {s : ι → Set α} (h : ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ.restrict (s i)))
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (⋃ i, s i))
-- aestronglyMeasurable_iUnion_iff
@[simp]
theorem _root_.aestronglyMeasurable_iUnion_iff [PseudoMetrizableSpace β] {s : ι → Set α} :
    AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
      ∀ i, AEStronglyMeasurable f (μ.restrict (s i)) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_4} [Countable ι] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] {s : ι → Set α}
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (⋃ i, s i)) ↔
  ∀ (i : ι), MeasureTheory.AEStronglyMeasurable f (μ.restrict (s i))
-- aestronglyMeasurable_union_iff
@[simp]
theorem _root_.aestronglyMeasurable_union_iff [PseudoMetrizableSpace β] {s t : Set α} :
    AEStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
      AEStronglyMeasurable f (μ.restrict s) ∧ AEStronglyMeasurable f (μ.restrict t) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] {s t : Set α}
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (s ∪ t)) ↔
  MeasureTheory.AEStronglyMeasurable f (μ.restrict s) ∧ MeasureTheory.AEStronglyMeasurable f (μ.restrict t)
-- MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff
theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
    {a b : α} :
    AEStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
      AEStronglyMeasurable f (μ.restrict <| Ioc a b) ∧
        AEStronglyMeasurable f (μ.restrict <| Ioc b a) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [LinearOrder α] [TopologicalSpace.PseudoMetrizableSpace β] {f : α → β} {a b : α}
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (Ι a b)) ↔
  MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc a b)) ∧
    MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc b a))
-- MeasureTheory.AEStronglyMeasurable.smul_measure
@[measurability]
theorem smul_measure {R : Type*} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    (h : AEStronglyMeasurable f μ) (c : R) : AEStronglyMeasurable f (c • μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {R : Type u_5} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (h : MeasureTheory.AEStronglyMeasurable f μ) (c : R)
Goal: MeasureTheory.AEStronglyMeasurable f (c • μ)
-- aestronglyMeasurable_const_smul_iff
theorem _root_.aestronglyMeasurable_const_smul_iff (c : G) :
    AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {G : Type u_6} [Group G] [MulAction G β] [ContinuousConstSMul G β] (c : G)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ
-- IsUnit.aestronglyMeasurable_const_smul_iff
nonrec theorem _root_.IsUnit.aestronglyMeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
    AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {M : Type u_5} [Monoid M] [MulAction M β] [ContinuousConstSMul M β] {c : M} (hc : IsUnit c)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ
-- aestronglyMeasurable_const_smul_iff₀
theorem _root_.aestronglyMeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
    AEStronglyMeasurable (fun x => c • f x) μ ↔ AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {G₀ : Type u_7} [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β] {c : G₀} (hc : c ≠ 0)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk
theorem finStronglyMeasurable_mk (hf : AEFinStronglyMeasurable f μ) :
    FinStronglyMeasurable (hf.mk f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: MeasureTheory.FinStronglyMeasurable (MeasureTheory.AEFinStronglyMeasurable.mk f hf) μ
-- MeasureTheory.AEFinStronglyMeasurable.ae_eq_mk
theorem ae_eq_mk (hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: f =ᵐ[μ] MeasureTheory.AEFinStronglyMeasurable.mk f hf
-- MeasureTheory.AEFinStronglyMeasurable.aemeasurable
@[aesop 10% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {β} [Zero β] [MeasurableSpace β] [TopologicalSpace β]
    [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEFinStronglyMeasurable f μ) :
    AEMeasurable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [Zero β] [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: AEMeasurable f μ
-- MeasureTheory.AEFinStronglyMeasurable.mul
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f * g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [MonoidWithZero β] [ContinuousMul β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ)
Goal: MeasureTheory.AEFinStronglyMeasurable (f * g) μ
-- MeasureTheory.AEFinStronglyMeasurable.add
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f + g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [AddMonoid β] [ContinuousAdd β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ)
Goal: MeasureTheory.AEFinStronglyMeasurable (f + g) μ
-- MeasureTheory.AEFinStronglyMeasurable.neg
@[measurability]
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : AEFinStronglyMeasurable f μ) :
    AEFinStronglyMeasurable (-f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [AddGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: MeasureTheory.AEFinStronglyMeasurable (-f) μ
-- MeasureTheory.AEFinStronglyMeasurable.sub
@[measurability]
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f - g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [AddGroup β] [ContinuousSub β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ)
Goal: MeasureTheory.AEFinStronglyMeasurable (f - g) μ
-- MeasureTheory.AEFinStronglyMeasurable.const_smul
@[measurability]
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
    [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : AEFinStronglyMeasurable f μ) (c : 𝕜) :
    AEFinStronglyMeasurable (c • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {𝕜 : Type u_5} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜] [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (c : 𝕜)
Goal: MeasureTheory.AEFinStronglyMeasurable (c • f) μ
-- MeasureTheory.AEFinStronglyMeasurable.sup
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊔ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [Zero β] [SemilatticeSup β] [ContinuousSup β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ)
Goal: MeasureTheory.AEFinStronglyMeasurable (f ⊔ g) μ