ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- aemeasurable_zero_measure
@[simp, measurability]
theorem aemeasurable_zero_measure : AEMeasurable f (0 : Measure α) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β}
Goal: AEMeasurable f 0
-- aemeasurable_id''
@[fun_prop]
theorem aemeasurable_id'' (μ : Measure α) {m : MeasurableSpace α} (hm : m ≤ m0) :
    @AEMeasurable α α m m0 id μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) {m : MeasurableSpace α} (hm : m ≤ m0)
Goal: AEMeasurable id μ
-- aemeasurable_of_map_neZero
lemma aemeasurable_of_map_neZero {mβ : MeasurableSpace β} {μ : Measure α}
    {f : α → β} (h : NeZero (μ.map f)) :
    AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {f : α → β} (h : NeZero (MeasureTheory.Measure.map f μ))
Goal: AEMeasurable f μ
-- AEMeasurable.mono_ac
lemma mono_ac (hf : AEMeasurable f ν) (hμν : μ ≪ ν) : AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ ν : MeasureTheory.Measure α} (hf : AEMeasurable f ν) (hμν : μ ≪ ν)
Goal: AEMeasurable f μ
-- AEMeasurable.mono_measure
theorem mono_measure (h : AEMeasurable f μ) (h' : ν ≤ μ) : AEMeasurable f ν 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ ν : MeasureTheory.Measure α} (h : AEMeasurable f μ) (h' : ν ≤ μ)
Goal: AEMeasurable f ν
-- AEMeasurable.mono_set
theorem mono_set {s t} (h : s ⊆ t) (ht : AEMeasurable f (μ.restrict t)) :
    AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s t : Set α} (h : s ⊆ t) (ht : AEMeasurable f (μ.restrict t))
Goal: AEMeasurable f (μ.restrict s)
-- AEMeasurable.mono'
@[fun_prop]
protected theorem mono' (h : AEMeasurable f μ) (h' : ν ≪ μ) : AEMeasurable f ν 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ ν : MeasureTheory.Measure α} (h : AEMeasurable f μ) (h' : ν ≪ μ)
Goal: AEMeasurable f ν
-- AEMeasurable.ae_mem_imp_eq_mk
theorem ae_mem_imp_eq_mk {s} (h : AEMeasurable f (μ.restrict s)) :
    ∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s : Set α} (h : AEMeasurable f (μ.restrict s))
Goal: ∀ᵐ (x : α) ∂μ, x ∈ s → f x = AEMeasurable.mk f h x
-- AEMeasurable.ae_inf_principal_eq_mk
theorem ae_inf_principal_eq_mk {s} (h : AEMeasurable f (μ.restrict s)) : f =ᶠ[ae μ ⊓ 𝓟 s] h.mk f 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s : Set α} (h : AEMeasurable f (μ.restrict s))
Goal: f =ᶠ[MeasureTheory.ae μ ⊓ 𝓟 s] AEMeasurable.mk f h
-- AEMeasurable.sum_measure
@[measurability]
theorem sum_measure [Countable ι] {μ : ι → Measure α} (h : ∀ i, AEMeasurable f (μ i)) :
    AEMeasurable f (sum μ) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} [Countable ι] {μ : ι → MeasureTheory.Measure α} (h : ∀ (i : ι), AEMeasurable f (μ i))
Goal: AEMeasurable f (MeasureTheory.Measure.sum μ)
-- aemeasurable_sum_measure_iff
@[simp]
theorem _root_.aemeasurable_sum_measure_iff [Countable ι] {μ : ι → Measure α} :
    AEMeasurable f (sum μ) ↔ ∀ i, AEMeasurable f (μ i) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} [Countable ι] {μ : ι → MeasureTheory.Measure α}
Goal: AEMeasurable f (MeasureTheory.Measure.sum μ) ↔ ∀ (i : ι), AEMeasurable f (μ i)
-- aemeasurable_add_measure_iff
@[simp]
theorem _root_.aemeasurable_add_measure_iff :
    AEMeasurable f (μ + ν) ↔ AEMeasurable f μ ∧ AEMeasurable f ν 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ ν : MeasureTheory.Measure α}
Goal: AEMeasurable f (μ + ν) ↔ AEMeasurable f μ ∧ AEMeasurable f ν
-- AEMeasurable.add_measure
@[measurability]
theorem add_measure {f : α → β} (hμ : AEMeasurable f μ) (hν : AEMeasurable f ν) :
    AEMeasurable f (μ + ν) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ ν : MeasureTheory.Measure α} {f : α → β} (hμ : AEMeasurable f μ) (hν : AEMeasurable f ν)
Goal: AEMeasurable f (μ + ν)
-- AEMeasurable.iUnion
@[measurability]
protected theorem iUnion [Countable ι] {s : ι → Set α}
    (h : ∀ i, AEMeasurable f (μ.restrict (s i))) : AEMeasurable f (μ.restrict (⋃ i, s i)) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α} (h : ∀ (i : ι), AEMeasurable f (μ.restrict (s i)))
Goal: AEMeasurable f (μ.restrict (⋃ i, s i))
-- aemeasurable_iUnion_iff
@[simp]
theorem _root_.aemeasurable_iUnion_iff [Countable ι] {s : ι → Set α} :
    AEMeasurable f (μ.restrict (⋃ i, s i)) ↔ ∀ i, AEMeasurable f (μ.restrict (s i)) 
Arguments: {ι : Type u_1} {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α}
Goal: AEMeasurable f (μ.restrict (⋃ i, s i)) ↔ ∀ (i : ι), AEMeasurable f (μ.restrict (s i))
-- aemeasurable_union_iff
@[simp]
theorem _root_.aemeasurable_union_iff {s t : Set α} :
    AEMeasurable f (μ.restrict (s ∪ t)) ↔
      AEMeasurable f (μ.restrict s) ∧ AEMeasurable f (μ.restrict t) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s t : Set α}
Goal: AEMeasurable f (μ.restrict (s ∪ t)) ↔ AEMeasurable f (μ.restrict s) ∧ AEMeasurable f (μ.restrict t)
-- AEMeasurable.smul_measure
@[measurability]
theorem smul_measure [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    (h : AEMeasurable f μ) (c : R) : AEMeasurable f (c • μ) 
Arguments: {α : Type u_2} {β : Type u_3} {R : Type u_6} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (h : AEMeasurable f μ) (c : R)
Goal: AEMeasurable f (c • μ)
-- AEMeasurable.comp_aemeasurable
theorem comp_aemeasurable {f : α → δ} {g : δ → β} (hg : AEMeasurable g (μ.map f))
    (hf : AEMeasurable f μ) : AEMeasurable (g ∘ f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} {g : δ → β} (hg : AEMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: AEMeasurable (g ∘ f) μ
-- AEMeasurable.comp_aemeasurable'
@[fun_prop]
theorem comp_aemeasurable' {f : α → δ} {g : δ → β} (hg : AEMeasurable g (μ.map f))
    (hf : AEMeasurable f μ) : AEMeasurable (fun x ↦ g (f x)) μ 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} {g : δ → β} (hg : AEMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => g (f x)) μ
-- AEMeasurable.comp_measurable
theorem comp_measurable {f : α → δ} {g : δ → β} (hg : AEMeasurable g (μ.map f))
    (hf : Measurable f) : AEMeasurable (g ∘ f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} {g : δ → β} (hg : AEMeasurable g (MeasureTheory.Measure.map f μ)) (hf : Measurable f)
Goal: AEMeasurable (g ∘ f) μ
-- AEMeasurable.comp_quasiMeasurePreserving
theorem comp_quasiMeasurePreserving {ν : Measure δ} {f : α → δ} {g : δ → β} (hg : AEMeasurable g ν)
    (hf : QuasiMeasurePreserving f μ ν) : AEMeasurable (g ∘ f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure δ} {f : α → δ} {g : δ → β} (hg : AEMeasurable g ν) (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν)
Goal: AEMeasurable (g ∘ f) μ
-- AEMeasurable.map_map_of_aemeasurable
theorem map_map_of_aemeasurable {g : β → γ} {f : α → β} (hg : AEMeasurable g (Measure.map f μ))
    (hf : AEMeasurable f μ) : (μ.map f).map g = μ.map (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {g : β → γ} {f : α → β} (hg : AEMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.Measure.map g (MeasureTheory.Measure.map f μ) = MeasureTheory.Measure.map (g ∘ f) μ
-- AEMeasurable.prod_mk
@[fun_prop, measurability]
theorem prod_mk {f : α → β} {g : α → γ} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun x => (f x, g x)) μ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → β} {g : α → γ} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun x => (f x, g x)) μ
-- AEMeasurable.exists_ae_eq_range_subset
theorem exists_ae_eq_range_subset (H : AEMeasurable f μ) {t : Set β} (ht : ∀ᵐ x ∂μ, f x ∈ t)
    (h₀ : t.Nonempty) : ∃ g, Measurable g ∧ range g ⊆ t ∧ f =ᵐ[μ] g 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (H : AEMeasurable f μ) {t : Set β} (ht : ∀ᵐ (x : α) ∂μ, f x ∈ t) (h₀ : t.Nonempty)
Goal: ∃ g, Measurable g ∧ Set.range g ⊆ t ∧ f =ᵐ[μ] g
-- AEMeasurable.exists_measurable_nonneg
theorem exists_measurable_nonneg {β} [Preorder β] [Zero β] {mβ : MeasurableSpace β} {f : α → β}
    (hf : AEMeasurable f μ) (f_nn : ∀ᵐ t ∂μ, 0 ≤ f t) : ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᵐ[μ] g 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [Preorder β] [Zero β] {mβ : MeasurableSpace β} {f : α → β} (hf : AEMeasurable f μ) (f_nn : ∀ᵐ (t : α) ∂μ, 0 ≤ f t)
Goal: ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᵐ[μ] g
-- AEMeasurable.subtype_mk
theorem subtype_mk (h : AEMeasurable f μ) {s : Set β} {hfs : ∀ x, f x ∈ s} :
    AEMeasurable (codRestrict f s hfs) μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : AEMeasurable f μ) {s : Set β} {hfs : ∀ (x : α), f x ∈ s}
Goal: AEMeasurable (Set.codRestrict f s hfs) μ
-- aemeasurable_const'
theorem aemeasurable_const' (h : ∀ᵐ (x) (y) ∂μ, f x = f y) : AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y)
Goal: AEMeasurable f μ
-- aemeasurable_uIoc_iff
theorem aemeasurable_uIoc_iff [LinearOrder α] {f : α → β} {a b : α} :
    (AEMeasurable f <| μ.restrict <| Ι a b) ↔
      (AEMeasurable f <| μ.restrict <| Ioc a b) ∧ (AEMeasurable f <| μ.restrict <| Ioc b a) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [LinearOrder α] {f : α → β} {a b : α}
Goal: AEMeasurable f (μ.restrict (Ι a b)) ↔
  AEMeasurable f (μ.restrict (Set.Ioc a b)) ∧ AEMeasurable f (μ.restrict (Set.Ioc b a))
-- aemeasurable_iff_measurable
theorem aemeasurable_iff_measurable [μ.IsComplete] : AEMeasurable f μ ↔ Measurable f 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [μ.IsComplete]
Goal: AEMeasurable f μ ↔ Measurable f
-- MeasurableEmbedding.aemeasurable_map_iff
theorem MeasurableEmbedding.aemeasurable_map_iff {g : β → γ} (hf : MeasurableEmbedding f) :
    AEMeasurable g (μ.map f) ↔ AEMeasurable (g ∘ f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {μ : MeasureTheory.Measure α} {g : β → γ} (hf : MeasurableEmbedding f)
Goal: AEMeasurable g (MeasureTheory.Measure.map f μ) ↔ AEMeasurable (g ∘ f) μ
-- MeasurableEmbedding.aemeasurable_comp_iff
theorem MeasurableEmbedding.aemeasurable_comp_iff {g : β → γ} (hg : MeasurableEmbedding g)
    {μ : Measure α} : AEMeasurable (g ∘ f) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {g : β → γ} (hg : MeasurableEmbedding g) {μ : MeasureTheory.Measure α}
Goal: AEMeasurable (g ∘ f) μ ↔ AEMeasurable f μ
-- aemeasurable_restrict_iff_comap_subtype
theorem aemeasurable_restrict_iff_comap_subtype {s : Set α} (hs : MeasurableSet s) {μ : Measure α}
    {f : α → β} : AEMeasurable f (μ.restrict s) ↔ AEMeasurable (f ∘ (↑) : s → β) (comap (↑) μ) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {s : Set α} (hs : MeasurableSet s) {μ : MeasureTheory.Measure α} {f : α → β}
Goal: AEMeasurable f (μ.restrict s) ↔ AEMeasurable (f ∘ Subtype.val) (MeasureTheory.Measure.comap Subtype.val μ)
-- aemeasurable_one
@[to_additive] -- @[to_additive (attr := simp)] -- Porting note (#10618): simp can prove this
theorem aemeasurable_one [One β] : AEMeasurable (fun _ : α => (1 : β)) μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [One β]
Goal: AEMeasurable (fun x => 1) μ
-- aemeasurable_zero
@[to_additive] -- @[to_additive (attr := simp)] -- Porting note (#10618): simp can prove this
theorem aemeasurable_one [One β] : AEMeasurable (fun _ : α => (1 : β)) μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [Zero β]
Goal: AEMeasurable (fun x => 0) μ
-- aemeasurable_smul_measure_iff
@[simp]
theorem aemeasurable_smul_measure_iff {c : ℝ≥0∞} (hc : c ≠ 0) :
    AEMeasurable f (c • μ) ↔ AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {c : ℝ≥0∞} (hc : c ≠ 0)
Goal: AEMeasurable f (c • μ) ↔ AEMeasurable f μ
-- aemeasurable_of_aemeasurable_trim
theorem aemeasurable_of_aemeasurable_trim {α} {m m0 : MeasurableSpace α} {μ : Measure α}
    (hm : m ≤ m0) {f : α → β} (hf : AEMeasurable f (μ.trim hm)) : AEMeasurable f μ 
Arguments: {β : Type u_3} [MeasurableSpace β] {α : Type u_7} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → β} (hf : AEMeasurable f (μ.trim hm))
Goal: AEMeasurable f μ
-- aemeasurable_restrict_of_measurable_subtype
theorem aemeasurable_restrict_of_measurable_subtype {s : Set α} (hs : MeasurableSet s)
    (hf : Measurable fun x : s => f x) : AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (hf : Measurable fun x => f ↑x)
Goal: AEMeasurable f (μ.restrict s)
-- aemeasurable_map_equiv_iff
theorem aemeasurable_map_equiv_iff (e : α ≃ᵐ β) {f : β → γ} :
    AEMeasurable f (μ.map e) ↔ AEMeasurable (f ∘ e) μ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} (e : α ≃ᵐ β) {f : β → γ}
Goal: AEMeasurable f (MeasureTheory.Measure.map (⇑e) μ) ↔ AEMeasurable (f ∘ ⇑e) μ
-- AEMeasurable.restrict
theorem AEMeasurable.restrict (hfm : AEMeasurable f μ) {s} : AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (hfm : AEMeasurable f μ) {s : Set α}
Goal: AEMeasurable f (μ.restrict s)
-- aemeasurable_Ioi_of_forall_Ioc
theorem aemeasurable_Ioi_of_forall_Ioc {β} {mβ : MeasurableSpace β} [LinearOrder α]
    [(atTop : Filter α).IsCountablyGenerated] {x : α} {g : α → β}
    (g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))) :
    AEMeasurable g (μ.restrict (Ioi x)) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} {mβ : MeasurableSpace β} [LinearOrder α] [Filter.atTop.IsCountablyGenerated] {x : α} {g : α → β} (g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Set.Ioc x t)))
Goal: AEMeasurable g (μ.restrict (Set.Ioi x))
-- aemeasurable_indicator_iff
theorem aemeasurable_indicator_iff {s} (hs : MeasurableSet s) :
    AEMeasurable (indicator s f) μ ↔ AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Zero β] {s : Set α} (hs : MeasurableSet s)
Goal: AEMeasurable (s.indicator f) μ ↔ AEMeasurable f (μ.restrict s)
-- aemeasurable_indicator_iff₀
theorem aemeasurable_indicator_iff₀ {s} (hs : NullMeasurableSet s μ) :
    AEMeasurable (indicator s f) μ ↔ AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Zero β] {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: AEMeasurable (s.indicator f) μ ↔ AEMeasurable f (μ.restrict s)
-- aemeasurable_indicator_const_iff
lemma aemeasurable_indicator_const_iff {s} [MeasurableSingletonClass β] (b : β) [NeZero b] :
    AEMeasurable (s.indicator (fun _ ↦ b)) μ ↔ NullMeasurableSet s μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} [Zero β] {s : Set α} [MeasurableSingletonClass β] (b : β) [NeZero b]
Goal: AEMeasurable (s.indicator fun x => b) μ ↔ MeasureTheory.NullMeasurableSet s μ
-- AEMeasurable.indicator
@[measurability]
theorem AEMeasurable.indicator (hfm : AEMeasurable f μ) {s} (hs : MeasurableSet s) :
    AEMeasurable (s.indicator f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Zero β] (hfm : AEMeasurable f μ) {s : Set α} (hs : MeasurableSet s)
Goal: AEMeasurable (s.indicator f) μ
-- AEMeasurable.indicator₀
theorem AEMeasurable.indicator₀ (hfm : AEMeasurable f μ) {s} (hs : NullMeasurableSet s μ) :
    AEMeasurable (s.indicator f) μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Zero β] (hfm : AEMeasurable f μ) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: AEMeasurable (s.indicator f) μ
-- MeasureTheory.Measure.restrict_map_of_aemeasurable
theorem MeasureTheory.Measure.restrict_map_of_aemeasurable {f : α → δ} (hf : AEMeasurable f μ)
    {s : Set δ} (hs : MeasurableSet s) : (μ.map f).restrict s = (μ.restrict <| f ⁻¹' s).map f 
Arguments: {α : Type u_2} {δ : Type u_5} {m0 : MeasurableSpace α} [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} (hf : AEMeasurable f μ) {s : Set δ} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.map f μ).restrict s = MeasureTheory.Measure.map f (μ.restrict (f ⁻¹' s))
-- MeasureTheory.Measure.map_mono_of_aemeasurable
theorem MeasureTheory.Measure.map_mono_of_aemeasurable {f : α → δ} (h : μ ≤ ν)
    (hf : AEMeasurable f ν) : μ.map f ≤ ν.map f 
Arguments: {α : Type u_2} {δ : Type u_5} {m0 : MeasurableSpace α} [MeasurableSpace δ] {μ ν : MeasureTheory.Measure α} {f : α → δ} (h : μ ≤ ν) (hf : AEMeasurable f ν)
Goal: MeasureTheory.Measure.map f μ ≤ MeasureTheory.Measure.map f ν
-- MeasureTheory.NullMeasurable.aemeasurable
lemma MeasureTheory.NullMeasurable.aemeasurable {f : α → β}
    [hc : MeasurableSpace.CountablyGenerated β] (h : NullMeasurable f μ) : AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} [hc : MeasurableSpace.CountablyGenerated β] (h : MeasureTheory.NullMeasurable f μ)
Goal: AEMeasurable f μ
-- MeasureTheory.NullMeasurable.aemeasurable_of_aerange
lemma MeasureTheory.NullMeasurable.aemeasurable_of_aerange {f : α → β} {t : Set β}
    [MeasurableSpace.CountablyGenerated t] (h : NullMeasurable f μ) (hft : ∀ᵐ x ∂μ, f x ∈ t) :
    AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} {t : Set β} [MeasurableSpace.CountablyGenerated ↑t] (h : MeasureTheory.NullMeasurable f μ) (hft : ∀ᵐ (x : α) ∂μ, f x ∈ t)
Goal: AEMeasurable f μ
-- MeasureTheory.Measure.map_sum
lemma map_sum {ι : Type*} {m : ι → Measure α} {f : α → β} (hf : AEMeasurable f (Measure.sum m)) :
    Measure.map f (Measure.sum m) = Measure.sum (fun i ↦ Measure.map f (m i)) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {ι : Type u_7} {m : ι → MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f (MeasureTheory.Measure.sum m))
Goal: MeasureTheory.Measure.map f (MeasureTheory.Measure.sum m) =
  MeasureTheory.Measure.sum fun i => MeasureTheory.Measure.map f (m i)
-- AddCircle.closedBall_ae_eq_ball
theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =ᵐ[volume] ball x ε 
Arguments: {T : ℝ} [hT : Fact (0 < T)] {x : AddCircle T} {ε : ℝ}
Goal: Metric.closedBall x ε =ᵐ[MeasureTheory.volume] Metric.ball x ε
-- AddCircle.isAddFundamentalDomain_of_ae_ball
theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCircle T)
    (hu : IsOfFinAddOrder u) (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
    IsAddFundamentalDomain (AddSubgroup.zmultiples u) I 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (I : Set (AddCircle T)) (u x : AddCircle T) (hu : IsOfFinAddOrder u) (hI : I =ᵐ[MeasureTheory.volume] Metric.ball x (T / (2 * ↑(addOrderOf u))))
Goal: MeasureTheory.IsAddFundamentalDomain (↥(AddSubgroup.zmultiples u)) I MeasureTheory.volume
-- AddCircle.volume_of_add_preimage_eq
theorem volume_of_add_preimage_eq (s I : Set <| AddCircle T) (u x : AddCircle T)
    (hu : IsOfFinAddOrder u) (hs : (u +ᵥ s : Set <| AddCircle T) =ᵐ[volume] s)
    (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
    volume s = addOrderOf u • volume (s ∩ I) 
Arguments: {T : ℝ} [hT : Fact (0 < T)] (s I : Set (AddCircle T)) (u x : AddCircle T) (hu : IsOfFinAddOrder u) (hs : u +ᵥ s =ᵐ[MeasureTheory.volume] s) (hI : I =ᵐ[MeasureTheory.volume] Metric.ball x (T / (2 * ↑(addOrderOf u))))
Goal: MeasureTheory.volume s = addOrderOf u • MeasureTheory.volume (s ∩ I)
-- MeasurableEquiv.smul_toEquiv
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [Group G] [MulAction G α] [MeasurableSMul G α] (c : G)
Goal: (MeasurableEquiv.smul c).toEquiv = MulAction.toPerm c
-- MeasurableEquiv.vadd_apply
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [AddGroup G] [AddAction G α] [MeasurableVAdd G α] (c : G)
Goal: ⇑(MeasurableEquiv.vadd c) = fun x => c +ᵥ x
-- MeasurableEquiv.smul_apply
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [Group G] [MulAction G α] [MeasurableSMul G α] (c : G)
Goal: ⇑(MeasurableEquiv.smul c) = fun x => c • x
-- MeasurableEquiv.vadd_toEquiv
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
      "If an additive group `G` acts on `α` by measurable maps, then each element `c : G`
      defines a measurable automorphism of `α`." ]
def smul (c : G) : α ≃ᵐ α 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [AddGroup G] [AddAction G α] [MeasurableVAdd G α] (c : G)
Goal: (MeasurableEquiv.vadd c).toEquiv = AddAction.toPerm c
-- measurableEmbedding_const_smul
@[to_additive]
theorem _root_.measurableEmbedding_const_smul (c : G) : MeasurableEmbedding (c • · : α → α) 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [Group G] [MulAction G α] [MeasurableSMul G α] (c : G)
Goal: MeasurableEmbedding fun x => c • x