ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div
theorem ae_tendsto_lintegral_nnnorm_sub_div {f : α → E} (hf : LocallyIntegrable f μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → E} (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
-- VitaliFamily.eventually_filterAt_integrableOn
theorem eventually_filterAt_integrableOn (x : α) {f : α → E} (hf : LocallyIntegrable f μ) :
    ∀ᶠ a in v.filterAt x, IntegrableOn f a μ 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {E : Type u_2} [NormedAddCommGroup E] (x : α) {f : α → E} (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: ∀ᶠ (a : Set α) in v.filterAt x, MeasureTheory.IntegrableOn f a μ
-- VitaliFamily.ae_tendsto_average_norm_sub
theorem ae_tendsto_average_norm_sub {f : α → E} (hf : LocallyIntegrable f μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ⨍ y in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → E} (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
-- VitaliFamily.ae_tendsto_average
theorem ae_tendsto_average [NormedSpace ℝ E] [CompleteSpace E] {f : α → E}
    (hf : LocallyIntegrable f μ) :
    ∀ᵐ x ∂μ, Tendsto (fun a => ⨍ y in a, f y ∂μ) (v.filterAt x) (𝓝 (f x)) 
Arguments: {α : Type u_1} [MetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (v : VitaliFamily μ) {E : Type u_2} [NormedAddCommGroup E] [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} (hf : MeasureTheory.LocallyIntegrable f μ)
Goal: ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ⨍ (y : α) in a, f y ∂μ) (v.filterAt x) (𝓝 (f x))
-- MeasureTheory.OuterMeasure.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : OuterMeasure α) = 0 
Arguments: {α : Type u_1}
Goal: ⇑0 = 0
-- MeasureTheory.OuterMeasure.coe_add
@[simp]
theorem coe_add (m₁ m₂ : OuterMeasure α) : ⇑(m₁ + m₂) = m₁ + m₂ 
Arguments: {α : Type u_1} (m₁ m₂ : MeasureTheory.OuterMeasure α)
Goal: ⇑(m₁ + m₂) = ⇑m₁ + ⇑m₂
-- MeasureTheory.OuterMeasure.add_apply
theorem add_apply (m₁ m₂ : OuterMeasure α) (s : Set α) : (m₁ + m₂) s = m₁ s + m₂ s 
Arguments: {α : Type u_1} (m₁ m₂ : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (m₁ + m₂) s = m₁ s + m₂ s
-- MeasureTheory.OuterMeasure.coe_smul
@[simp]
theorem coe_smul (c : R) (m : OuterMeasure α) : ⇑(c • m) = c • ⇑m 
Arguments: {α : Type u_1} {R : Type u_3} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R) (m : MeasureTheory.OuterMeasure α)
Goal: ⇑(c • m) = c • ⇑m
-- MeasureTheory.OuterMeasure.smul_apply
theorem smul_apply (c : R) (m : OuterMeasure α) (s : Set α) : (c • m) s = c • m s 
Arguments: {α : Type u_1} {R : Type u_3} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R) (m : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (c • m) s = c • m s
-- MeasureTheory.OuterMeasure.coeFnAddMonoidHom_apply
@[simps]
def coeFnAddMonoidHom : OuterMeasure α →+ Set α → ℝ≥0∞ 
Arguments: {α : Type u_1}
Goal: ∀ (a : MeasureTheory.OuterMeasure α) (a_1 : Set α), MeasureTheory.OuterMeasure.coeFnAddMonoidHom a a_1 = a a_1
-- MeasureTheory.OuterMeasure.coe_bot
@[simp]
theorem coe_bot : (⊥ : OuterMeasure α) = 0 
Arguments: {α : Type u_1}
Goal: ⊥ = 0
-- MeasureTheory.OuterMeasure.univ_eq_zero_iff
theorem univ_eq_zero_iff (m : OuterMeasure α) : m univ = 0 ↔ m = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: m Set.univ = 0 ↔ m = 0
-- MeasureTheory.OuterMeasure.sSup_apply
@[simp]
theorem sSup_apply (ms : Set (OuterMeasure α)) (s : Set α) :
    (sSup ms) s = ⨆ m ∈ ms, (m : OuterMeasure α) s 
Arguments: {α : Type u_1} (ms : Set (MeasureTheory.OuterMeasure α)) (s : Set α)
Goal: (sSup ms) s = ⨆ m ∈ ms, m s
-- MeasureTheory.OuterMeasure.iSup_apply
@[simp]
theorem iSup_apply {ι} (f : ι → OuterMeasure α) (s : Set α) : (⨆ i : ι, f i) s = ⨆ i, f i s 
Arguments: {α : Type u_1} {ι : Sort u_3} (f : ι → MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (⨆ i, f i) s = ⨆ i, (f i) s
-- MeasureTheory.OuterMeasure.coe_iSup
@[norm_cast]
theorem coe_iSup {ι} (f : ι → OuterMeasure α) : ⇑(⨆ i, f i) = ⨆ i, ⇑(f i) 
Arguments: {α : Type u_1} {ι : Sort u_3} (f : ι → MeasureTheory.OuterMeasure α)
Goal: ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)
-- MeasureTheory.OuterMeasure.sup_apply
@[simp]
theorem sup_apply (m₁ m₂ : OuterMeasure α) (s : Set α) : (m₁ ⊔ m₂) s = m₁ s ⊔ m₂ s 
Arguments: {α : Type u_1} (m₁ m₂ : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (m₁ ⊔ m₂) s = m₁ s ⊔ m₂ s
-- MeasureTheory.OuterMeasure.smul_iSup
theorem smul_iSup {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    {ι : Sort*} (f : ι → OuterMeasure α) (c : R) :
    (c • ⨆ i, f i) = ⨆ i, c • f i 
Arguments: {α : Type u_1} {R : Type u_3} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] {ι : Sort u_4} (f : ι → MeasureTheory.OuterMeasure α) (c : R)
Goal: c • ⨆ i, f i = ⨆ i, c • f i
-- MeasureTheory.OuterMeasure.mono''
@[mono, gcongr]
theorem mono'' {m₁ m₂ : OuterMeasure α} {s₁ s₂ : Set α} (hm : m₁ ≤ m₂) (hs : s₁ ⊆ s₂) :
    m₁ s₁ ≤ m₂ s₂ 
Arguments: {α : Type u_1} {m₁ m₂ : MeasureTheory.OuterMeasure α} {s₁ s₂ : Set α} (hm : m₁ ≤ m₂) (hs : s₁ ⊆ s₂)
Goal: m₁ s₁ ≤ m₂ s₂
-- MeasureTheory.OuterMeasure.map_apply
@[simp]
theorem map_apply {β} (f : α → β) (m : OuterMeasure α) (s : Set β) : map f m s = m (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure α) (s : Set β)
Goal: ((MeasureTheory.OuterMeasure.map f) m) s = m (f ⁻¹' s)
-- MeasureTheory.OuterMeasure.map_id
@[simp]
theorem map_id (m : OuterMeasure α) : map id m = m 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.map id) m = m
-- MeasureTheory.OuterMeasure.map_map
@[simp]
theorem map_map {β γ} (f : α → β) (g : β → γ) (m : OuterMeasure α) :
    map g (map f m) = map (g ∘ f) m 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_4} (f : α → β) (g : β → γ) (m : MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.map g) ((MeasureTheory.OuterMeasure.map f) m) = (MeasureTheory.OuterMeasure.map (g ∘ f)) m
-- MeasureTheory.OuterMeasure.map_mono
@[mono]
theorem map_mono {β} (f : α → β) : Monotone (map f) 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β)
Goal: Monotone ⇑(MeasureTheory.OuterMeasure.map f)
-- MeasureTheory.OuterMeasure.map_sup
@[simp]
theorem map_sup {β} (f : α → β) (m m' : OuterMeasure α) : map f (m ⊔ m') = map f m ⊔ map f m' 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β) (m m' : MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.map f) (m ⊔ m') =
  (MeasureTheory.OuterMeasure.map f) m ⊔ (MeasureTheory.OuterMeasure.map f) m'
-- MeasureTheory.OuterMeasure.map_iSup
@[simp]
theorem map_iSup {β ι} (f : α → β) (m : ι → OuterMeasure α) : map f (⨆ i, m i) = ⨆ i, map f (m i) 
Arguments: {α : Type u_1} {β : Type u_3} {ι : Sort u_4} (f : α → β) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.map f) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.map f) (m i)
-- MeasureTheory.OuterMeasure.dirac_apply
@[simp]
theorem dirac_apply (a : α) (s : Set α) : dirac a s = indicator s (fun _ => 1) a 
Arguments: {α : Type u_1} (a : α) (s : Set α)
Goal: (MeasureTheory.OuterMeasure.dirac a) s = s.indicator (fun x => 1) a
-- MeasureTheory.OuterMeasure.sum_apply
@[simp]
theorem sum_apply {ι} (f : ι → OuterMeasure α) (s : Set α) : sum f s = ∑' i, f i s 
Arguments: {α : Type u_1} {ι : Type u_3} (f : ι → MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (MeasureTheory.OuterMeasure.sum f) s = ∑' (i : ι), (f i) s
-- MeasureTheory.OuterMeasure.smul_dirac_apply
theorem smul_dirac_apply (a : ℝ≥0∞) (b : α) (s : Set α) :
    (a • dirac b) s = indicator s (fun _ => a) b 
Arguments: {α : Type u_1} (a : ℝ≥0∞) (b : α) (s : Set α)
Goal: (a • MeasureTheory.OuterMeasure.dirac b) s = s.indicator (fun x => a) b
-- MeasureTheory.OuterMeasure.comap_apply
@[simp]
theorem comap_apply {β} (f : α → β) (m : OuterMeasure β) (s : Set α) : comap f m s = m (f '' s) 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure β) (s : Set α)
Goal: ((MeasureTheory.OuterMeasure.comap f) m) s = m (f '' s)
-- MeasureTheory.OuterMeasure.comap_mono
@[mono]
theorem comap_mono {β} (f : α → β) : Monotone (comap f) 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β)
Goal: Monotone ⇑(MeasureTheory.OuterMeasure.comap f)
-- MeasureTheory.OuterMeasure.comap_iSup
@[simp]
theorem comap_iSup {β ι} (f : α → β) (m : ι → OuterMeasure β) :
    comap f (⨆ i, m i) = ⨆ i, comap f (m i) 
Arguments: {α : Type u_1} {β : Type u_3} {ι : Sort u_4} (f : α → β) (m : ι → MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.comap f) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.comap f) (m i)
-- MeasureTheory.OuterMeasure.restrict_apply
@[simp]
theorem restrict_apply (s t : Set α) (m : OuterMeasure α) : restrict s m t = m (t ∩ s) 
Arguments: {α : Type u_1} (s t : Set α) (m : MeasureTheory.OuterMeasure α)
Goal: ((MeasureTheory.OuterMeasure.restrict s) m) t = m (t ∩ s)
-- MeasureTheory.OuterMeasure.restrict_mono
@[mono]
theorem restrict_mono {s t : Set α} (h : s ⊆ t) {m m' : OuterMeasure α} (hm : m ≤ m') :
    restrict s m ≤ restrict t m' 
Arguments: {α : Type u_1} {s t : Set α} (h : s ⊆ t) {m m' : MeasureTheory.OuterMeasure α} (hm : m ≤ m')
Goal: (MeasureTheory.OuterMeasure.restrict s) m ≤ (MeasureTheory.OuterMeasure.restrict t) m'
-- MeasureTheory.OuterMeasure.restrict_univ
@[simp]
theorem restrict_univ (m : OuterMeasure α) : restrict univ m = m 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.restrict Set.univ) m = m
-- MeasureTheory.OuterMeasure.restrict_empty
@[simp]
theorem restrict_empty (m : OuterMeasure α) : restrict ∅ m = 0 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.restrict ∅) m = 0
-- MeasureTheory.OuterMeasure.restrict_iSup
@[simp]
theorem restrict_iSup {ι} (s : Set α) (m : ι → OuterMeasure α) :
    restrict s (⨆ i, m i) = ⨆ i, restrict s (m i) 
Arguments: {α : Type u_1} {ι : Sort u_3} (s : Set α) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.restrict s) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.restrict s) (m i)
-- MeasureTheory.OuterMeasure.map_comap
theorem map_comap {β} (f : α → β) (m : OuterMeasure β) : map f (comap f m) = restrict (range f) m 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) =
  (MeasureTheory.OuterMeasure.restrict (Set.range f)) m
-- MeasureTheory.OuterMeasure.map_comap_le
theorem map_comap_le {β} (f : α → β) (m : OuterMeasure β) : map f (comap f m) ≤ m 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) ≤ m
-- MeasureTheory.OuterMeasure.restrict_le_self
theorem restrict_le_self (m : OuterMeasure α) (s : Set α) : restrict s m ≤ m 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (MeasureTheory.OuterMeasure.restrict s) m ≤ m
-- MeasureTheory.OuterMeasure.map_le_restrict_range
@[simp]
theorem map_le_restrict_range {β} {ma : OuterMeasure α} {mb : OuterMeasure β} {f : α → β} :
    map f ma ≤ restrict (range f) mb ↔ map f ma ≤ mb 
Arguments: {α : Type u_1} {β : Type u_3} {ma : MeasureTheory.OuterMeasure α} {mb : MeasureTheory.OuterMeasure β} {f : α → β}
Goal: (MeasureTheory.OuterMeasure.map f) ma ≤ (MeasureTheory.OuterMeasure.restrict (Set.range f)) mb ↔
  (MeasureTheory.OuterMeasure.map f) ma ≤ mb
-- MeasureTheory.OuterMeasure.map_comap_of_surjective
theorem map_comap_of_surjective {β} {f : α → β} (hf : Surjective f) (m : OuterMeasure β) :
    map f (comap f m) = m 
Arguments: {α : Type u_1} {β : Type u_3} {f : α → β} (hf : Function.Surjective f) (m : MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) = m
-- MeasureTheory.OuterMeasure.le_comap_map
theorem le_comap_map {β} (f : α → β) (m : OuterMeasure α) : m ≤ comap f (map f m) 
Arguments: {α : Type u_1} {β : Type u_3} (f : α → β) (m : MeasureTheory.OuterMeasure α)
Goal: m ≤ (MeasureTheory.OuterMeasure.comap f) ((MeasureTheory.OuterMeasure.map f) m)
-- MeasureTheory.OuterMeasure.comap_map
theorem comap_map {β} {f : α → β} (hf : Injective f) (m : OuterMeasure α) : comap f (map f m) = m 
Arguments: {α : Type u_1} {β : Type u_3} {f : α → β} (hf : Function.Injective f) (m : MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.comap f) ((MeasureTheory.OuterMeasure.map f) m) = m
-- MeasureTheory.OuterMeasure.top_apply
@[simp]
theorem top_apply {s : Set α} (h : s.Nonempty) : (⊤ : OuterMeasure α) s = ∞ 
Arguments: {α : Type u_1} {s : Set α} (h : s.Nonempty)
Goal: ⊤ s = ⊤
-- MeasureTheory.OuterMeasure.top_apply'
theorem top_apply' (s : Set α) : (⊤ : OuterMeasure α) s = ⨅ h : s = ∅, 0 
Arguments: {α : Type u_1} (s : Set α)
Goal: ⊤ s = ⨅ (_ : s = ∅), 0
-- MeasureTheory.OuterMeasure.comap_top
@[simp]
theorem comap_top (f : α → β) : comap f ⊤ = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β)
Goal: (MeasureTheory.OuterMeasure.comap f) ⊤ = ⊤
-- MeasureTheory.OuterMeasure.map_top
theorem map_top (f : α → β) : map f ⊤ = restrict (range f) ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β)
Goal: (MeasureTheory.OuterMeasure.map f) ⊤ = (MeasureTheory.OuterMeasure.restrict (Set.range f)) ⊤
-- MeasureTheory.OuterMeasure.map_top_of_surjective
@[simp]
theorem map_top_of_surjective (f : α → β) (hf : Surjective f) : map f ⊤ = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} (f : α → β) (hf : Function.Surjective f)
Goal: (MeasureTheory.OuterMeasure.map f) ⊤ = ⊤
-- MeasureTheory.lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero
theorem lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    -- Porting note: needed to add explicit casts in the next two hypotheses
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ = 0) :
    f =ᵐ[μ] (0 : α → E') 
Arguments: {α : Type u_1} {E' : Type u_2} {𝕜 : Type u_4} {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] (hm : m ≤ m0) (f : ↥(MeasureTheory.lpMeas E' 𝕜 m p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑↑f) s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑↑f x ∂μ = 0)
Goal: ↑↑↑f =ᵐ[μ] 0
-- MeasureTheory.Lp.ae_eq_zero_of_forall_setIntegral_eq_zero'
theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
    (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
    (hf_meas : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E' : Type u_2} (𝕜 : Type u_4) {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp E' p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑f) s μ) (hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = 0) (hf_meas : MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ)
Goal: ↑↑f =ᵐ[μ] 0
-- MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq'
theorem Lp.ae_eq_of_forall_setIntegral_eq' (hm : m ≤ m0) (f g : Lp E' p μ) (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
    (hfg : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    (hf_meas : AEStronglyMeasurable' m f μ) (hg_meas : AEStronglyMeasurable' m g μ) :
    f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E' : Type u_2} (𝕜 : Type u_4) {p : ℝ≥0∞} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [RCLike 𝕜] [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] [CompleteSpace E'] [NormedSpace ℝ E'] (hm : m ≤ m0) (f g : ↥(MeasureTheory.Lp E' p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑f) s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn (↑↑g) s μ) (hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ) (hf_meas : MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ) (hg_meas : MeasureTheory.AEStronglyMeasurable' m (↑↑g) μ)
Goal: ↑↑f =ᵐ[μ] ↑↑g
-- MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite'
theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    {f g : α → F'} (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
    (hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
    (hfg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
    (hfm : AEStronglyMeasurable' m f μ) (hgm : AEStronglyMeasurable' m g μ) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {F' : Type u_3} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] {f g : α → F'} (hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn g s μ) (hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ) (hfm : MeasureTheory.AEStronglyMeasurable' m f μ) (hgm : MeasureTheory.AEStronglyMeasurable' m g μ)
Goal: f =ᵐ[μ] g
-- MeasureTheory.integral_norm_le_of_forall_fin_meas_integral_eq
theorem integral_norm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
    (hf : StronglyMeasurable f) (hfi : IntegrableOn f s μ) (hg : StronglyMeasurable[m] g)
    (hgi : IntegrableOn g s μ)
    (hgf : ∀ t, MeasurableSet[m] t → μ t < ∞ → ∫ x in t, g x ∂μ = ∫ x in t, f x ∂μ)
    (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) : (∫ x in s, ‖g x‖ ∂μ) ≤ ∫ x in s, ‖f x‖ ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) {f g : α → ℝ} (hf : MeasureTheory.StronglyMeasurable f) (hfi : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.StronglyMeasurable g) (hgi : MeasureTheory.IntegrableOn g s μ) (hgf : ∀ (t : Set α), MeasurableSet t → μ t < ⊤ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤)
Goal: ∫ (x : α) in s, ‖g x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ
-- MeasureTheory.lintegral_nnnorm_le_of_forall_fin_meas_integral_eq
theorem lintegral_nnnorm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
    (hf : StronglyMeasurable f) (hfi : IntegrableOn f s μ) (hg : StronglyMeasurable[m] g)
    (hgi : IntegrableOn g s μ)
    (hgf : ∀ t, MeasurableSet[m] t → μ t < ∞ → ∫ x in t, g x ∂μ = ∫ x in t, f x ∂μ)
    (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) : (∫⁻ x in s, ‖g x‖₊ ∂μ) ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) {f g : α → ℝ} (hf : MeasureTheory.StronglyMeasurable f) (hfi : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.StronglyMeasurable g) (hgi : MeasureTheory.IntegrableOn g s μ) (hgf : ∀ (t : Set α), MeasurableSet t → μ t < ⊤ → ∫ (x : α) in t, g x ∂μ = ∫ (x : α) in t, f x ∂μ) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤)
Goal: ∫⁻ (x : α) in s, ↑‖g x‖₊ ∂μ ≤ ∫⁻ (x : α) in s, ↑‖f x‖₊ ∂μ
-- MeasureTheory.SimpleFunc.finite_range'
structure SimpleFunc.{u, v} (α : Type u) [MeasurableSpace α] (β : Type v) where
  toFun : α → β
  measurableSet_fiber' : ∀ x, MeasurableSet (toFun ⁻¹' {x})
  finite_range' : (Set.range toFun).Finite
Arguments: {α : Type u} [MeasurableSpace α] {β : Type v} (self : MeasureTheory.SimpleFunc α β)
Goal: (Set.range ↑self).Finite
-- MeasureTheory.SimpleFunc.measurableSet_fiber'
structure SimpleFunc.{u, v} (α : Type u) [MeasurableSpace α] (β : Type v) where
  toFun : α → β
  measurableSet_fiber' : ∀ x, MeasurableSet (toFun ⁻¹' {x})
  finite_range' : (Set.range toFun).Finite
Arguments: {α : Type u} [MeasurableSpace α] {β : Type v} (self : MeasureTheory.SimpleFunc α β) (x : β)
Goal: MeasurableSet (↑self ⁻¹' {x})
-- MeasureTheory.SimpleFunc.coe_injective
theorem coe_injective ⦃f g : α →ₛ β⦄ (H : (f : α → β) = g) : f = g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] ⦃f g : MeasureTheory.SimpleFunc α β⦄ (H : ↑f = ↑g)
Goal: f = g
-- MeasureTheory.SimpleFunc.ext
@[ext]
theorem ext {f g : α →ₛ β} (H : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {f g : MeasureTheory.SimpleFunc α β} (H : ∀ (a : α), ↑f a = ↑g a)
Goal: f = g
-- MeasureTheory.SimpleFunc.finite_range
theorem finite_range (f : α →ₛ β) : (Set.range f).Finite 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β)
Goal: (Set.range ↑f).Finite
-- MeasureTheory.SimpleFunc.measurableSet_fiber
theorem measurableSet_fiber (f : α →ₛ β) (x : β) : MeasurableSet (f ⁻¹' {x}) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (x : β)
Goal: MeasurableSet (↑f ⁻¹' {x})
-- MeasureTheory.SimpleFunc.apply_mk
theorem apply_mk (f : α → β) (h h') (x : α) : SimpleFunc.mk f h h' x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : α → β) (h : ∀ (x : β), MeasurableSet (f ⁻¹' {x})) (h' : (Set.range f).Finite) (x : α)
Goal: ↑{ toFun := f, measurableSet_fiber' := h, finite_range' := h' } x = f x
-- MeasureTheory.SimpleFunc.mem_range
@[simp]
theorem mem_range {f : α →ₛ β} {b} : b ∈ f.range ↔ b ∈ range f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {f : MeasureTheory.SimpleFunc α β} {b : β}
Goal: b ∈ f.range ↔ b ∈ Set.range ↑f
-- MeasureTheory.SimpleFunc.mem_range_self
theorem mem_range_self (f : α →ₛ β) (x : α) : f x ∈ f.range 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β) (x : α)
Goal: ↑f x ∈ f.range
-- MeasureTheory.SimpleFunc.coe_range
@[simp]
theorem coe_range (f : α →ₛ β) : (↑f.range : Set β) = Set.range f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α β)
Goal: ↑f.range = Set.range ↑f
-- MeasureTheory.SimpleFunc.mem_range_of_measure_ne_zero
theorem mem_range_of_measure_ne_zero {f : α →ₛ β} {x : β} {μ : Measure α} (H : μ (f ⁻¹' {x}) ≠ 0) :
    x ∈ f.range 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {f : MeasureTheory.SimpleFunc α β} {x : β} {μ : MeasureTheory.Measure α} (H : μ (↑f ⁻¹' {x}) ≠ 0)
Goal: x ∈ f.range
-- MeasureTheory.SimpleFunc.forall_mem_range
theorem forall_mem_range {f : α →ₛ β} {p : β → Prop} : (∀ y ∈ f.range, p y) ↔ ∀ x, p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {f : MeasureTheory.SimpleFunc α β} {p : β → Prop}
Goal: (∀ y ∈ f.range, p y) ↔ ∀ (x : α), p (↑f x)
-- MeasureTheory.SimpleFunc.exists_range_iff
theorem exists_range_iff {f : α →ₛ β} {p : β → Prop} : (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] {f : MeasureTheory.SimpleFunc α β} {p : β → Prop}
Goal: (∃ y ∈ f.range, p y) ↔ ∃ x, p (↑f x)