ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Integrable.edist_toL1_toL1
@[simp]
theorem edist_toL1_toL1 (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
    edist (hf.toL1 f) (hg.toL1 g) = ∫⁻ a, edist (f a) (g a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f g : α → β) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: edist (MeasureTheory.Integrable.toL1 f hf) (MeasureTheory.Integrable.toL1 g hg) = ∫⁻ (a : α), edist (f a) (g a) ∂μ
-- MeasureTheory.Integrable.edist_toL1_zero
@[simp]
theorem edist_toL1_zero (f : α → β) (hf : Integrable f μ) :
    edist (hf.toL1 f) 0 = ∫⁻ a, edist (f a) 0 ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] (f : α → β) (hf : MeasureTheory.Integrable f μ)
Goal: edist (MeasureTheory.Integrable.toL1 f hf) 0 = ∫⁻ (a : α), edist (f a) 0 ∂μ
-- MeasureTheory.Integrable.toL1_smul
theorem toL1_smul (f : α → β) (hf : Integrable f μ) (k : 𝕜) :
    toL1 (fun a => k • f a) (hf.smul k) = k • toL1 f hf 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] (f : α → β) (hf : MeasureTheory.Integrable f μ) (k : 𝕜)
Goal: MeasureTheory.Integrable.toL1 (fun a => k • f a) ⋯ = k • MeasureTheory.Integrable.toL1 f hf
-- MeasureTheory.Integrable.toL1_smul'
theorem toL1_smul' (f : α → β) (hf : Integrable f μ) (k : 𝕜) :
    toL1 (k • f) (hf.smul k) = k • toL1 f hf 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup β] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β] (f : α → β) (hf : MeasureTheory.Integrable f μ) (k : 𝕜)
Goal: MeasureTheory.Integrable.toL1 (k • f) ⋯ = k • MeasureTheory.Integrable.toL1 f hf
-- MeasureTheory.HasFiniteIntegral.restrict
lemma HasFiniteIntegral.restrict (h : HasFiniteIntegral f μ) {s : Set α} :
    HasFiniteIntegral f (μ.restrict s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {f : α → E} (h : MeasureTheory.HasFiniteIntegral f μ) {s : Set α}
Goal: MeasureTheory.HasFiniteIntegral f (μ.restrict s)
-- MeasureTheory.Integrable.restrict
lemma Integrable.restrict (f_intble : Integrable f μ) {s : Set α} :
    Integrable f (μ.restrict s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {f : α → E} (f_intble : MeasureTheory.Integrable f μ) {s : Set α}
Goal: MeasureTheory.Integrable f (μ.restrict s)
-- ContinuousLinearMap.integrable_comp
theorem ContinuousLinearMap.integrable_comp {φ : α → H} (L : H →L[𝕜] E) (φ_int : Integrable φ μ) :
    Integrable (fun a : α => L (φ a)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {H : Type u_7} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : α → H} (L : H →L[𝕜] E) (φ_int : MeasureTheory.Integrable φ μ)
Goal: MeasureTheory.Integrable (fun a => L (φ a)) μ
-- ContinuousLinearEquiv.integrable_comp_iff
@[simp]
theorem ContinuousLinearEquiv.integrable_comp_iff {φ : α → H} (L : H ≃L[𝕜] E) :
    Integrable (fun a : α ↦ L (φ a)) μ ↔ Integrable φ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {H : Type u_7} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : α → H} (L : H ≃L[𝕜] E)
Goal: MeasureTheory.Integrable (fun a => L (φ a)) μ ↔ MeasureTheory.Integrable φ μ
-- LinearIsometryEquiv.integrable_comp_iff
@[simp]
theorem LinearIsometryEquiv.integrable_comp_iff {φ : α → H} (L : H ≃ₗᵢ[𝕜] E) :
    Integrable (fun a : α ↦ L (φ a)) μ ↔ Integrable φ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {H : Type u_7} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : α → H} (L : H ≃ₗᵢ[𝕜] E)
Goal: MeasureTheory.Integrable (fun a => L (φ a)) μ ↔ MeasureTheory.Integrable φ μ
-- MeasureTheory.Integrable.apply_continuousLinearMap
theorem MeasureTheory.Integrable.apply_continuousLinearMap {φ : α → H →L[𝕜] E}
    (φ_int : Integrable φ μ) (v : H) : Integrable (fun a => φ a v) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedAddCommGroup E] {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {H : Type u_7} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : α → H →L[𝕜] E} (φ_int : MeasureTheory.Integrable φ μ) (v : H)
Goal: MeasureTheory.Integrable (fun a => (φ a) v) μ
-- MeasureTheory.Integrable.fst
lemma Integrable.fst {f : α → E × F} (hf : Integrable f μ) : Integrable (fun x ↦ (f x).1) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : α → E × F} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => (f x).1) μ
-- MeasureTheory.Integrable.snd
lemma Integrable.snd {f : α → E × F} (hf : Integrable f μ) : Integrable (fun x ↦ (f x).2) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : α → E × F} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => (f x).2) μ
-- MeasureTheory.integrable_prod
lemma integrable_prod {f : α → E × F} :
    Integrable f μ ↔ Integrable (fun x ↦ (f x).1) μ ∧ Integrable (fun x ↦ (f x).2) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} {F : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : α → E × F}
Goal: MeasureTheory.Integrable f μ ↔
  MeasureTheory.Integrable (fun x => (f x).1) μ ∧ MeasureTheory.Integrable (fun x => (f x).2) μ
-- MeasurableSpace.self_subset_generateMeasurableRec
theorem self_subset_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
    s ⊆ generateMeasurableRec s i 
Arguments: {α : Type u} (s : Set (Set α)) (i : (Quotient.out (Cardinal.aleph 1).ord).α)
Goal: s ⊆ MeasurableSpace.generateMeasurableRec s i
-- MeasurableSpace.empty_mem_generateMeasurableRec
theorem empty_mem_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
    ∅ ∈ generateMeasurableRec s i 
Arguments: {α : Type u} (s : Set (Set α)) (i : (Quotient.out (Cardinal.aleph 1).ord).α)
Goal: ∅ ∈ MeasurableSpace.generateMeasurableRec s i
-- MeasurableSpace.compl_mem_generateMeasurableRec
theorem compl_mem_generateMeasurableRec {s : Set (Set α)} {i j : ω₁} (h : j < i) {t : Set α}
    (ht : t ∈ generateMeasurableRec s j) : tᶜ ∈ generateMeasurableRec s i 
Arguments: {α : Type u} {s : Set (Set α)} {i j : (Quotient.out (Cardinal.aleph 1).ord).α} (h : j < i) {t : Set α} (ht : t ∈ MeasurableSpace.generateMeasurableRec s j)
Goal: tᶜ ∈ MeasurableSpace.generateMeasurableRec s i
-- MeasurableSpace.iUnion_mem_generateMeasurableRec
theorem iUnion_mem_generateMeasurableRec {s : Set (Set α)} {i : ω₁} {f : ℕ → Set α}
    (hf : ∀ n, ∃ j < i, f n ∈ generateMeasurableRec s j) :
    (⋃ n, f n) ∈ generateMeasurableRec s i 
Arguments: {α : Type u} {s : Set (Set α)} {i : (Quotient.out (Cardinal.aleph 1).ord).α} {f : ℕ → Set α} (hf : ∀ (n : ℕ), ∃ j < i, f n ∈ MeasurableSpace.generateMeasurableRec s j)
Goal: ⋃ n, f n ∈ MeasurableSpace.generateMeasurableRec s i
-- MeasurableSpace.generateMeasurableRec_subset
theorem generateMeasurableRec_subset (s : Set (Set α)) {i j : ω₁} (h : i ≤ j) :
    generateMeasurableRec s i ⊆ generateMeasurableRec s j 
Arguments: {α : Type u} (s : Set (Set α)) {i j : (Quotient.out (Cardinal.aleph 1).ord).α} (h : i ≤ j)
Goal: MeasurableSpace.generateMeasurableRec s i ⊆ MeasurableSpace.generateMeasurableRec s j
-- MeasurableSpace.cardinal_generateMeasurableRec_le
theorem cardinal_generateMeasurableRec_le (s : Set (Set α)) (i : ω₁) :
    #(generateMeasurableRec s i) ≤ max #s 2 ^ aleph0.{u} 
Arguments: {α : Type u} (s : Set (Set α)) (i : (Quotient.out (Cardinal.aleph 1).ord).α)
Goal: #↑(MeasurableSpace.generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀
-- MeasurableSpace.generateMeasurable_eq_rec
theorem generateMeasurable_eq_rec (s : Set (Set α)) :
    { t | GenerateMeasurable s t } =
        ⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i 
Arguments: {α : Type u} (s : Set (Set α))
Goal: {t | MeasurableSpace.GenerateMeasurable s t} = ⋃ i, MeasurableSpace.generateMeasurableRec s i
-- MeasurableSpace.cardinal_generateMeasurable_le
theorem cardinal_generateMeasurable_le (s : Set (Set α)) :
    #{ t | GenerateMeasurable s t } ≤ max #s 2 ^ aleph0.{u} 
Arguments: {α : Type u} (s : Set (Set α))
Goal: #↑{t | MeasurableSpace.GenerateMeasurable s t} ≤ max (#↑s) 2 ^ ℵ₀
-- MeasurableSpace.cardinalMeasurableSet_le
theorem cardinalMeasurableSet_le (s : Set (Set α)) :
    #{ t | @MeasurableSet α (generateFrom s) t } ≤ max #s 2 ^ aleph0.{u} 
Arguments: {α : Type u} (s : Set (Set α))
Goal: #↑{t | MeasurableSet t} ≤ max (#↑s) 2 ^ ℵ₀
-- MeasurableSpace.cardinal_generateMeasurable_le_continuum
theorem cardinal_generateMeasurable_le_continuum {s : Set (Set α)} (hs : #s ≤ 𝔠) :
    #{ t | GenerateMeasurable s t } ≤ 𝔠 
Arguments: {α : Type u} {s : Set (Set α)} (hs : #↑s ≤ 𝔠)
Goal: #↑{t | MeasurableSpace.GenerateMeasurable s t} ≤ 𝔠
-- MeasurableSpace.cardinal_measurableSet_le_continuum
theorem cardinal_measurableSet_le_continuum {s : Set (Set α)} :
    #s ≤ 𝔠 → #{ t | @MeasurableSet α (generateFrom s) t } ≤ 𝔠 
Arguments: {α : Type u} {s : Set (Set α)}
Goal: #↑s ≤ 𝔠 → #↑{t | MeasurableSet t} ≤ 𝔠
-- MeasureTheory.eLpNorm_eq_eLpNorm'
theorem eLpNorm_eq_eLpNorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
    eLpNorm f p μ = eLpNorm' f (ENNReal.toReal p) μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) {f : α → F}
Goal: MeasureTheory.eLpNorm f p μ = MeasureTheory.eLpNorm' f p.toReal μ
-- MeasureTheory.eLpNorm_nnreal_eq_eLpNorm'
lemma eLpNorm_nnreal_eq_eLpNorm' {f : α → F} {p : ℝ≥0} (hp : p ≠ 0) :
    eLpNorm f p μ = eLpNorm' f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {p : ℝ≥0} (hp : p ≠ 0)
Goal: MeasureTheory.eLpNorm f (↑p) μ = MeasureTheory.eLpNorm' f (↑p) μ
-- MeasureTheory.eLpNorm_eq_lintegral_rpow_nnnorm
theorem eLpNorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
    eLpNorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) {f : α → F}
Goal: MeasureTheory.eLpNorm f p μ = (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ (1 / p.toReal)
-- MeasureTheory.eLpNorm_nnreal_eq_lintegral
lemma eLpNorm_nnreal_eq_lintegral {f : α → F} {p : ℝ≥0} (hp : p ≠ 0) :
    eLpNorm f p μ = (∫⁻ x, ‖f x‖₊ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {p : ℝ≥0} (hp : p ≠ 0)
Goal: MeasureTheory.eLpNorm f (↑p) μ = (∫⁻ (x : α), ↑‖f x‖₊ ^ ↑p ∂μ) ^ (1 / ↑p)
-- MeasureTheory.eLpNorm_one_eq_lintegral_nnnorm
theorem eLpNorm_one_eq_lintegral_nnnorm {f : α → F} : eLpNorm f 1 μ = ∫⁻ x, ‖f x‖₊ ∂μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: MeasureTheory.eLpNorm f 1 μ = ∫⁻ (x : α), ↑‖f x‖₊ ∂μ
-- MeasureTheory.eLpNorm_exponent_top
@[simp]
theorem eLpNorm_exponent_top {f : α → F} : eLpNorm f ∞ μ = eLpNormEssSup f μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: MeasureTheory.eLpNorm f ⊤ μ = MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.Memℒp.aestronglyMeasurable
theorem Memℒp.aestronglyMeasurable {f : α → E} {p : ℝ≥0∞} (h : Memℒp f p μ) :
    AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} {p : ℝ≥0∞} (h : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.lintegral_rpow_nnnorm_eq_rpow_eLpNorm'
theorem lintegral_rpow_nnnorm_eq_rpow_eLpNorm' {f : α → F} (hq0_lt : 0 < q) :
    ∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ = eLpNorm' f q μ ^ q 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} (hq0_lt : 0 < q)
Goal: ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ = MeasureTheory.eLpNorm' f q μ ^ q
-- MeasureTheory.eLpNorm_nnreal_pow_eq_lintegral
lemma eLpNorm_nnreal_pow_eq_lintegral {f : α → F} {p : ℝ≥0} (hp : p ≠ 0) :
    eLpNorm f p μ ^ (p : ℝ) = ∫⁻ x, ‖f x‖₊ ^ (p : ℝ) ∂μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {p : ℝ≥0} (hp : p ≠ 0)
Goal: MeasureTheory.eLpNorm f (↑p) μ ^ ↑p = ∫⁻ (x : α), ↑‖f x‖₊ ^ ↑p ∂μ
-- MeasureTheory.Memℒp.eLpNorm_lt_top
theorem Memℒp.eLpNorm_lt_top {f : α → E} (hfp : Memℒp f p μ) : eLpNorm f p μ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hfp : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.eLpNorm f p μ < ⊤
-- MeasureTheory.Memℒp.eLpNorm_ne_top
theorem Memℒp.eLpNorm_ne_top {f : α → E} (hfp : Memℒp f p μ) : eLpNorm f p μ ≠ ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hfp : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.eLpNorm f p μ ≠ ⊤
-- MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top
theorem lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top {f : α → F} (hq0_lt : 0 < q)
    (hfq : eLpNorm' f q μ < ∞) : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} (hq0_lt : 0 < q) (hfq : MeasureTheory.eLpNorm' f q μ < ⊤)
Goal: ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ < ⊤
-- MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top
theorem lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hfp : eLpNorm f p μ < ∞) : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hfp : MeasureTheory.eLpNorm f p μ < ⊤)
Goal: ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ < ⊤
-- MeasureTheory.eLpNorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top
theorem eLpNorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) : eLpNorm f p μ < ∞ ↔ (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.eLpNorm f p μ < ⊤ ↔ ∫⁻ (a : α), ↑‖f a‖₊ ^ p.toReal ∂μ < ⊤
-- MeasureTheory.eLpNorm'_exponent_zero
@[simp]
theorem eLpNorm'_exponent_zero {f : α → F} : eLpNorm' f 0 μ = 1 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: MeasureTheory.eLpNorm' f 0 μ = 1
-- MeasureTheory.eLpNorm_exponent_zero
@[simp]
theorem eLpNorm_exponent_zero {f : α → F} : eLpNorm f 0 μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: MeasureTheory.eLpNorm f 0 μ = 0
-- MeasureTheory.memℒp_zero_iff_aestronglyMeasurable
@[simp]
theorem memℒp_zero_iff_aestronglyMeasurable {f : α → E} :
    Memℒp f 0 μ ↔ AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E}
Goal: MeasureTheory.Memℒp f 0 μ ↔ MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.eLpNorm'_zero
@[simp]
theorem eLpNorm'_zero (hp0_lt : 0 < q) : eLpNorm' (0 : α → F) q μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (hp0_lt : 0 < q)
Goal: MeasureTheory.eLpNorm' 0 q μ = 0
-- MeasureTheory.eLpNorm'_zero'
@[simp]
theorem eLpNorm'_zero' (hq0_ne : q ≠ 0) (hμ : μ ≠ 0) : eLpNorm' (0 : α → F) q μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (hq0_ne : q ≠ 0) (hμ : μ ≠ 0)
Goal: MeasureTheory.eLpNorm' 0 q μ = 0
-- MeasureTheory.eLpNormEssSup_zero
@[simp]
theorem eLpNormEssSup_zero : eLpNormEssSup (0 : α → F) μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F]
Goal: MeasureTheory.eLpNormEssSup 0 μ = 0
-- MeasureTheory.eLpNorm_zero
@[simp]
theorem eLpNorm_zero : eLpNorm (0 : α → F) p μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F]
Goal: MeasureTheory.eLpNorm 0 p μ = 0
-- MeasureTheory.eLpNorm_zero'
@[simp]
theorem eLpNorm_zero' : eLpNorm (fun _ : α => (0 : F)) p μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F]
Goal: MeasureTheory.eLpNorm (fun x => 0) p μ = 0
-- MeasureTheory.zero_memℒp
theorem zero_memℒp : Memℒp (0 : α → E) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E]
Goal: MeasureTheory.Memℒp 0 p μ
-- MeasureTheory.zero_mem_ℒp'
theorem zero_mem_ℒp' : Memℒp (fun _ : α => (0 : E)) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E]
Goal: MeasureTheory.Memℒp (fun x => 0) p μ
-- MeasureTheory.eLpNorm'_measure_zero_of_pos
theorem eLpNorm'_measure_zero_of_pos {f : α → F} (hq_pos : 0 < q) :
    eLpNorm' f q (0 : Measure α) = 0 
Arguments: {α : Type u_1} {F : Type u_3} {q : ℝ} [NormedAddCommGroup F] [MeasurableSpace α] {f : α → F} (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm' f q 0 = 0
-- MeasureTheory.eLpNorm'_measure_zero_of_exponent_zero
theorem eLpNorm'_measure_zero_of_exponent_zero {f : α → F} : eLpNorm' f 0 (0 : Measure α) = 1 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [MeasurableSpace α] {f : α → F}
Goal: MeasureTheory.eLpNorm' f 0 0 = 1
-- MeasureTheory.eLpNorm'_measure_zero_of_neg
theorem eLpNorm'_measure_zero_of_neg {f : α → F} (hq_neg : q < 0) :
    eLpNorm' f q (0 : Measure α) = ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {q : ℝ} [NormedAddCommGroup F] [MeasurableSpace α] {f : α → F} (hq_neg : q < 0)
Goal: MeasureTheory.eLpNorm' f q 0 = ⊤
-- MeasureTheory.eLpNormEssSup_measure_zero
@[simp]
theorem eLpNormEssSup_measure_zero {f : α → F} : eLpNormEssSup f (0 : Measure α) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [MeasurableSpace α] {f : α → F}
Goal: MeasureTheory.eLpNormEssSup f 0 = 0
-- MeasureTheory.eLpNorm_measure_zero
@[simp]
theorem eLpNorm_measure_zero {f : α → F} : eLpNorm f p (0 : Measure α) = 0 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] [MeasurableSpace α] {f : α → F}
Goal: MeasureTheory.eLpNorm f p 0 = 0
-- MeasureTheory.eLpNorm'_neg
@[simp]
theorem eLpNorm'_neg {f : α → F} : eLpNorm' (-f) q μ = eLpNorm' f q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: MeasureTheory.eLpNorm' (-f) q μ = MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.eLpNorm_neg
@[simp]
theorem eLpNorm_neg {f : α → F} : eLpNorm (-f) p μ = eLpNorm f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: MeasureTheory.eLpNorm (-f) p μ = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Memℒp.neg
theorem Memℒp.neg {f : α → E} (hf : Memℒp f p μ) : Memℒp (-f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (-f) p μ
-- MeasureTheory.memℒp_neg_iff
theorem memℒp_neg_iff {f : α → E} : Memℒp (-f) p μ ↔ Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E}
Goal: MeasureTheory.Memℒp (-f) p μ ↔ MeasureTheory.Memℒp f p μ
-- MeasureTheory.eLpNorm_indicator_eq_restrict
theorem eLpNorm_indicator_eq_restrict {f : α → E} {s : Set α} (hs : MeasurableSet s) :
    eLpNorm (s.indicator f) p μ = eLpNorm f p (μ.restrict s) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.eLpNorm (s.indicator f) p μ = MeasureTheory.eLpNorm f p (μ.restrict s)
-- MeasureTheory.eLpNorm'_const
theorem eLpNorm'_const (c : F) (hq_pos : 0 < q) :
    eLpNorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / q) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (c : F) (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm' (fun x => c) q μ = ↑‖c‖₊ * μ Set.univ ^ (1 / q)
-- MeasureTheory.eLpNorm'_const'
theorem eLpNorm'_const' [IsFiniteMeasure μ] (c : F) (hc_ne_zero : c ≠ 0) (hq_ne_zero : q ≠ 0) :
    eLpNorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / q) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [MeasureTheory.IsFiniteMeasure μ] (c : F) (hc_ne_zero : c ≠ 0) (hq_ne_zero : q ≠ 0)
Goal: MeasureTheory.eLpNorm' (fun x => c) q μ = ↑‖c‖₊ * μ Set.univ ^ (1 / q)
-- MeasureTheory.eLpNormEssSup_const
theorem eLpNormEssSup_const (c : F) (hμ : μ ≠ 0) :
    eLpNormEssSup (fun _ : α => c) μ = (‖c‖₊ : ℝ≥0∞) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (c : F) (hμ : μ ≠ 0)
Goal: MeasureTheory.eLpNormEssSup (fun x => c) μ = ↑‖c‖₊
-- MeasureTheory.eLpNorm'_const_of_isProbabilityMeasure
theorem eLpNorm'_const_of_isProbabilityMeasure (c : F) (hq_pos : 0 < q) [IsProbabilityMeasure μ] :
    eLpNorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (c : F) (hq_pos : 0 < q) [MeasureTheory.IsProbabilityMeasure μ]
Goal: MeasureTheory.eLpNorm' (fun x => c) q μ = ↑‖c‖₊
-- MeasureTheory.eLpNorm_const
theorem eLpNorm_const (c : F) (h0 : p ≠ 0) (hμ : μ ≠ 0) :
    eLpNorm (fun _ : α => c) p μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / ENNReal.toReal p) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (c : F) (h0 : p ≠ 0) (hμ : μ ≠ 0)
Goal: MeasureTheory.eLpNorm (fun x => c) p μ = ↑‖c‖₊ * μ Set.univ ^ (1 / p.toReal)
-- MeasureTheory.eLpNorm_const'
theorem eLpNorm_const' (c : F) (h0 : p ≠ 0) (h_top : p ≠ ∞) :
    eLpNorm (fun _ : α => c) p μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / ENNReal.toReal p) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (c : F) (h0 : p ≠ 0) (h_top : p ≠ ⊤)
Goal: MeasureTheory.eLpNorm (fun x => c) p μ = ↑‖c‖₊ * μ Set.univ ^ (1 / p.toReal)