ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.eLpNorm_const_lt_top_iff
theorem eLpNorm_const_lt_top_iff {p : ℝ≥0∞} {c : F} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    eLpNorm (fun _ : α => c) p μ < ∞ ↔ c = 0 ∨ μ Set.univ < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {p : ℝ≥0∞} {c : F} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.eLpNorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ μ Set.univ < ⊤
-- MeasureTheory.memℒp_const
theorem memℒp_const (c : E) [IsFiniteMeasure μ] : Memℒp (fun _ : α => c) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (c : E) [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.Memℒp (fun x => c) p μ
-- MeasureTheory.memℒp_top_const
theorem memℒp_top_const (c : E) : Memℒp (fun _ : α => c) ∞ μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (c : E)
Goal: MeasureTheory.Memℒp (fun x => c) ⊤ μ
-- MeasureTheory.memℒp_const_iff
theorem memℒp_const_iff {p : ℝ≥0∞} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    Memℒp (fun _ : α => c) p μ ↔ c = 0 ∨ μ Set.univ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.Memℒp (fun x => c) p μ ↔ c = 0 ∨ μ Set.univ < ⊤
-- MeasureTheory.eLpNorm'_mono_nnnorm_ae
lemma eLpNorm'_mono_nnnorm_ae {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ ‖g x‖₊) :
    eLpNorm' f q μ ≤ eLpNorm' g q μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ ‖g x‖₊)
Goal: MeasureTheory.eLpNorm' f q μ ≤ MeasureTheory.eLpNorm' g q μ
-- MeasureTheory.eLpNorm'_mono_ae
theorem eLpNorm'_mono_ae {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) :
    eLpNorm' f q μ ≤ eLpNorm' g q μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ ‖g x‖)
Goal: MeasureTheory.eLpNorm' f q μ ≤ MeasureTheory.eLpNorm' g q μ
-- MeasureTheory.eLpNorm'_congr_nnnorm_ae
theorem eLpNorm'_congr_nnnorm_ae {f g : α → F} (hfg : ∀ᵐ x ∂μ, ‖f x‖₊ = ‖g x‖₊) :
    eLpNorm' f q μ = eLpNorm' g q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f g : α → F} (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ = ‖g x‖₊)
Goal: MeasureTheory.eLpNorm' f q μ = MeasureTheory.eLpNorm' g q μ
-- MeasureTheory.eLpNorm'_congr_norm_ae
theorem eLpNorm'_congr_norm_ae {f g : α → F} (hfg : ∀ᵐ x ∂μ, ‖f x‖ = ‖g x‖) :
    eLpNorm' f q μ = eLpNorm' g q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f g : α → F} (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖ = ‖g x‖)
Goal: MeasureTheory.eLpNorm' f q μ = MeasureTheory.eLpNorm' g q μ
-- MeasureTheory.eLpNorm'_congr_ae
theorem eLpNorm'_congr_ae {f g : α → F} (hfg : f =ᵐ[μ] g) : eLpNorm' f q μ = eLpNorm' g q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f g : α → F} (hfg : f =ᵐ[μ] g)
Goal: MeasureTheory.eLpNorm' f q μ = MeasureTheory.eLpNorm' g q μ
-- MeasureTheory.eLpNormEssSup_congr_ae
theorem eLpNormEssSup_congr_ae {f g : α → F} (hfg : f =ᵐ[μ] g) :
    eLpNormEssSup f μ = eLpNormEssSup g μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f g : α → F} (hfg : f =ᵐ[μ] g)
Goal: MeasureTheory.eLpNormEssSup f μ = MeasureTheory.eLpNormEssSup g μ
-- MeasureTheory.eLpNormEssSup_mono_nnnorm_ae
theorem eLpNormEssSup_mono_nnnorm_ae {f g : α → F} (hfg : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ ‖g x‖₊) :
    eLpNormEssSup f μ ≤ eLpNormEssSup g μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f g : α → F} (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ ‖g x‖₊)
Goal: MeasureTheory.eLpNormEssSup f μ ≤ MeasureTheory.eLpNormEssSup g μ
-- MeasureTheory.eLpNorm_mono_nnnorm_ae
theorem eLpNorm_mono_nnnorm_ae {f : α → F} {g : α → G} (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ ‖g x‖₊) :
    eLpNorm f p μ ≤ eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ ‖g x‖₊)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_mono_ae
theorem eLpNorm_mono_ae {f : α → F} {g : α → G} (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) :
    eLpNorm f p μ ≤ eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ ‖g x‖)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_mono_ae_real
theorem eLpNorm_mono_ae_real {f : α → F} {g : α → ℝ} (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ g x) :
    eLpNorm f p μ ≤ eLpNorm g 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} {g : α → ℝ} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ g x)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_mono_nnnorm
theorem eLpNorm_mono_nnnorm {f : α → F} {g : α → G} (h : ∀ x, ‖f x‖₊ ≤ ‖g x‖₊) :
    eLpNorm f p μ ≤ eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (h : ∀ (x : α), ‖f x‖₊ ≤ ‖g x‖₊)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_mono
theorem eLpNorm_mono {f : α → F} {g : α → G} (h : ∀ x, ‖f x‖ ≤ ‖g x‖) :
    eLpNorm f p μ ≤ eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (h : ∀ (x : α), ‖f x‖ ≤ ‖g x‖)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_mono_real
theorem eLpNorm_mono_real {f : α → F} {g : α → ℝ} (h : ∀ x, ‖f x‖ ≤ g x) :
    eLpNorm f p μ ≤ eLpNorm g 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} {g : α → ℝ} (h : ∀ (x : α), ‖f x‖ ≤ g x)
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNormEssSup_le_of_ae_nnnorm_bound
theorem eLpNormEssSup_le_of_ae_nnnorm_bound {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) :
    eLpNormEssSup f μ ≤ C 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C)
Goal: MeasureTheory.eLpNormEssSup f μ ≤ ↑C
-- MeasureTheory.eLpNormEssSup_le_of_ae_bound
theorem eLpNormEssSup_le_of_ae_bound {f : α → F} {C : ℝ} (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
    eLpNormEssSup f μ ≤ ENNReal.ofReal C 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {C : ℝ} (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: MeasureTheory.eLpNormEssSup f μ ≤ ENNReal.ofReal C
-- MeasureTheory.eLpNormEssSup_lt_top_of_ae_nnnorm_bound
theorem eLpNormEssSup_lt_top_of_ae_nnnorm_bound {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) :
    eLpNormEssSup f μ < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C)
Goal: MeasureTheory.eLpNormEssSup f μ < ⊤
-- MeasureTheory.eLpNormEssSup_lt_top_of_ae_bound
theorem eLpNormEssSup_lt_top_of_ae_bound {f : α → F} {C : ℝ} (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
    eLpNormEssSup f μ < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {C : ℝ} (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: MeasureTheory.eLpNormEssSup f μ < ⊤
-- MeasureTheory.eLpNorm_le_of_ae_nnnorm_bound
theorem eLpNorm_le_of_ae_nnnorm_bound {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) :
    eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C)
Goal: MeasureTheory.eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹
-- MeasureTheory.eLpNorm_le_of_ae_bound
theorem eLpNorm_le_of_ae_bound {f : α → F} {C : ℝ} (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
    eLpNorm f p μ ≤ μ Set.univ ^ p.toReal⁻¹ * ENNReal.ofReal C 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} {C : ℝ} (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: MeasureTheory.eLpNorm f p μ ≤ μ Set.univ ^ p.toReal⁻¹ * ENNReal.ofReal C
-- MeasureTheory.eLpNorm_congr_nnnorm_ae
theorem eLpNorm_congr_nnnorm_ae {f : α → F} {g : α → G} (hfg : ∀ᵐ x ∂μ, ‖f x‖₊ = ‖g x‖₊) :
    eLpNorm f p μ = eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ = ‖g x‖₊)
Goal: MeasureTheory.eLpNorm f p μ = MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_congr_norm_ae
theorem eLpNorm_congr_norm_ae {f : α → F} {g : α → G} (hfg : ∀ᵐ x ∂μ, ‖f x‖ = ‖g x‖) :
    eLpNorm f p μ = eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖ = ‖g x‖)
Goal: MeasureTheory.eLpNorm f p μ = MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_indicator_sub_indicator
theorem eLpNorm_indicator_sub_indicator (s t : Set α) (f : α → E) :
    eLpNorm (s.indicator f - t.indicator f) p μ = eLpNorm ((s ∆ t).indicator f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (s t : Set α) (f : α → E)
Goal: MeasureTheory.eLpNorm (s.indicator f - t.indicator f) p μ = MeasureTheory.eLpNorm ((symmDiff s t).indicator f) p μ
-- MeasureTheory.eLpNorm'_norm
@[simp]
theorem eLpNorm'_norm {f : α → F} :
    eLpNorm' (fun a => ‖f a‖) 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' (fun a => ‖f a‖) q μ = MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.eLpNorm_norm
@[simp]
theorem eLpNorm_norm (f : α → F) : eLpNorm (fun x => ‖f x‖) 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 (fun x => ‖f x‖) p μ = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm'_norm_rpow
theorem eLpNorm'_norm_rpow (f : α → F) (p q : ℝ) (hq_pos : 0 < q) :
    eLpNorm' (fun x => ‖f x‖ ^ q) p μ = eLpNorm' f (p * q) μ ^ q 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (f : α → F) (p q : ℝ) (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm' (fun x => ‖f x‖ ^ q) p μ = MeasureTheory.eLpNorm' f (p * q) μ ^ q
-- MeasureTheory.eLpNorm_norm_rpow
theorem eLpNorm_norm_rpow (f : α → F) (hq_pos : 0 < q) :
    eLpNorm (fun x => ‖f x‖ ^ q) p μ = eLpNorm f (p * ENNReal.ofReal q) μ ^ q 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (f : α → F) (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm (fun x => ‖f x‖ ^ q) p μ = MeasureTheory.eLpNorm f (p * ENNReal.ofReal q) μ ^ q
-- MeasureTheory.eLpNorm_congr_ae
theorem eLpNorm_congr_ae {f g : α → F} (hfg : f =ᵐ[μ] g) : eLpNorm f p μ = eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f g : α → F} (hfg : f =ᵐ[μ] g)
Goal: MeasureTheory.eLpNorm f p μ = MeasureTheory.eLpNorm g p μ
-- MeasureTheory.memℒp_congr_ae
theorem memℒp_congr_ae {f g : α → E} (hfg : f =ᵐ[μ] g) : Memℒp f p μ ↔ Memℒp g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : α → E} (hfg : f =ᵐ[μ] g)
Goal: MeasureTheory.Memℒp f p μ ↔ MeasureTheory.Memℒp g p μ
-- MeasureTheory.Memℒp.ae_eq
theorem Memℒp.ae_eq {f g : α → E} (hfg : f =ᵐ[μ] g) (hf_Lp : Memℒp f p μ) : Memℒp g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : α → E} (hfg : f =ᵐ[μ] g) (hf_Lp : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp g p μ
-- MeasureTheory.Memℒp.of_le
theorem Memℒp.of_le {f : α → E} {g : α → F} (hg : Memℒp g p μ) (hf : AEStronglyMeasurable f μ)
    (hfg : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} (hg : MeasureTheory.Memℒp g p μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ ‖g x‖)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.Memℒp.mono'
theorem Memℒp.mono' {f : α → E} {g : α → ℝ} (hg : Memℒp g p μ) (hf : AEStronglyMeasurable f μ)
    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : 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} {g : α → ℝ} (hg : MeasureTheory.Memℒp g p μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ g a)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.Memℒp.congr_norm
theorem Memℒp.congr_norm {f : α → E} {g : α → F} (hf : Memℒp f p μ) (hg : AEStronglyMeasurable g μ)
    (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Memℒp g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
Goal: MeasureTheory.Memℒp g p μ
-- MeasureTheory.memℒp_congr_norm
theorem memℒp_congr_norm {f : α → E} {g : α → F} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : Memℒp f p μ ↔ Memℒp g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
Goal: MeasureTheory.Memℒp f p μ ↔ MeasureTheory.Memℒp g p μ
-- MeasureTheory.memℒp_top_of_bound
theorem memℒp_top_of_bound {f : α → E} (hf : AEStronglyMeasurable f μ) (C : ℝ)
    (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : Memℒp f ∞ μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (C : ℝ) (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: MeasureTheory.Memℒp f ⊤ μ
-- MeasureTheory.Memℒp.of_bound
theorem Memℒp.of_bound [IsFiniteMeasure μ] {f : α → E} (hf : AEStronglyMeasurable f μ) (C : ℝ)
    (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : 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] [MeasureTheory.IsFiniteMeasure μ] {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (C : ℝ) (hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.eLpNorm'_mono_measure
@[mono]
theorem eLpNorm'_mono_measure (f : α → F) (hμν : ν ≤ μ) (hq : 0 ≤ q) :
    eLpNorm' f q ν ≤ eLpNorm' f q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup F] (f : α → F) (hμν : ν ≤ μ) (hq : 0 ≤ q)
Goal: MeasureTheory.eLpNorm' f q ν ≤ MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.eLpNormEssSup_mono_measure
@[mono]
theorem eLpNormEssSup_mono_measure (f : α → F) (hμν : ν ≪ μ) :
    eLpNormEssSup f ν ≤ eLpNormEssSup f μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup F] (f : α → F) (hμν : ν ≪ μ)
Goal: MeasureTheory.eLpNormEssSup f ν ≤ MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNorm_mono_measure
@[mono]
theorem eLpNorm_mono_measure (f : α → F) (hμν : ν ≤ μ) : eLpNorm f p ν ≤ eLpNorm f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup F] (f : α → F) (hμν : ν ≤ μ)
Goal: MeasureTheory.eLpNorm f p ν ≤ MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Memℒp.mono_measure
theorem Memℒp.mono_measure {f : α → E} (hμν : ν ≤ μ) (hf : Memℒp f p μ) : Memℒp f p ν 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hμν : ν ≤ μ) (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp f p ν
-- MeasureTheory.eLpNorm_restrict_le
lemma eLpNorm_restrict_le (f : α → F) (p : ℝ≥0∞) (μ : Measure α) (s : Set α) :
    eLpNorm f p (μ.restrict s) ≤ eLpNorm f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} [NormedAddCommGroup F] (f : α → F) (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) (s : Set α)
Goal: MeasureTheory.eLpNorm f p (μ.restrict s) ≤ MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_restrict_eq_of_support_subset
theorem eLpNorm_restrict_eq_of_support_subset {s : Set α} {f : α → F} (hsf : f.support ⊆ s) :
    eLpNorm f p (μ.restrict s) = 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] {s : Set α} {f : α → F} (hsf : Function.support f ⊆ s)
Goal: MeasureTheory.eLpNorm f p (μ.restrict s) = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Memℒp.restrict
theorem Memℒp.restrict (s : Set α) {f : α → E} (hf : Memℒp f p μ) : Memℒp 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] (s : Set α) {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp f p (μ.restrict s)
-- MeasureTheory.eLpNorm'_smul_measure
theorem eLpNorm'_smul_measure {p : ℝ} (hp : 0 ≤ p) {f : α → F} (c : ℝ≥0∞) :
    eLpNorm' f p (c • μ) = c ^ (1 / p) * eLpNorm' f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {p : ℝ} (hp : 0 ≤ p) {f : α → F} (c : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm' f p (c • μ) = c ^ (1 / p) * MeasureTheory.eLpNorm' f p μ
-- MeasureTheory.eLpNormEssSup_smul_measure
theorem eLpNormEssSup_smul_measure {f : α → F} {c : ℝ≥0∞} (hc : c ≠ 0) :
    eLpNormEssSup f (c • 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} {c : ℝ≥0∞} (hc : c ≠ 0)
Goal: MeasureTheory.eLpNormEssSup f (c • μ) = MeasureTheory.eLpNormEssSup f μ
-- _private.Mathlib.MeasureTheory.Function.LpSeminorm.Basic.0.MeasureTheory.eLpNorm_smul_measure_of_ne_zero_of_ne_top
private theorem eLpNorm_smul_measure_of_ne_zero_of_ne_top {p : ℝ≥0∞} (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) {f : α → F} (c : ℝ≥0∞) :
    eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {p : ℝ≥0∞} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) {f : α → F} (c : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f p (c • μ) = c ^ (1 / p).toReal • MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_smul_measure_of_ne_zero
theorem eLpNorm_smul_measure_of_ne_zero {p : ℝ≥0∞} {f : α → F} {c : ℝ≥0∞} (hc : c ≠ 0) :
    eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {p : ℝ≥0∞} {f : α → F} {c : ℝ≥0∞} (hc : c ≠ 0)
Goal: MeasureTheory.eLpNorm f p (c • μ) = c ^ (1 / p).toReal • MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_smul_measure_of_ne_top
theorem eLpNorm_smul_measure_of_ne_top {p : ℝ≥0∞} (hp_ne_top : p ≠ ∞) {f : α → F} (c : ℝ≥0∞) :
    eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {p : ℝ≥0∞} (hp_ne_top : p ≠ ⊤) {f : α → F} (c : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f p (c • μ) = c ^ (1 / p).toReal • MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_one_smul_measure
theorem eLpNorm_one_smul_measure {f : α → F} (c : ℝ≥0∞) :
    eLpNorm f 1 (c • μ) = c * eLpNorm f 1 μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} (c : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f 1 (c • μ) = c * MeasureTheory.eLpNorm f 1 μ
-- MeasureTheory.Memℒp.of_measure_le_smul
theorem Memℒp.of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (hμ'_le : μ' ≤ c • μ)
    {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] {μ' : MeasureTheory.Measure α} (c : ℝ≥0∞) (hc : c ≠ ⊤) (hμ'_le : μ' ≤ c • μ) {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp f p μ'
-- MeasureTheory.Memℒp.smul_measure
theorem Memℒp.smul_measure {f : α → E} {c : ℝ≥0∞} (hf : Memℒp f p μ) (hc : c ≠ ∞) :
    Memℒp f p (c • 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} {c : ℝ≥0∞} (hf : MeasureTheory.Memℒp f p μ) (hc : c ≠ ⊤)
Goal: MeasureTheory.Memℒp f p (c • μ)
-- MeasureTheory.eLpNorm_one_add_measure
theorem eLpNorm_one_add_measure (f : α → F) (μ ν : Measure α) :
    eLpNorm f 1 (μ + ν) = eLpNorm f 1 μ + eLpNorm f 1 ν 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} [NormedAddCommGroup F] (f : α → F) (μ ν : MeasureTheory.Measure α)
Goal: MeasureTheory.eLpNorm f 1 (μ + ν) = MeasureTheory.eLpNorm f 1 μ + MeasureTheory.eLpNorm f 1 ν
-- MeasureTheory.eLpNorm_le_add_measure_right
theorem eLpNorm_le_add_measure_right (f : α → F) (μ ν : Measure α) {p : ℝ≥0∞} :
    eLpNorm f p μ ≤ eLpNorm f p (μ + ν) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} [NormedAddCommGroup F] (f : α → F) (μ ν : MeasureTheory.Measure α) {p : ℝ≥0∞}
Goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm f p (μ + ν)
-- MeasureTheory.eLpNorm_le_add_measure_left
theorem eLpNorm_le_add_measure_left (f : α → F) (μ ν : Measure α) {p : ℝ≥0∞} :
    eLpNorm f p ν ≤ eLpNorm f p (μ + ν) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} [NormedAddCommGroup F] (f : α → F) (μ ν : MeasureTheory.Measure α) {p : ℝ≥0∞}
Goal: MeasureTheory.eLpNorm f p ν ≤ MeasureTheory.eLpNorm f p (μ + ν)
-- MeasureTheory.Memℒp.left_of_add_measure
theorem Memℒp.left_of_add_measure {f : α → E} (h : Memℒp f p (μ + ν)) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (h : MeasureTheory.Memℒp f p (μ + ν))
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.Memℒp.right_of_add_measure
theorem Memℒp.right_of_add_measure {f : α → E} (h : Memℒp f p (μ + ν)) : Memℒp f p ν 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ ν : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (h : MeasureTheory.Memℒp f p (μ + ν))
Goal: MeasureTheory.Memℒp f p ν