ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Lp.dist_def
theorem dist_def (f g : Lp E p μ) : dist f g = (eLpNorm (⇑f - ⇑g) p μ).toReal 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: dist f g = (MeasureTheory.eLpNorm (↑↑f - ↑↑g) p μ).toReal
-- MeasureTheory.Lp.edist_def
theorem edist_def (f g : Lp E p μ) : edist f g = eLpNorm (⇑f - ⇑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 : ↥(MeasureTheory.Lp E p μ))
Goal: edist f g = MeasureTheory.eLpNorm (↑↑f - ↑↑g) p μ
-- MeasureTheory.Lp.edist_dist
protected theorem edist_dist (f g : Lp E p μ) : edist f g = .ofReal (dist f g) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: edist f g = ENNReal.ofReal (dist f g)
-- MeasureTheory.Lp.dist_edist
protected theorem dist_edist (f g : Lp E p μ) : dist f g = (edist f g).toReal 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: dist f g = (edist f g).toReal
-- MeasureTheory.Lp.dist_eq_norm
theorem dist_eq_norm (f g : Lp E p μ) : dist f g = ‖f - g‖ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: dist f g = ‖f - g‖
-- MeasureTheory.Lp.edist_toLp_toLp
@[simp]
theorem edist_toLp_toLp (f g : α → E) (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    edist (hf.toLp f) (hg.toLp g) = eLpNorm (f - 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) (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: edist (MeasureTheory.Memℒp.toLp f hf) (MeasureTheory.Memℒp.toLp g hg) = MeasureTheory.eLpNorm (f - g) p μ
-- MeasureTheory.Lp.edist_toLp_zero
@[simp]
theorem edist_toLp_zero (f : α → E) (hf : Memℒp f p μ) : edist (hf.toLp f) 0 = 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) (hf : MeasureTheory.Memℒp f p μ)
Goal: edist (MeasureTheory.Memℒp.toLp f hf) 0 = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Lp.nnnorm_zero
@[simp]
theorem nnnorm_zero : ‖(0 : Lp E p μ)‖₊ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E]
Goal: ‖0‖₊ = 0
-- MeasureTheory.Lp.norm_zero
@[simp]
theorem norm_zero : ‖(0 : Lp E p μ)‖ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E]
Goal: ‖0‖ = 0
-- MeasureTheory.Lp.norm_measure_zero
@[simp]
theorem norm_measure_zero (f : Lp E p (0 : MeasureTheory.Measure α)) : ‖f‖ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p 0))
Goal: ‖f‖ = 0
-- MeasureTheory.Lp.norm_exponent_zero
@[simp] theorem norm_exponent_zero (f : Lp E 0 μ) : ‖f‖ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E 0 μ))
Goal: ‖f‖ = 0
-- MeasureTheory.Lp.nnnorm_eq_zero_iff
theorem nnnorm_eq_zero_iff {f : Lp E p μ} (hp : 0 < p) : ‖f‖₊ = 0 ↔ f = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ↥(MeasureTheory.Lp E p μ)} (hp : 0 < p)
Goal: ‖f‖₊ = 0 ↔ f = 0
-- MeasureTheory.Lp.norm_eq_zero_iff
theorem norm_eq_zero_iff {f : Lp E p μ} (hp : 0 < p) : ‖f‖ = 0 ↔ f = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ↥(MeasureTheory.Lp E p μ)} (hp : 0 < p)
Goal: ‖f‖ = 0 ↔ f = 0
-- MeasureTheory.Lp.eq_zero_iff_ae_eq_zero
theorem eq_zero_iff_ae_eq_zero {f : Lp E p μ} : f = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : ↥(MeasureTheory.Lp E p μ)}
Goal: f = 0 ↔ ↑↑f =ᵐ[μ] 0
-- MeasureTheory.Lp.nnnorm_neg
@[simp]
theorem nnnorm_neg (f : Lp E p μ) : ‖-f‖₊ = ‖f‖₊ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖-f‖₊ = ‖f‖₊
-- MeasureTheory.Lp.norm_neg
@[simp]
theorem norm_neg (f : Lp E p μ) : ‖-f‖ = ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖-f‖ = ‖f‖
-- MeasureTheory.Lp.nnnorm_le_mul_nnnorm_of_ae_le_mul
theorem nnnorm_le_mul_nnnorm_of_ae_le_mul {c : ℝ≥0} {f : Lp E p μ} {g : Lp F p μ}
    (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) : ‖f‖₊ ≤ c * ‖g‖₊ 
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] {c : ℝ≥0} {f : ↥(MeasureTheory.Lp E p μ)} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊)
Goal: ‖f‖₊ ≤ c * ‖g‖₊
-- MeasureTheory.Lp.norm_le_mul_norm_of_ae_le_mul
theorem norm_le_mul_norm_of_ae_le_mul {c : ℝ} {f : Lp E p μ} {g : Lp F p μ}
    (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ c * ‖g x‖) : ‖f‖ ≤ c * ‖g‖ 
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] {c : ℝ} {f : ↥(MeasureTheory.Lp E p μ)} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ c * ‖↑↑g x‖)
Goal: ‖f‖ ≤ c * ‖g‖
-- MeasureTheory.Lp.norm_le_norm_of_ae_le
theorem norm_le_norm_of_ae_le {f : Lp E p μ} {g : Lp F p μ} (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) :
    ‖f‖ ≤ ‖g‖ 
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 : ↥(MeasureTheory.Lp E p μ)} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖)
Goal: ‖f‖ ≤ ‖g‖
-- MeasureTheory.Lp.mem_Lp_of_nnnorm_ae_le_mul
theorem mem_Lp_of_nnnorm_ae_le_mul {c : ℝ≥0} {f : α →ₘ[μ] E} {g : Lp F p μ}
    (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) : f ∈ Lp E 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] {c : ℝ≥0} {f : α →ₘ[μ] E} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖₊ ≤ c * ‖↑↑g x‖₊)
Goal: f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.mem_Lp_of_ae_le_mul
theorem mem_Lp_of_ae_le_mul {c : ℝ} {f : α →ₘ[μ] E} {g : Lp F p μ}
    (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ c * ‖g x‖) : f ∈ Lp E 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] {c : ℝ} {f : α →ₘ[μ] E} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖ ≤ c * ‖↑↑g x‖)
Goal: f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.mem_Lp_of_nnnorm_ae_le
theorem mem_Lp_of_nnnorm_ae_le {f : α →ₘ[μ] E} {g : Lp F p μ} (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ ‖g x‖₊) :
    f ∈ Lp E 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 →ₘ[μ] E} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖₊ ≤ ‖↑↑g x‖₊)
Goal: f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.mem_Lp_of_ae_le
theorem mem_Lp_of_ae_le {f : α →ₘ[μ] E} {g : Lp F p μ} (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) :
    f ∈ Lp E 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 →ₘ[μ] E} {g : ↥(MeasureTheory.Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖ ≤ ‖↑↑g x‖)
Goal: f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.mem_Lp_of_ae_nnnorm_bound
theorem mem_Lp_of_ae_nnnorm_bound [IsFiniteMeasure μ] {f : α →ₘ[μ] E} (C : ℝ≥0)
    (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) : f ∈ Lp E 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 →ₘ[μ] E} (C : ℝ≥0) (hfC : ∀ᵐ (x : α) ∂μ, ‖↑f x‖₊ ≤ C)
Goal: f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.mem_Lp_of_ae_bound
theorem mem_Lp_of_ae_bound [IsFiniteMeasure μ] {f : α →ₘ[μ] E} (C : ℝ) (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
    f ∈ Lp E 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 →ₘ[μ] E} (C : ℝ) (hfC : ∀ᵐ (x : α) ∂μ, ‖↑f x‖ ≤ C)
Goal: f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.nnnorm_le_of_ae_bound
theorem nnnorm_le_of_ae_bound [IsFiniteMeasure μ] {f : Lp E p μ} {C : ℝ≥0}
    (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) : ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C 
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 : ↥(MeasureTheory.Lp E p μ)} {C : ℝ≥0} (hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C)
Goal: ‖f‖₊ ≤ MeasureTheory.measureUnivNNReal μ ^ p.toReal⁻¹ * C
-- MeasureTheory.Lp.norm_le_of_ae_bound
theorem norm_le_of_ae_bound [IsFiniteMeasure μ] {f : Lp E p μ} {C : ℝ} (hC : 0 ≤ C)
    (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : ‖f‖ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C 
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 : ↥(MeasureTheory.Lp E p μ)} {C : ℝ} (hC : 0 ≤ C) (hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ C)
Goal: ‖f‖ ≤ ↑(MeasureTheory.measureUnivNNReal μ) ^ p.toReal⁻¹ * C
-- MeasureTheory.Lp.const_smul_mem_Lp
theorem const_smul_mem_Lp (c : 𝕜) (f : Lp E p μ) : c • (f : α →ₘ[μ] E) ∈ Lp E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (f : ↥(MeasureTheory.Lp E p μ))
Goal: c • ↑f ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.coe_LpSubmodule
theorem coe_LpSubmodule : (LpSubmodule E p μ 𝕜).toAddSubgroup = Lp E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E]
Goal: (MeasureTheory.Lp.LpSubmodule E p μ 𝕜).toAddSubgroup = MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.coeFn_smul
theorem coeFn_smul (c : 𝕜) (f : Lp E p μ) : ⇑(c • f) =ᵐ[μ] c • ⇑f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(c • f) =ᵐ[μ] c • ↑↑f
-- MeasureTheory.Memℒp.toLp_const_smul
theorem toLp_const_smul {f : α → E} (c : 𝕜) (hf : Memℒp f p μ) :
    (hf.const_smul c).toLp (c • f) = c • hf.toLp f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {𝕜 : Type u_5} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] {f : α → E} (c : 𝕜) (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp.toLp (c • f) ⋯ = c • MeasureTheory.Memℒp.toLp f hf
-- MeasureTheory.eLpNormEssSup_indicator_le
theorem eLpNormEssSup_indicator_le (s : Set α) (f : α → G) :
    eLpNormEssSup (s.indicator f) μ ≤ eLpNormEssSup f μ 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] (s : Set α) (f : α → G)
Goal: MeasureTheory.eLpNormEssSup (s.indicator f) μ ≤ MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNormEssSup_indicator_const_le
theorem eLpNormEssSup_indicator_const_le (s : Set α) (c : G) :
    eLpNormEssSup (s.indicator fun _ : α => c) μ ≤ ‖c‖₊ 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] (s : Set α) (c : G)
Goal: MeasureTheory.eLpNormEssSup (s.indicator fun x => c) μ ≤ ↑‖c‖₊
-- MeasureTheory.eLpNormEssSup_indicator_const_eq
theorem eLpNormEssSup_indicator_const_eq (s : Set α) (c : G) (hμs : μ s ≠ 0) :
    eLpNormEssSup (s.indicator fun _ : α => c) μ = ‖c‖₊ 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] (s : Set α) (c : G) (hμs : μ s ≠ 0)
Goal: MeasureTheory.eLpNormEssSup (s.indicator fun x => c) μ = ↑‖c‖₊
-- MeasureTheory.eLpNorm_indicator_le
theorem eLpNorm_indicator_le (f : α → E) : eLpNorm (s.indicator 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] {s : Set α} (f : α → E)
Goal: MeasureTheory.eLpNorm (s.indicator f) p μ ≤ MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_indicator_const₀
lemma eLpNorm_indicator_const₀ {c : G} (hs : NullMeasurableSet s μ) (hp : p ≠ 0) (hp_top : p ≠ ∞) :
    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖₊ * μ s ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {s : Set α} {c : G} (hs : MeasureTheory.NullMeasurableSet s μ) (hp : p ≠ 0) (hp_top : p ≠ ⊤)
Goal: MeasureTheory.eLpNorm (s.indicator fun x => c) p μ = ↑‖c‖₊ * μ s ^ (1 / p.toReal)
-- MeasureTheory.eLpNorm_indicator_const
theorem eLpNorm_indicator_const {c : G} (hs : MeasurableSet s) (hp : p ≠ 0) (hp_top : p ≠ ∞) :
    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖₊ * μ s ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {s : Set α} {c : G} (hs : MeasurableSet s) (hp : p ≠ 0) (hp_top : p ≠ ⊤)
Goal: MeasureTheory.eLpNorm (s.indicator fun x => c) p μ = ↑‖c‖₊ * μ s ^ (1 / p.toReal)
-- MeasureTheory.eLpNorm_indicator_const'
theorem eLpNorm_indicator_const' {c : G} (hs : MeasurableSet s) (hμs : μ s ≠ 0) (hp : p ≠ 0) :
    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖₊ * μ s ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {s : Set α} {c : G} (hs : MeasurableSet s) (hμs : μ s ≠ 0) (hp : p ≠ 0)
Goal: MeasureTheory.eLpNorm (s.indicator fun x => c) p μ = ↑‖c‖₊ * μ s ^ (1 / p.toReal)
-- MeasureTheory.eLpNorm_indicator_const_le
theorem eLpNorm_indicator_const_le (c : G) (p : ℝ≥0∞) :
    eLpNorm (s.indicator fun _ => c) p μ ≤ ‖c‖₊ * μ s ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup G] {s : Set α} (c : G) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm (s.indicator fun x => c) p μ ≤ ↑‖c‖₊ * μ s ^ (1 / p.toReal)
-- MeasureTheory.Memℒp.indicator
theorem Memℒp.indicator (hs : MeasurableSet s) (hf : Memℒp f p μ) : Memℒp (s.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] {f : α → E} {s : Set α} (hs : MeasurableSet s) (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (s.indicator f) p μ
-- MeasureTheory.eLpNormEssSup_indicator_eq_eLpNormEssSup_restrict
theorem eLpNormEssSup_indicator_eq_eLpNormEssSup_restrict {f : α → F} (hs : MeasurableSet s) :
    eLpNormEssSup (s.indicator f) μ = eLpNormEssSup f (μ.restrict s) 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {s : Set α} {f : α → F} (hs : MeasurableSet s)
Goal: MeasureTheory.eLpNormEssSup (s.indicator f) μ = MeasureTheory.eLpNormEssSup f (μ.restrict s)
-- MeasureTheory.eLpNorm_indicator_eq_eLpNorm_restrict
theorem eLpNorm_indicator_eq_eLpNorm_restrict {f : α → F} (hs : MeasurableSet s) :
    eLpNorm (s.indicator f) p μ = eLpNorm f p (μ.restrict s) 
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} (hs : MeasurableSet s)
Goal: MeasureTheory.eLpNorm (s.indicator f) p μ = MeasureTheory.eLpNorm f p (μ.restrict s)
-- MeasureTheory.memℒp_indicator_iff_restrict
theorem memℒp_indicator_iff_restrict (hs : MeasurableSet s) :
    Memℒp (s.indicator 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] {f : α → E} {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.Memℒp (s.indicator f) p μ ↔ MeasureTheory.Memℒp f p (μ.restrict s)
-- MeasureTheory.Memℒp.memℒp_of_exponent_le_of_measure_support_ne_top
theorem Memℒp.memℒp_of_exponent_le_of_measure_support_ne_top
    {p q : ℝ≥0∞} {f : α → E} (hfq : Memℒp f q μ) {s : Set α} (hf : ∀ x, x ∉ s → f x = 0)
    (hs : μ s ≠ ∞) (hpq : p ≤ q) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p q : ℝ≥0∞} {f : α → E} (hfq : MeasureTheory.Memℒp f q μ) {s : Set α} (hf : ∀ x ∉ s, f x = 0) (hs : μ s ≠ ⊤) (hpq : p ≤ q)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.memℒp_indicator_const
theorem memℒp_indicator_const (p : ℝ≥0∞) (hs : MeasurableSet s) (c : E) (hμsc : c = 0 ∨ μ s ≠ ∞) :
    Memℒp (s.indicator fun _ => c) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} (p : ℝ≥0∞) (hs : MeasurableSet s) (c : E) (hμsc : c = 0 ∨ μ s ≠ ⊤)
Goal: MeasureTheory.Memℒp (s.indicator fun x => c) p μ
-- MeasureTheory.exists_eLpNorm_indicator_le
theorem exists_eLpNorm_indicator_le (hp : p ≠ ∞) (c : E) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → eLpNorm (s.indicator fun _ => c) p μ ≤ ε 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hp : p ≠ ⊤) (c : E) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ η, 0 < η ∧ ∀ (s : Set α), μ s ≤ ↑η → MeasureTheory.eLpNorm (s.indicator fun x => c) p μ ≤ ε
-- MeasureTheory.Memℒp.piecewise
protected lemma Memℒp.piecewise [DecidablePred (· ∈ s)] {g}
    (hs : MeasurableSet s) (hf : Memℒp f p (μ.restrict s)) (hg : Memℒp g p (μ.restrict sᶜ)) :
    Memℒp (s.piecewise f 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 : α → E} {s : Set α} [DecidablePred fun x => x ∈ s] {g : α → E} (hs : MeasurableSet s) (hf : MeasureTheory.Memℒp f p (μ.restrict s)) (hg : MeasureTheory.Memℒp g p (μ.restrict sᶜ))
Goal: MeasureTheory.Memℒp (s.piecewise f g) p μ
-- HasCompactSupport.memℒp_of_bound
theorem _root_.HasCompactSupport.memℒp_of_bound {f : X → E} (hf : HasCompactSupport f)
    (h2f : AEStronglyMeasurable f μ) (C : ℝ) (hfC : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : Memℒp f p μ 
Arguments: {E : Type u_2} {p : ℝ≥0∞} [NormedAddCommGroup E] {X : Type u_5} [TopologicalSpace X] [MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {f : X → E} (hf : HasCompactSupport f) (h2f : MeasureTheory.AEStronglyMeasurable f μ) (C : ℝ) (hfC : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C)
Goal: MeasureTheory.Memℒp f p μ
-- Continuous.memℒp_of_hasCompactSupport
theorem _root_.Continuous.memℒp_of_hasCompactSupport [OpensMeasurableSpace X]
    {f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f) : Memℒp f p μ 
Arguments: {E : Type u_2} {p : ℝ≥0∞} [NormedAddCommGroup E] {X : Type u_5} [TopologicalSpace X] [MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [OpensMeasurableSpace X] {f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.indicatorConstLp_add
theorem indicatorConstLp_add {c' : E} :
    indicatorConstLp p hs hμs c + indicatorConstLp p hs hμs c' =
    indicatorConstLp p hs hμs (c + c') 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c c' : E}
Goal: MeasureTheory.indicatorConstLp p hs hμs c + MeasureTheory.indicatorConstLp p hs hμs c' =
  MeasureTheory.indicatorConstLp p hs hμs (c + c')
-- MeasureTheory.indicatorConstLp_sub
theorem indicatorConstLp_sub {c' : E} :
    indicatorConstLp p hs hμs c - indicatorConstLp p hs hμs c' =
    indicatorConstLp p hs hμs (c - c') 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c c' : E}
Goal: MeasureTheory.indicatorConstLp p hs hμs c - MeasureTheory.indicatorConstLp p hs hμs c' =
  MeasureTheory.indicatorConstLp p hs hμs (c - c')
-- MeasureTheory.indicatorConstLp_coeFn
theorem indicatorConstLp_coeFn : ⇑(indicatorConstLp p hs hμs c) =ᵐ[μ] s.indicator fun _ => c 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E}
Goal: ↑↑(MeasureTheory.indicatorConstLp p hs hμs c) =ᵐ[μ] s.indicator fun x => c
-- MeasureTheory.indicatorConstLp_coeFn_mem
theorem indicatorConstLp_coeFn_mem : ∀ᵐ x : α ∂μ, x ∈ s → indicatorConstLp p hs hμs c x = c 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E}
Goal: ∀ᵐ (x : α) ∂μ, x ∈ s → ↑↑(MeasureTheory.indicatorConstLp p hs hμs c) x = c
-- MeasureTheory.indicatorConstLp_coeFn_nmem
theorem indicatorConstLp_coeFn_nmem : ∀ᵐ x : α ∂μ, x ∉ s → indicatorConstLp p hs hμs c x = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E}
Goal: ∀ᵐ (x : α) ∂μ, x ∉ s → ↑↑(MeasureTheory.indicatorConstLp p hs hμs c) x = 0
-- MeasureTheory.norm_indicatorConstLp
theorem norm_indicatorConstLp (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: ‖MeasureTheory.indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)
-- MeasureTheory.norm_indicatorConstLp_top
theorem norm_indicatorConstLp_top (hμs_ne_zero : μ s ≠ 0) :
    ‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} (hμs_ne_zero : μ s ≠ 0)
Goal: ‖MeasureTheory.indicatorConstLp ⊤ hs hμs c‖ = ‖c‖
-- MeasureTheory.norm_indicatorConstLp'
theorem norm_indicatorConstLp' (hp_pos : p ≠ 0) (hμs_pos : μ s ≠ 0) :
    ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} (hp_pos : p ≠ 0) (hμs_pos : μ s ≠ 0)
Goal: ‖MeasureTheory.indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)
-- MeasureTheory.norm_indicatorConstLp_le
theorem norm_indicatorConstLp_le :
    ‖indicatorConstLp p hs hμs c‖ ≤ ‖c‖ * (μ s).toReal ^ (1 / p.toReal) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E}
Goal: ‖MeasureTheory.indicatorConstLp p hs hμs c‖ ≤ ‖c‖ * (μ s).toReal ^ (1 / p.toReal)