ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Memℒp.norm
theorem Memℒp.norm {f : α → E} (h : Memℒp f p μ) : Memℒp (fun x => ‖f x‖) 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} (h : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun x => ‖f x‖) p μ
-- MeasureTheory.memℒp_norm_iff
theorem memℒp_norm_iff {f : α → E} (hf : AEStronglyMeasurable f μ) :
    Memℒp (fun x => ‖f x‖) 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.AEStronglyMeasurable f μ)
Goal: MeasureTheory.Memℒp (fun x => ‖f x‖) p μ ↔ MeasureTheory.Memℒp f p μ
-- MeasureTheory.eLpNorm'_eq_zero_of_ae_zero
theorem eLpNorm'_eq_zero_of_ae_zero {f : α → F} (hq0_lt : 0 < q) (hf_zero : f =ᵐ[μ] 0) :
    eLpNorm' f q μ = 0 
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) (hf_zero : f =ᵐ[μ] 0)
Goal: MeasureTheory.eLpNorm' f q μ = 0
-- MeasureTheory.eLpNorm'_eq_zero_of_ae_zero'
theorem eLpNorm'_eq_zero_of_ae_zero' (hq0_ne : q ≠ 0) (hμ : μ ≠ 0) {f : α → F}
    (hf_zero : f =ᵐ[μ] 0) :
    eLpNorm' 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) {f : α → F} (hf_zero : f =ᵐ[μ] 0)
Goal: MeasureTheory.eLpNorm' f q μ = 0
-- MeasureTheory.ae_eq_zero_of_eLpNorm'_eq_zero
theorem ae_eq_zero_of_eLpNorm'_eq_zero {f : α → E} (hq0 : 0 ≤ q) (hf : AEStronglyMeasurable f μ)
    (h : eLpNorm' f q μ = 0) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hq0 : 0 ≤ q) (hf : MeasureTheory.AEStronglyMeasurable f μ) (h : MeasureTheory.eLpNorm' f q μ = 0)
Goal: f =ᵐ[μ] 0
-- MeasureTheory.eLpNorm'_eq_zero_iff
theorem eLpNorm'_eq_zero_iff (hq0_lt : 0 < q) {f : α → E} (hf : AEStronglyMeasurable f μ) :
    eLpNorm' f q μ = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hq0_lt : 0 < q) {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.eLpNorm' f q μ = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.coe_nnnorm_ae_le_eLpNormEssSup
theorem coe_nnnorm_ae_le_eLpNormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :
    ∀ᵐ x ∂μ, (‖f x‖₊ : ℝ≥0∞) ≤ eLpNormEssSup f μ 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F]
Goal: ∀ {x : MeasurableSpace α} (f : α → F) (μ : MeasureTheory.Measure α),
  ∀ᵐ (x_1 : α) ∂μ, ↑‖f x_1‖₊ ≤ MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNormEssSup_eq_zero_iff
@[simp]
theorem eLpNormEssSup_eq_zero_iff {f : α → F} : eLpNormEssSup f μ = 0 ↔ f =ᵐ[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.eLpNormEssSup f μ = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.eLpNorm_eq_zero_iff
theorem eLpNorm_eq_zero_iff {f : α → E} (hf : AEStronglyMeasurable f μ) (h0 : p ≠ 0) :
    eLpNorm f p μ = 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 : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (h0 : p ≠ 0)
Goal: MeasureTheory.eLpNorm f p μ = 0 ↔ f =ᵐ[μ] 0
-- MeasureTheory.ae_le_eLpNormEssSup
theorem ae_le_eLpNormEssSup {f : α → F} : ∀ᵐ y ∂μ, ‖f y‖₊ ≤ 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: ∀ᵐ (y : α) ∂μ, ↑‖f y‖₊ ≤ MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.meas_eLpNormEssSup_lt
theorem meas_eLpNormEssSup_lt {f : α → F} : μ { y | eLpNormEssSup f μ < ‖f y‖₊ } = 0 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F}
Goal: μ {y | MeasureTheory.eLpNormEssSup f μ < ↑‖f y‖₊} = 0
-- MeasureTheory.eLpNormEssSup_piecewise
lemma eLpNormEssSup_piecewise {s : Set α} (f g : α → E) [DecidablePred (· ∈ s)]
    (hs : MeasurableSet s) :
    eLpNormEssSup (Set.piecewise s f g) μ
      = max (eLpNormEssSup f (μ.restrict s)) (eLpNormEssSup g (μ.restrict sᶜ)) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} (f g : α → E) [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s)
Goal: MeasureTheory.eLpNormEssSup (s.piecewise f g) μ =
  max (MeasureTheory.eLpNormEssSup f (μ.restrict s)) (MeasureTheory.eLpNormEssSup g (μ.restrict sᶜ))
-- MeasureTheory.eLpNorm_top_piecewise
lemma eLpNorm_top_piecewise {s : Set α} (f g : α → E) [DecidablePred (· ∈ s)]
    (hs : MeasurableSet s) :
    eLpNorm (Set.piecewise s f g) ∞ μ
      = max (eLpNorm f ∞ (μ.restrict s)) (eLpNorm g ∞ (μ.restrict sᶜ)) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {s : Set α} (f g : α → E) [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s)
Goal: MeasureTheory.eLpNorm (s.piecewise f g) ⊤ μ =
  max (MeasureTheory.eLpNorm f ⊤ (μ.restrict s)) (MeasureTheory.eLpNorm g ⊤ (μ.restrict sᶜ))
-- MeasureTheory.eLpNormEssSup_map_measure
theorem eLpNormEssSup_map_measure (hg : AEStronglyMeasurable g (Measure.map f μ))
    (hf : AEMeasurable f μ) : eLpNormEssSup g (Measure.map f μ) = eLpNormEssSup (g ∘ f) μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.eLpNormEssSup g (MeasureTheory.Measure.map f μ) = MeasureTheory.eLpNormEssSup (g ∘ f) μ
-- MeasureTheory.eLpNorm_map_measure
theorem eLpNorm_map_measure (hg : AEStronglyMeasurable g (Measure.map f μ))
    (hf : AEMeasurable f μ) : eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.eLpNorm g p (MeasureTheory.Measure.map f μ) = MeasureTheory.eLpNorm (g ∘ f) p μ
-- MeasureTheory.memℒp_map_measure_iff
theorem memℒp_map_measure_iff (hg : AEStronglyMeasurable g (Measure.map f μ))
    (hf : AEMeasurable f μ) : Memℒp g p (Measure.map f μ) ↔ Memℒp (g ∘ f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.Memℒp g p (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.Memℒp (g ∘ f) p μ
-- MeasureTheory.Memℒp.comp_of_map
theorem Memℒp.comp_of_map (hg : Memℒp g p (Measure.map f μ)) (hf : AEMeasurable f μ) :
    Memℒp (g ∘ f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E} (hg : MeasureTheory.Memℒp g p (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: MeasureTheory.Memℒp (g ∘ f) p μ
-- MeasureTheory.eLpNorm_comp_measurePreserving
theorem eLpNorm_comp_measurePreserving {ν : MeasureTheory.Measure β} (hg : AEStronglyMeasurable g ν)
    (hf : MeasurePreserving f μ ν) : eLpNorm (g ∘ f) p μ = eLpNorm g p ν 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E} {ν : MeasureTheory.Measure β} (hg : MeasureTheory.AEStronglyMeasurable g ν) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: MeasureTheory.eLpNorm (g ∘ f) p μ = MeasureTheory.eLpNorm g p ν
-- MeasureTheory.AEEqFun.eLpNorm_compMeasurePreserving
theorem AEEqFun.eLpNorm_compMeasurePreserving {ν : MeasureTheory.Measure β} (g : β →ₘ[ν] E)
    (hf : MeasurePreserving f μ ν) :
    eLpNorm (g.compMeasurePreserving f hf) p μ = eLpNorm g p ν 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {ν : MeasureTheory.Measure β} (g : β →ₘ[ν] E) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: MeasureTheory.eLpNorm (↑(g.compMeasurePreserving f hf)) p μ = MeasureTheory.eLpNorm (↑g) p ν
-- MeasureTheory.Memℒp.comp_measurePreserving
theorem Memℒp.comp_measurePreserving {ν : MeasureTheory.Measure β} (hg : Memℒp g p ν)
    (hf : MeasurePreserving f μ ν) : Memℒp (g ∘ f) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E} {ν : MeasureTheory.Measure β} (hg : MeasureTheory.Memℒp g p ν) (hf : MeasureTheory.MeasurePreserving f μ ν)
Goal: MeasureTheory.Memℒp (g ∘ f) p μ
-- MeasurableEmbedding.eLpNormEssSup_map_measure
theorem _root_.MeasurableEmbedding.eLpNormEssSup_map_measure {g : β → F}
    (hf : MeasurableEmbedding f) : eLpNormEssSup g (Measure.map f μ) = eLpNormEssSup (g ∘ f) μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → F} (hf : MeasurableEmbedding f)
Goal: MeasureTheory.eLpNormEssSup g (MeasureTheory.Measure.map f μ) = MeasureTheory.eLpNormEssSup (g ∘ f) μ
-- MeasurableEmbedding.eLpNorm_map_measure
theorem _root_.MeasurableEmbedding.eLpNorm_map_measure {g : β → F} (hf : MeasurableEmbedding f) :
    eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → F} (hf : MeasurableEmbedding f)
Goal: MeasureTheory.eLpNorm g p (MeasureTheory.Measure.map f μ) = MeasureTheory.eLpNorm (g ∘ f) p μ
-- MeasurableEmbedding.memℒp_map_measure_iff
theorem _root_.MeasurableEmbedding.memℒp_map_measure_iff {g : β → F} (hf : MeasurableEmbedding f) :
    Memℒp g p (Measure.map f μ) ↔ Memℒp (g ∘ f) p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → F} (hf : MeasurableEmbedding f)
Goal: MeasureTheory.Memℒp g p (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.Memℒp (g ∘ f) p μ
-- MeasurableEquiv.memℒp_map_measure_iff
theorem _root_.MeasurableEquiv.memℒp_map_measure_iff (f : α ≃ᵐ β) {g : β → F} :
    Memℒp g p (Measure.map f μ) ↔ Memℒp (g ∘ f) p μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {β : Type u_5} {mβ : MeasurableSpace β} (f : α ≃ᵐ β) {g : β → F}
Goal: MeasureTheory.Memℒp g p (MeasureTheory.Measure.map (⇑f) μ) ↔ MeasureTheory.Memℒp (g ∘ ⇑f) p μ
-- MeasureTheory.eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul
theorem eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ≥0}
    (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) {p : ℝ} (hp : 0 < p) :
    eLpNorm' f p μ ≤ c • eLpNorm' g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} {c : ℝ≥0} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) {p : ℝ} (hp : 0 < p)
Goal: MeasureTheory.eLpNorm' f p μ ≤ c • MeasureTheory.eLpNorm' g p μ
-- MeasureTheory.eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul
theorem eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ≥0}
    (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) : eLpNormEssSup f μ ≤ c • eLpNormEssSup g μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} {c : ℝ≥0} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊)
Goal: MeasureTheory.eLpNormEssSup f μ ≤ c • MeasureTheory.eLpNormEssSup g μ
-- MeasureTheory.eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul
theorem eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ≥0}
    (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) (p : ℝ≥0∞) : eLpNorm f p μ ≤ c • eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} {c : ℝ≥0} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f p μ ≤ c • MeasureTheory.eLpNorm g p μ
-- _private.Mathlib.MeasureTheory.Function.LpSeminorm.Basic.0.MeasureTheory.le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg
private theorem le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg {α} [LinearOrderedSemiring α]
    {a b c : α} (ha : 0 ≤ a) (hb : b < 0) (hc : 0 ≤ c) : a ≤ b * c ↔ a = 0 ∧ c = 0 
Arguments: {α : Type u_5} [LinearOrderedSemiring α] {a b c : α} (ha : 0 ≤ a) (hb : b < 0) (hc : 0 ≤ c)
Goal: a ≤ b * c ↔ a = 0 ∧ c = 0
-- MeasureTheory.eLpNorm_eq_zero_and_zero_of_ae_le_mul_neg
theorem eLpNorm_eq_zero_and_zero_of_ae_le_mul_neg {f : α → F} {g : α → G} {c : ℝ}
    (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ c * ‖g x‖) (hc : c < 0) (p : ℝ≥0∞) :
    eLpNorm f p μ = 0 ∧ eLpNorm g p μ = 0 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} {c : ℝ} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c * ‖g x‖) (hc : c < 0) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f p μ = 0 ∧ MeasureTheory.eLpNorm g p μ = 0
-- MeasureTheory.eLpNorm_le_mul_eLpNorm_of_ae_le_mul
theorem eLpNorm_le_mul_eLpNorm_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ}
    (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ c * ‖g x‖) (p : ℝ≥0∞) :
    eLpNorm f p μ ≤ ENNReal.ofReal c * eLpNorm g p μ 
Arguments: {α : Type u_1} {F : Type u_3} {G : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] [NormedAddCommGroup G] {f : α → F} {g : α → G} {c : ℝ} (h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c * ‖g x‖) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm f p μ ≤ ENNReal.ofReal c * MeasureTheory.eLpNorm g p μ
-- MeasureTheory.Memℒp.of_nnnorm_le_mul
theorem Memℒp.of_nnnorm_le_mul {f : α → E} {g : α → F} {c : ℝ≥0} (hg : Memℒp g p μ)
    (hf : AEStronglyMeasurable f μ) (hfg : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖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} {c : ℝ≥0} (hg : MeasureTheory.Memℒp g p μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.Memℒp.of_le_mul
theorem Memℒp.of_le_mul {f : α → E} {g : α → F} {c : ℝ} (hg : Memℒp g p μ)
    (hf : AEStronglyMeasurable f μ) (hfg : ∀ᵐ x ∂μ, ‖f x‖ ≤ c * ‖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} {c : ℝ} (hg : MeasureTheory.Memℒp g p μ) (hf : MeasureTheory.AEStronglyMeasurable f μ) (hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c * ‖g x‖)
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.eLpNorm'_const_smul_le
theorem eLpNorm'_const_smul_le (c : 𝕜) (f : α → F) (hq_pos : 0 < q) :
    eLpNorm' (c • f) q μ ≤ ‖c‖₊ • eLpNorm' f q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {𝕜 : Type u_5} [NormedRing 𝕜] [MulActionWithZero 𝕜 F] [BoundedSMul 𝕜 F] (c : 𝕜) (f : α → F) (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm' (c • f) q μ ≤ ‖c‖₊ • MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.eLpNormEssSup_const_smul_le
theorem eLpNormEssSup_const_smul_le (c : 𝕜) (f : α → F) :
    eLpNormEssSup (c • f) μ ≤ ‖c‖₊ • eLpNormEssSup f μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {𝕜 : Type u_5} [NormedRing 𝕜] [MulActionWithZero 𝕜 F] [BoundedSMul 𝕜 F] (c : 𝕜) (f : α → F)
Goal: MeasureTheory.eLpNormEssSup (c • f) μ ≤ ‖c‖₊ • MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNorm_const_smul_le
theorem eLpNorm_const_smul_le (c : 𝕜) (f : α → F) : eLpNorm (c • f) p μ ≤ ‖c‖₊ • 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] {𝕜 : Type u_5} [NormedRing 𝕜] [MulActionWithZero 𝕜 F] [BoundedSMul 𝕜 F] (c : 𝕜) (f : α → F)
Goal: MeasureTheory.eLpNorm (c • f) p μ ≤ ‖c‖₊ • MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Memℒp.const_smul
theorem Memℒp.const_smul {f : α → E} (hf : Memℒp f p μ) (c : 𝕜) : Memℒp (c • f) 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] [MulActionWithZero 𝕜 E] [BoundedSMul 𝕜 E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ) (c : 𝕜)
Goal: MeasureTheory.Memℒp (c • f) p μ
-- MeasureTheory.Memℒp.const_mul
theorem Memℒp.const_mul {R} [NormedRing R] {f : α → R} (hf : Memℒp f p μ) (c : R) :
    Memℒp (fun x => c * f x) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {R : Type u_6} [NormedRing R] {f : α → R} (hf : MeasureTheory.Memℒp f p μ) (c : R)
Goal: MeasureTheory.Memℒp (fun x => c * f x) p μ
-- MeasureTheory.eLpNorm'_const_smul
theorem eLpNorm'_const_smul {f : α → F} (c : 𝕜) (hq_pos : 0 < q) :
    eLpNorm' (c • f) q μ = ‖c‖₊ • eLpNorm' f q μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {𝕜 : Type u_5} [NormedDivisionRing 𝕜] [Module 𝕜 F] [BoundedSMul 𝕜 F] {f : α → F} (c : 𝕜) (hq_pos : 0 < q)
Goal: MeasureTheory.eLpNorm' (c • f) q μ = ‖c‖₊ • MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.eLpNormEssSup_const_smul
theorem eLpNormEssSup_const_smul (c : 𝕜) (f : α → F) :
    eLpNormEssSup (c • f) μ = (‖c‖₊ : ℝ≥0∞) * eLpNormEssSup f μ 
Arguments: {α : Type u_1} {F : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {𝕜 : Type u_5} [NormedDivisionRing 𝕜] [Module 𝕜 F] [BoundedSMul 𝕜 F] (c : 𝕜) (f : α → F)
Goal: MeasureTheory.eLpNormEssSup (c • f) μ = ↑‖c‖₊ * MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNorm_const_smul
theorem eLpNorm_const_smul (c : 𝕜) (f : α → F) :
    eLpNorm (c • f) p μ = (‖c‖₊ : ℝ≥0∞) * 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] {𝕜 : Type u_5} [NormedDivisionRing 𝕜] [Module 𝕜 F] [BoundedSMul 𝕜 F] (c : 𝕜) (f : α → F)
Goal: MeasureTheory.eLpNorm (c • f) p μ = ↑‖c‖₊ * MeasureTheory.eLpNorm f p μ
-- MeasureTheory.le_eLpNorm_of_bddBelow
theorem le_eLpNorm_of_bddBelow (hp : p ≠ 0) (hp' : p ≠ ∞) {f : α → F} (C : ℝ≥0) {s : Set α}
    (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → C ≤ ‖f x‖₊) :
    C • μ s ^ (1 / p.toReal) ≤ 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] (hp : p ≠ 0) (hp' : p ≠ ⊤) {f : α → F} (C : ℝ≥0) {s : Set α} (hs : MeasurableSet s) (hf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖f x‖₊)
Goal: C • μ s ^ (1 / p.toReal) ≤ MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Memℒp.re
theorem Memℒp.re (hf : Memℒp f p μ) : Memℒp (fun x => RCLike.re (f x)) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [RCLike 𝕜] {f : α → 𝕜} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun x => RCLike.re (f x)) p μ
-- MeasureTheory.Memℒp.im
theorem Memℒp.im (hf : Memℒp f p μ) : Memℒp (fun x => RCLike.im (f x)) p μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {𝕜 : Type u_5} [RCLike 𝕜] {f : α → 𝕜} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun x => RCLike.im (f x)) p μ
-- MeasureTheory.ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd
theorem ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd {p : ℝ≥0∞} {f : ℕ → α → E}
    (hfmeas : ∀ n, Measurable (f n)) (hbdd : ∀ n, eLpNorm (f n) p μ ≤ R) :
    ∀ᵐ x ∂μ, liminf (fun n => ((‖f n x‖₊ : ℝ≥0∞) ^ p.toReal : ℝ≥0∞)) atTop < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [MeasurableSpace E] [OpensMeasurableSpace E] {R : ℝ≥0} {p : ℝ≥0∞} {f : ℕ → α → E} (hfmeas : ∀ (n : ℕ), Measurable (f n)) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) p μ ≤ ↑R)
Goal: ∀ᵐ (x : α) ∂μ, Filter.liminf (fun n => ↑‖f n x‖₊ ^ p.toReal) Filter.atTop < ⊤
-- MeasureTheory.ae_bdd_liminf_atTop_of_eLpNorm_bdd
theorem ae_bdd_liminf_atTop_of_eLpNorm_bdd {p : ℝ≥0∞} (hp : p ≠ 0) {f : ℕ → α → E}
    (hfmeas : ∀ n, Measurable (f n)) (hbdd : ∀ n, eLpNorm (f n) p μ ≤ R) :
    ∀ᵐ x ∂μ, liminf (fun n => (‖f n x‖₊ : ℝ≥0∞)) atTop < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [MeasurableSpace E] [OpensMeasurableSpace E] {R : ℝ≥0} {p : ℝ≥0∞} (hp : p ≠ 0) {f : ℕ → α → E} (hfmeas : ∀ (n : ℕ), Measurable (f n)) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) p μ ≤ ↑R)
Goal: ∀ᵐ (x : α) ∂μ, Filter.liminf (fun n => ↑‖f n x‖₊) Filter.atTop < ⊤
-- Continuous.memℒp_top_of_hasCompactSupport
theorem _root_.Continuous.memℒp_top_of_hasCompactSupport
    {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
    {f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f) (μ : Measure X) : Memℒp f ⊤ μ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] {X : Type u_5} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] {f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f) (μ : MeasureTheory.Measure X)
Goal: MeasureTheory.Memℒp f ⊤ μ
-- MeasureTheory.Memℒp.exists_eLpNorm_indicator_compl_lt
theorem Memℒp.exists_eLpNorm_indicator_compl_lt {β : Type*} [NormedAddCommGroup β] (hp_top : p ≠ ∞)
    {f : α → β} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ s : Set α, MeasurableSet s ∧ μ s < ∞ ∧ eLpNorm (sᶜ.indicator f) p μ < ε 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {β : Type u_5} [NormedAddCommGroup β] (hp_top : p ≠ ⊤) {f : α → β} (hf : MeasureTheory.Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ s, MeasurableSet s ∧ μ s < ⊤ ∧ MeasureTheory.eLpNorm (sᶜ.indicator f) p μ < ε
-- MeasurableSpace.map_def
lemma map_def {s : Set β} : MeasurableSet[m.map f] s ↔ MeasurableSet[m] (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : α → β} {s : Set β}
Goal: MeasurableSet s ↔ MeasurableSet (f ⁻¹' s)
-- MeasurableSpace.map_id
@[simp]
theorem map_id : m.map id = m 
Arguments: {α : Type u_1} {m : MeasurableSpace α}
Goal: MeasurableSpace.map id m = m
-- MeasurableSpace.map_comp
@[simp]
theorem map_comp {f : α → β} {g : β → γ} : (m.map f).map g = m.map (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {f : α → β} {g : β → γ}
Goal: MeasurableSpace.map g (MeasurableSpace.map f m) = MeasurableSpace.map (g ∘ f) m
-- MeasurableSpace.comap_eq_generateFrom
theorem comap_eq_generateFrom (m : MeasurableSpace β) (f : α → β) :
    m.comap f = generateFrom { t | ∃ s, MeasurableSet s ∧ f ⁻¹' s = t } 
Arguments: {α : Type u_1} {β : Type u_2} (m : MeasurableSpace β) (f : α → β)
Goal: MeasurableSpace.comap f m = MeasurableSpace.generateFrom {t | ∃ s, MeasurableSet s ∧ f ⁻¹' s = t}
-- MeasurableSpace.comap_id
@[simp]
theorem comap_id : m.comap id = m 
Arguments: {α : Type u_1} {m : MeasurableSpace α}
Goal: MeasurableSpace.comap id m = m
-- MeasurableSpace.comap_comp
@[simp]
theorem comap_comp {f : β → α} {g : γ → β} : (m.comap f).comap g = m.comap (f ∘ g) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {f : β → α} {g : γ → β}
Goal: MeasurableSpace.comap g (MeasurableSpace.comap f m) = MeasurableSpace.comap (f ∘ g) m