ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.nnnorm_indicatorConstLp_le
theorem nnnorm_indicatorConstLp_le :
    ‖indicatorConstLp p hs hμs c‖₊ ≤ ‖c‖₊ * (μ s).toNNReal ^ (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).toNNReal ^ (1 / p.toReal)
-- MeasureTheory.ennnorm_indicatorConstLp_le
theorem ennnorm_indicatorConstLp_le :
    (‖indicatorConstLp p hs hμs c‖₊ : ℝ≥0∞) ≤ ‖c‖₊ * (μ s) ^ (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 ^ (1 / p.toReal)
-- MeasureTheory.edist_indicatorConstLp_eq_nnnorm
theorem edist_indicatorConstLp_eq_nnnorm {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ∞} :
    edist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) =
      ‖indicatorConstLp p (hs.symmDiff ht) (measure_symmDiff_ne_top hμs hμt) 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} {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ⊤}
Goal: edist (MeasureTheory.indicatorConstLp p hs hμs c) (MeasureTheory.indicatorConstLp p ht hμt c) =
  ↑‖MeasureTheory.indicatorConstLp p ⋯ ⋯ c‖₊
-- MeasureTheory.dist_indicatorConstLp_eq_norm
theorem dist_indicatorConstLp_eq_norm {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ∞} :
    dist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) =
      ‖indicatorConstLp p (hs.symmDiff ht) (measure_symmDiff_ne_top hμs hμt) 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} {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ⊤}
Goal: dist (MeasureTheory.indicatorConstLp p hs hμs c) (MeasureTheory.indicatorConstLp p ht hμt c) =
  ‖MeasureTheory.indicatorConstLp p ⋯ ⋯ c‖
-- MeasureTheory.tendsto_indicatorConstLp_set
theorem tendsto_indicatorConstLp_set [hp₁ : Fact (1 ≤ p)] {β : Type*} {l : Filter β} {t : β → Set α}
    {ht : ∀ b, MeasurableSet (t b)} {hμt : ∀ b, μ (t b) ≠ ∞} (hp : p ≠ ∞)
    (h : Tendsto (fun b ↦ μ (t b ∆ s)) l (𝓝 0)) :
    Tendsto (fun b ↦ indicatorConstLp p (ht b) (hμt b) c) l (𝓝 (indicatorConstLp p hs hμs 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} [hp₁ : Fact (1 ≤ p)] {β : Type u_5} {l : Filter β} {t : β → Set α} {ht : ∀ (b : β), MeasurableSet (t b)} {hμt : ∀ (b : β), μ (t b) ≠ ⊤} (hp : p ≠ ⊤) (h : Filter.Tendsto (fun b => μ (t b ∆ s)) l (𝓝 0))
Goal: Filter.Tendsto (fun b => MeasureTheory.indicatorConstLp p ⋯ ⋯ c) l (𝓝 (MeasureTheory.indicatorConstLp p hs hμs c))
-- MeasureTheory.continuous_indicatorConstLp_set
theorem continuous_indicatorConstLp_set [Fact (1 ≤ p)] {X : Type*} [TopologicalSpace X]
    {s : X → Set α} {hs : ∀ x, MeasurableSet (s x)} {hμs : ∀ x, μ (s x) ≠ ∞} (hp : p ≠ ∞)
    (h : ∀ x, Tendsto (fun y ↦ μ (s y ∆ s x)) (𝓝 x) (𝓝 0)) :
    Continuous fun x ↦ indicatorConstLp p (hs x) (hμs x) c 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {c : E} [Fact (1 ≤ p)] {X : Type u_5} [TopologicalSpace X] {s : X → Set α} {hs : ∀ (x : X), MeasurableSet (s x)} {hμs : ∀ (x : X), μ (s x) ≠ ⊤} (hp : p ≠ ⊤) (h : ∀ (x : X), Filter.Tendsto (fun y => μ (s y ∆ s x)) (𝓝 x) (𝓝 0))
Goal: Continuous fun x => MeasureTheory.indicatorConstLp p ⋯ ⋯ c
-- MeasureTheory.indicatorConstLp_empty
@[simp]
theorem indicatorConstLp_empty :
    indicatorConstLp p MeasurableSet.empty (by simp : μ ∅ ≠ ∞) c = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {c : E}
Goal: MeasureTheory.indicatorConstLp p ⋯ ⋯ c = 0
-- MeasureTheory.indicatorConstLp_inj
theorem indicatorConstLp_inj {s t : Set α} (hs : MeasurableSet s) (hsμ : μ s ≠ ∞)
    (ht : MeasurableSet t) (htμ : μ t ≠ ∞) {c : E} (hc : c ≠ 0) :
    indicatorConstLp p hs hsμ c = indicatorConstLp p ht htμ c ↔ s =ᵐ[μ] t 
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} (hs : MeasurableSet s) (hsμ : μ s ≠ ⊤) (ht : MeasurableSet t) (htμ : μ t ≠ ⊤) {c : E} (hc : c ≠ 0)
Goal: MeasureTheory.indicatorConstLp p hs hsμ c = MeasureTheory.indicatorConstLp p ht htμ c ↔ s =ᵐ[μ] t
-- MeasureTheory.memℒp_add_of_disjoint
theorem memℒp_add_of_disjoint {f g : α → E} (h : Disjoint (support f) (support g))
    (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    Memℒp (f + g) p μ ↔ 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} (h : Disjoint (Function.support f) (Function.support g)) (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.Memℒp (f + g) p μ ↔ MeasureTheory.Memℒp f p μ ∧ MeasureTheory.Memℒp g p μ
-- MeasureTheory.indicatorConstLp_disjoint_union
theorem indicatorConstLp_disjoint_union {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t)
    (hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (c : E) :
    indicatorConstLp p (hs.union ht) (measure_union_ne_top hμs hμt) c =
      indicatorConstLp p hs hμs c + indicatorConstLp p ht hμt c 
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} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (hst : s ∩ t = ∅) (c : E)
Goal: MeasureTheory.indicatorConstLp p ⋯ ⋯ c =
  MeasureTheory.indicatorConstLp p hs hμs c + MeasureTheory.indicatorConstLp p ht hμt c
-- MeasureTheory.Lp.coeFn_const
lemma Lp.coeFn_const : Lp.const p μ c =ᵐ[μ] Function.const α 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] (c : E)
Goal: ↑↑((MeasureTheory.Lp.const p μ) c) =ᵐ[μ] Function.const α c
-- MeasureTheory.Lp.const_val
@[simp] lemma Lp.const_val : (Lp.const p μ c).1 = AEEqFun.const α 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] (c : E)
Goal: ↑((MeasureTheory.Lp.const p μ) c) = MeasureTheory.AEEqFun.const α c
-- MeasureTheory.Memℒp.toLp_const
@[simp]
lemma Memℒp.toLp_const : Memℒp.toLp _ (memℒp_const c) = Lp.const p μ 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] (c : E)
Goal: MeasureTheory.Memℒp.toLp (fun x => c) ⋯ = (MeasureTheory.Lp.const p μ) c
-- MeasureTheory.indicatorConstLp_univ
@[simp]
lemma indicatorConstLp_univ :
    indicatorConstLp p .univ (measure_ne_top μ _) c = Lp.const p μ 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] (c : E)
Goal: MeasureTheory.indicatorConstLp p ⋯ ⋯ c = (MeasureTheory.Lp.const p μ) c
-- MeasureTheory.Lp.norm_const
theorem Lp.norm_const [NeZero μ] (hp_zero : p ≠ 0) :
    ‖Lp.const p μ c‖ = ‖c‖ * (μ Set.univ).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] [MeasureTheory.IsFiniteMeasure μ] (c : E) [NeZero μ] (hp_zero : p ≠ 0)
Goal: ‖(MeasureTheory.Lp.const p μ) c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
-- MeasureTheory.Lp.norm_const'
theorem Lp.norm_const' (hp_zero : p ≠ 0) (hp_top : p ≠ ∞) :
    ‖Lp.const p μ c‖ = ‖c‖ * (μ Set.univ).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] [MeasureTheory.IsFiniteMeasure μ] (c : E) (hp_zero : p ≠ 0) (hp_top : p ≠ ⊤)
Goal: ‖(MeasureTheory.Lp.const p μ) c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
-- MeasureTheory.Lp.norm_const_le
theorem Lp.norm_const_le : ‖Lp.const p μ c‖ ≤ ‖c‖ * (μ Set.univ).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] [MeasureTheory.IsFiniteMeasure μ] (c : E)
Goal: ‖(MeasureTheory.Lp.const p μ) c‖ ≤ ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)
-- MeasureTheory.Lp.constₗ_apply
@[simps] protected def Lp.constₗ (𝕜 : Type*) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] :
    E →ₗ[𝕜] 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] (𝕜 : Type u_5) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (a : E)
Goal: (MeasureTheory.Lp.constₗ p μ 𝕜) a = (MeasureTheory.Lp.const p μ) a
-- MeasureTheory.Lp.constL_apply
@[simps! apply]
protected def Lp.constL (𝕜 : Type*) [NormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)] :
    E →L[𝕜] 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] (𝕜 : Type u_5) [NormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)] (a : E)
Goal: (MeasureTheory.Lp.constL p μ 𝕜) a = (MeasureTheory.Lp.const p μ) a
-- MeasureTheory.Lp.norm_constL_le
theorem Lp.norm_constL_le (𝕜 : Type*) [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    [Fact (1 ≤ p)] :
    ‖(Lp.constL p μ 𝕜 : E →L[𝕜] Lp E p μ)‖ ≤ (μ Set.univ).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] [MeasureTheory.IsFiniteMeasure μ] (𝕜 : Type u_5) [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)]
Goal: ‖MeasureTheory.Lp.constL p μ 𝕜‖ ≤ (μ Set.univ).toReal ^ (1 / p.toReal)
-- MeasureTheory.Memℒp.norm_rpow_div
theorem Memℒp.norm_rpow_div {f : α → E} (hf : Memℒp f p μ) (q : ℝ≥0∞) :
    Memℒp (fun x : α => ‖f x‖ ^ q.toReal) (p / q) 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) (q : ℝ≥0∞)
Goal: MeasureTheory.Memℒp (fun x => ‖f x‖ ^ q.toReal) (p / q) μ
-- MeasureTheory.memℒp_norm_rpow_iff
theorem memℒp_norm_rpow_iff {q : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) (q_zero : q ≠ 0)
    (q_top : q ≠ ∞) : Memℒp (fun x : α => ‖f x‖ ^ q.toReal) (p / q) 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] {q : ℝ≥0∞} {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (q_zero : q ≠ 0) (q_top : q ≠ ⊤)
Goal: MeasureTheory.Memℒp (fun x => ‖f x‖ ^ q.toReal) (p / q) μ ↔ MeasureTheory.Memℒp f p μ
-- MeasureTheory.Memℒp.norm_rpow
theorem Memℒp.norm_rpow {f : α → E} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
    Memℒp (fun x : α => ‖f x‖ ^ p.toReal) 1 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) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤)
Goal: MeasureTheory.Memℒp (fun x => ‖f x‖ ^ p.toReal) 1 μ
-- MeasureTheory.AEEqFun.compMeasurePreserving_mem_Lp
theorem AEEqFun.compMeasurePreserving_mem_Lp {β : Type*} [MeasurableSpace β]
    {μb : MeasureTheory.Measure β} {g : β →ₘ[μb] E} (hg : g ∈ Lp E p μb) {f : α → β}
    (hf : MeasurePreserving f μ μb) :
    g.compMeasurePreserving f hf ∈ 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] {β : Type u_5} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {g : β →ₘ[μb] E} (hg : g ∈ MeasureTheory.Lp E p μb) {f : α → β} (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: g.compMeasurePreserving f hf ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.compMeasurePreserving_val
@[simp]
theorem compMeasurePreserving_val (g : Lp E p μb) (hf : MeasurePreserving f μ μb) :
    (compMeasurePreserving f hf g).1 = g.1.compMeasurePreserving f hf 
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} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} (g : ↥(MeasureTheory.Lp E p μb)) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: ↑((MeasureTheory.Lp.compMeasurePreserving f hf) g) = (↑g).compMeasurePreserving f hf
-- MeasureTheory.Lp.coeFn_compMeasurePreserving
theorem coeFn_compMeasurePreserving (g : Lp E p μb) (hf : MeasurePreserving f μ μb) :
    compMeasurePreserving f hf g =ᵐ[μ] g ∘ f 
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} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} (g : ↥(MeasureTheory.Lp E p μb)) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: ↑↑((MeasureTheory.Lp.compMeasurePreserving f hf) g) =ᵐ[μ] ↑↑g ∘ f
-- MeasureTheory.Lp.norm_compMeasurePreserving
@[simp]
theorem norm_compMeasurePreserving (g : Lp E p μb) (hf : MeasurePreserving f μ μb) :
    ‖compMeasurePreserving f hf g‖ = ‖g‖ 
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} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} (g : ↥(MeasureTheory.Lp E p μb)) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: ‖(MeasureTheory.Lp.compMeasurePreserving f hf) g‖ = ‖g‖
-- MeasureTheory.Lp.isometry_compMeasurePreserving
theorem isometry_compMeasurePreserving [Fact (1 ≤ p)] (hf : MeasurePreserving f μ μb) :
    Isometry (compMeasurePreserving f hf : Lp E p μb → 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] {β : Type u_5} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} [Fact (1 ≤ p)] (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: Isometry ⇑(MeasureTheory.Lp.compMeasurePreserving f hf)
-- MeasureTheory.Lp.toLp_compMeasurePreserving
theorem toLp_compMeasurePreserving {g : β → E} (hg : Memℒp g p μb) (hf : MeasurePreserving f μ μb) :
    compMeasurePreserving f hf (hg.toLp g) = (hg.comp_measurePreserving hf).toLp _ 
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} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} {g : β → E} (hg : MeasureTheory.Memℒp g p μb) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: (MeasureTheory.Lp.compMeasurePreserving f hf) (MeasureTheory.Memℒp.toLp g hg) = MeasureTheory.Memℒp.toLp (g ∘ f) ⋯
-- MeasureTheory.Lp.indicatorConstLp_compMeasurePreserving
theorem indicatorConstLp_compMeasurePreserving {s : Set β} (hs : MeasurableSet s)
    (hμs : μb s ≠ ∞) (c : E) (hf : MeasurePreserving f μ μb) :
    Lp.compMeasurePreserving f hf (indicatorConstLp p hs hμs c) =
      indicatorConstLp p (hs.preimage hf.measurable)
        (by rwa [hf.measure_preimage hs.nullMeasurableSet]) c 
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} [MeasurableSpace β] {μb : MeasureTheory.Measure β} {f : α → β} {s : Set β} (hs : MeasurableSet s) (hμs : μb s ≠ ⊤) (c : E) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: (MeasureTheory.Lp.compMeasurePreserving f hf) (MeasureTheory.indicatorConstLp p hs hμs c) =
  MeasureTheory.indicatorConstLp p ⋯ ⋯ c
-- MeasureTheory.Lp.compMeasurePreservingₗ_apply
@[simps]
def compMeasurePreservingₗ (f : α → β) (hf : MeasurePreserving f μ μb) :
    Lp E p μb →ₗ[𝕜] 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] {β : Type u_5} [MeasurableSpace β] {μb : MeasureTheory.Measure β} (𝕜 : Type u_6) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : α → β) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: ∀ (a : ↥(MeasureTheory.Lp E p μb)),
  (MeasureTheory.Lp.compMeasurePreservingₗ 𝕜 f hf) a = (↑(MeasureTheory.Lp.compMeasurePreserving f hf)).toFun a
-- MeasureTheory.Lp.compMeasurePreservingₗᵢ_apply_coe
@[simps!]
def compMeasurePreservingₗᵢ [Fact (1 ≤ p)] (f : α → β) (hf : MeasurePreserving f μ μb) :
    Lp E p μb →ₗᵢ[𝕜] 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] {β : Type u_5} [MeasurableSpace β] {μb : MeasureTheory.Measure β} (𝕜 : Type u_6) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Fact (1 ≤ p)] (f : α → β) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: ∀ (a : ↥(MeasureTheory.Lp E p μb)),
  ↑((MeasureTheory.Lp.compMeasurePreservingₗᵢ 𝕜 f hf) a) = (↑a).compMeasurePreserving f hf
-- MeasureTheory.Lp.compMeasurePreservingₗᵢ_toFun_coe
@[simps!]
def compMeasurePreservingₗᵢ [Fact (1 ≤ p)] (f : α → β) (hf : MeasurePreserving f μ μb) :
    Lp E p μb →ₗᵢ[𝕜] 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] {β : Type u_5} [MeasurableSpace β] {μb : MeasureTheory.Measure β} (𝕜 : Type u_6) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Fact (1 ≤ p)] (f : α → β) (hf : MeasureTheory.MeasurePreserving f μ μb)
Goal: ∀ (a : ↥(MeasureTheory.Lp E p μb)),
  ↑((MeasureTheory.Lp.compMeasurePreservingₗᵢ 𝕜 f hf) a) = (↑a).compMeasurePreserving f hf
-- LipschitzWith.comp_memℒp
theorem LipschitzWith.comp_memℒp {α E F} {K} [MeasurableSpace α] {μ : Measure α}
    [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g)
    (g0 : g 0 = 0) (hL : Memℒp f p μ) : Memℒp (g ∘ f) p μ 
Arguments: {p : ℝ≥0∞} {α : Type u_5} {E : Type u_6} {F : Type u_7} {K : ℝ≥0} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g) (g0 : g 0 = 0) (hL : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (g ∘ f) p μ
-- MeasureTheory.Memℒp.of_comp_antilipschitzWith
theorem MeasureTheory.Memℒp.of_comp_antilipschitzWith {α E F} {K'} [MeasurableSpace α]
    {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F}
    (hL : Memℒp (g ∘ f) p μ) (hg : UniformContinuous g) (hg' : AntilipschitzWith K' g)
    (g0 : g 0 = 0) : Memℒp f p μ 
Arguments: {p : ℝ≥0∞} {α : Type u_5} {E : Type u_6} {F : Type u_7} {K' : ℝ≥0} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hL : MeasureTheory.Memℒp (g ∘ f) p μ) (hg : UniformContinuous g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0)
Goal: MeasureTheory.Memℒp f p μ
-- LipschitzWith.memℒp_comp_iff_of_antilipschitz
theorem memℒp_comp_iff_of_antilipschitz {α E F} {K K'} [MeasurableSpace α] {μ : Measure α}
    [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g)
    (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) : Memℒp (g ∘ f) p μ ↔ Memℒp f p μ 
Arguments: {p : ℝ≥0∞} {α : Type u_5} {E : Type u_6} {F : Type u_7} {K K' : ℝ≥0} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0)
Goal: MeasureTheory.Memℒp (g ∘ f) p μ ↔ MeasureTheory.Memℒp f p μ
-- LipschitzWith.coeFn_compLp
theorem coeFn_compLp (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : Lp E p μ) :
    hg.compLp g0 f =ᵐ[μ] g ∘ f 
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] {g : E → F} {c : ℝ≥0} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(hg.compLp g0 f) =ᵐ[μ] g ∘ ↑↑f
-- LipschitzWith.compLp_zero
@[simp]
theorem compLp_zero (hg : LipschitzWith c g) (g0 : g 0 = 0) : hg.compLp g0 (0 : Lp E p μ) = 0 
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] {g : E → F} {c : ℝ≥0} (hg : LipschitzWith c g) (g0 : g 0 = 0)
Goal: hg.compLp g0 0 = 0
-- LipschitzWith.norm_compLp_sub_le
theorem norm_compLp_sub_le (hg : LipschitzWith c g) (g0 : g 0 = 0) (f f' : Lp E p μ) :
    ‖hg.compLp g0 f - hg.compLp g0 f'‖ ≤ c * ‖f - f'‖ 
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] {g : E → F} {c : ℝ≥0} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f f' : ↥(MeasureTheory.Lp E p μ))
Goal: ‖hg.compLp g0 f - hg.compLp g0 f'‖ ≤ ↑c * ‖f - f'‖
-- LipschitzWith.norm_compLp_le
theorem norm_compLp_le (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : Lp E p μ) :
    ‖hg.compLp g0 f‖ ≤ c * ‖f‖ 
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] {g : E → F} {c : ℝ≥0} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖hg.compLp g0 f‖ ≤ ↑c * ‖f‖
-- LipschitzWith.lipschitzWith_compLp
theorem lipschitzWith_compLp [Fact (1 ≤ p)] (hg : LipschitzWith c g) (g0 : g 0 = 0) :
    LipschitzWith c (hg.compLp g0 : Lp E p μ → Lp 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] {g : E → F} {c : ℝ≥0} [Fact (1 ≤ p)] (hg : LipschitzWith c g) (g0 : g 0 = 0)
Goal: LipschitzWith c (hg.compLp g0)
-- LipschitzWith.continuous_compLp
theorem continuous_compLp [Fact (1 ≤ p)] (hg : LipschitzWith c g) (g0 : g 0 = 0) :
    Continuous (hg.compLp g0 : Lp E p μ → Lp 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] {g : E → F} {c : ℝ≥0} [Fact (1 ≤ p)] (hg : LipschitzWith c g) (g0 : g 0 = 0)
Goal: Continuous (hg.compLp g0)
-- ContinuousLinearMap.coeFn_compLp
theorem coeFn_compLp (L : E →L[𝕜] F) (f : Lp E p μ) : ∀ᵐ a ∂μ, (L.compLp f) a = L (f a) 
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] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ∀ᵐ (a : α) ∂μ, ↑↑(L.compLp f) a = L (↑↑f a)
-- ContinuousLinearMap.coeFn_compLp'
theorem coeFn_compLp' (L : E →L[𝕜] F) (f : Lp E p μ) : L.compLp f =ᵐ[μ] fun a => L (f a) 
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] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(L.compLp f) =ᵐ[μ] fun a => L (↑↑f a)
-- ContinuousLinearMap.comp_memℒp
theorem comp_memℒp (L : E →L[𝕜] F) (f : Lp E p μ) : Memℒp (L ∘ 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] {𝕜 : Type u_5} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ))
Goal: MeasureTheory.Memℒp (⇑L ∘ ↑↑f) p μ