ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.Tendsto.compMeasurePreservingLp
theorem Filter.Tendsto.compMeasurePreservingLp {α : Type*} {l : Filter α}
    {f : α → Lp E p ν} {f₀ : Lp E p ν} {g : α → C(X, Y)} {g₀ : C(X, Y)}
    (hf : Tendsto f l (𝓝 f₀)) (hg : Tendsto g l (𝓝 g₀))
    (hgm : ∀ a, MeasurePreserving (g a) μ ν) (hgm₀ : MeasurePreserving g₀ μ ν) (hp : p ≠ ∞) :
    Tendsto (fun a ↦ Lp.compMeasurePreserving (g a) (hgm a) (f a)) l
      (𝓝 (Lp.compMeasurePreserving g₀ hgm₀ f₀)) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] {E : Type u_4} [NormedAddCommGroup E] {p : ℝ≥0∞} [Fact (1 ≤ p)] {α : Type u_5} {l : Filter α} {f : α → ↥(MeasureTheory.Lp E p ν)} {f₀ : ↥(MeasureTheory.Lp E p ν)} {g : α → C(X, Y)} {g₀ : C(X, Y)} (hf : Filter.Tendsto f l (𝓝 f₀)) (hg : Filter.Tendsto g l (𝓝 g₀)) (hgm : ∀ (a : α), MeasureTheory.MeasurePreserving (⇑(g a)) μ ν) (hgm₀ : MeasureTheory.MeasurePreserving (⇑g₀) μ ν) (hp : p ≠ ⊤)
Goal: Filter.Tendsto (fun a => (MeasureTheory.Lp.compMeasurePreserving ⇑(g a) ⋯) (f a)) l
  (𝓝 ((MeasureTheory.Lp.compMeasurePreserving (⇑g₀) hgm₀) f₀))
-- ContinuousWithinAt.compMeasurePreservingLp
theorem ContinuousWithinAt.compMeasurePreservingLp (hf : ContinuousWithinAt f s z)
    (hg : ContinuousWithinAt g s z) (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    ContinuousWithinAt (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) s z 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] {E : Type u_4} [NormedAddCommGroup E] {p : ℝ≥0∞} [Fact (1 ≤ p)] {Z : Type u_5} [TopologicalSpace Z] {f : Z → ↥(MeasureTheory.Lp E p ν)} {g : Z → C(X, Y)} {s : Set Z} {z : Z} (hf : ContinuousWithinAt f s z) (hg : ContinuousWithinAt g s z) (hgm : ∀ (z : Z), MeasureTheory.MeasurePreserving (⇑(g z)) μ ν) (hp : p ≠ ⊤)
Goal: ContinuousWithinAt (fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)) s z
-- ContinuousAt.compMeasurePreservingLp
theorem ContinuousAt.compMeasurePreservingLp (hf : ContinuousAt f z)
    (hg : ContinuousAt g z) (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    ContinuousAt (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) z 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] {E : Type u_4} [NormedAddCommGroup E] {p : ℝ≥0∞} [Fact (1 ≤ p)] {Z : Type u_5} [TopologicalSpace Z] {f : Z → ↥(MeasureTheory.Lp E p ν)} {g : Z → C(X, Y)} {z : Z} (hf : ContinuousAt f z) (hg : ContinuousAt g z) (hgm : ∀ (z : Z), MeasureTheory.MeasurePreserving (⇑(g z)) μ ν) (hp : p ≠ ⊤)
Goal: ContinuousAt (fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)) z
-- ContinuousOn.compMeasurePreservingLp
theorem ContinuousOn.compMeasurePreservingLp (hf : ContinuousOn f s)
    (hg : ContinuousOn g s) (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    ContinuousOn (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) s 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] {E : Type u_4} [NormedAddCommGroup E] {p : ℝ≥0∞} [Fact (1 ≤ p)] {Z : Type u_5} [TopologicalSpace Z] {f : Z → ↥(MeasureTheory.Lp E p ν)} {g : Z → C(X, Y)} {s : Set Z} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (hgm : ∀ (z : Z), MeasureTheory.MeasurePreserving (⇑(g z)) μ ν) (hp : p ≠ ⊤)
Goal: ContinuousOn (fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)) s
-- Continuous.compMeasurePreservingLp
theorem Continuous.compMeasurePreservingLp (hf : Continuous f) (hg : Continuous g)
    (hgm : ∀ z, MeasurePreserving (g z) μ ν) (hp : p ≠ ∞) :
    Continuous (fun z ↦ Lp.compMeasurePreserving (g z) (hgm z) (f z)) 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] {E : Type u_4} [NormedAddCommGroup E] {p : ℝ≥0∞} [Fact (1 ≤ p)] {Z : Type u_5} [TopologicalSpace Z] {f : Z → ↥(MeasureTheory.Lp E p ν)} {g : Z → C(X, Y)} (hf : Continuous f) (hg : Continuous g) (hgm : ∀ (z : Z), MeasureTheory.MeasurePreserving (⇑(g z)) μ ν) (hp : p ≠ ⊤)
Goal: Continuous fun z => (MeasureTheory.Lp.compMeasurePreserving ⇑(g z) ⋯) (f z)
-- MeasureTheory.Memℒp.integrable_sq
theorem Memℒp.integrable_sq {f : α → ℝ} (h : Memℒp f 2 μ) : Integrable (fun x => f x ^ 2) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (h : MeasureTheory.Memℒp f 2 μ)
Goal: MeasureTheory.Integrable (fun x => f x ^ 2) μ
-- MeasureTheory.memℒp_two_iff_integrable_sq_norm
theorem memℒp_two_iff_integrable_sq_norm {f : α → F} (hf : AEStronglyMeasurable f μ) :
    Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ 
Arguments: {α : Type u_1} {F : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] {f : α → F} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.Memℒp f 2 μ ↔ MeasureTheory.Integrable (fun x => ‖f x‖ ^ 2) μ
-- MeasureTheory.memℒp_two_iff_integrable_sq
theorem memℒp_two_iff_integrable_sq {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
    Memℒp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.Memℒp f 2 μ ↔ MeasureTheory.Integrable (fun x => f x ^ 2) μ
-- MeasureTheory.Memℒp.const_inner
theorem Memℒp.const_inner (c : E) {f : α → E} (hf : Memℒp f p μ) : Memℒp (fun a => ⟪c, f a⟫) p μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {E : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (c : E) {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp (fun a => ⟪c, f a⟫_𝕜) p μ
-- MeasureTheory.Memℒp.inner_const
theorem Memℒp.inner_const {f : α → E} (hf : Memℒp f p μ) (c : E) : Memℒp (fun a => ⟪f a, c⟫) p μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {E : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ) (c : E)
Goal: MeasureTheory.Memℒp (fun a => ⟪f a, c⟫_𝕜) p μ
-- MeasureTheory.Integrable.const_inner
theorem Integrable.const_inner (c : E) (hf : Integrable f μ) :
    Integrable (fun x => ⟪c, f x⟫) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {f : α → E} (c : E) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => ⟪c, f x⟫_𝕜) μ
-- MeasureTheory.Integrable.inner_const
theorem Integrable.inner_const (hf : Integrable f μ) (c : E) :
    Integrable (fun x => ⟪f x, c⟫) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {f : α → E} (hf : MeasureTheory.Integrable f μ) (c : E)
Goal: MeasureTheory.Integrable (fun x => ⟪f x, c⟫_𝕜) μ
-- integral_inner
theorem _root_.integral_inner {f : α → E} (hf : Integrable f μ) (c : E) :
    ∫ x, ⟪c, f x⟫ ∂μ = ⟪c, ∫ x, f x ∂μ⟫ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedSpace ℝ E] {f : α → E} (hf : MeasureTheory.Integrable f μ) (c : E)
Goal: ∫ (x : α), ⟪c, f x⟫_𝕜 ∂μ = ⟪c, ∫ (x : α), f x ∂μ⟫_𝕜
-- integral_eq_zero_of_forall_integral_inner_eq_zero
theorem _root_.integral_eq_zero_of_forall_integral_inner_eq_zero (f : α → E) (hf : Integrable f μ)
    (hf_int : ∀ c : E, ∫ x, ⟪c, f x⟫ ∂μ = 0) : ∫ x, f x ∂μ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} (𝕜 : Type u_3) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedSpace ℝ E] (f : α → E) (hf : MeasureTheory.Integrable f μ) (hf_int : ∀ (c : E), ∫ (x : α), ⟪c, f x⟫_𝕜 ∂μ = 0)
Goal: ∫ (x : α), f x ∂μ = 0
-- MeasureTheory.L2.eLpNorm_rpow_two_norm_lt_top
theorem eLpNorm_rpow_two_norm_lt_top (f : Lp F 2 μ) :
    eLpNorm (fun x => ‖f x‖ ^ (2 : ℝ)) 1 μ < ∞ 
Arguments: {α : Type u_1} {F : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup F] (f : ↥(MeasureTheory.Lp F 2 μ))
Goal: MeasureTheory.eLpNorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤
-- MeasureTheory.L2.eLpNorm_inner_lt_top
theorem eLpNorm_inner_lt_top (f g : α →₂[μ] E) : eLpNorm (fun x : α => ⟪f x, g x⟫) 1 μ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f g : ↥(MeasureTheory.Lp E 2 μ))
Goal: MeasureTheory.eLpNorm (fun x => ⟪↑↑f x, ↑↑g x⟫_𝕜) 1 μ < ⊤
-- MeasureTheory.L2.inner_def
theorem inner_def (f g : α →₂[μ] E) : ⟪f, g⟫ = ∫ a : α, ⟪f a, g a⟫ ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f g : ↥(MeasureTheory.Lp E 2 μ))
Goal: ⟪f, g⟫_𝕜 = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫_𝕜 ∂μ
-- MeasureTheory.L2.integral_inner_eq_sq_eLpNorm
theorem integral_inner_eq_sq_eLpNorm (f : α →₂[μ] E) :
    ∫ a, ⟪f a, f a⟫ ∂μ = ENNReal.toReal (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ (2 : ℝ) ∂μ) 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫_𝕜 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal
-- _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.norm_sq_eq_inner'
private theorem norm_sq_eq_inner' (f : α →₂[μ] E) : ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫_𝕜
-- MeasureTheory.L2.mem_L1_inner
theorem mem_L1_inner (f g : α →₂[μ] E) :
    AEEqFun.mk (fun x => ⟪f x, g x⟫)
        ((Lp.aestronglyMeasurable f).inner (Lp.aestronglyMeasurable g)) ∈
      Lp 𝕜 1 μ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f g : ↥(MeasureTheory.Lp E 2 μ))
Goal: MeasureTheory.AEEqFun.mk (fun x => ⟪↑↑f x, ↑↑g x⟫_𝕜) ⋯ ∈ MeasureTheory.Lp 𝕜 1 μ
-- MeasureTheory.L2.integrable_inner
theorem integrable_inner (f g : α →₂[μ] E) : Integrable (fun x : α => ⟪f x, g x⟫) μ 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f g : ↥(MeasureTheory.Lp E 2 μ))
Goal: MeasureTheory.Integrable (fun x => ⟪↑↑f x, ↑↑g x⟫_𝕜) μ
-- _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.add_left'
private theorem add_left' (f f' g : α →₂[μ] E) : ⟪f + f', g⟫ = inner f g + inner f' g 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f f' g : ↥(MeasureTheory.Lp E 2 μ))
Goal: ⟪f + f', g⟫_𝕜 = ⟪f, g⟫_𝕜 + ⟪f', g⟫_𝕜
-- _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.smul_left'
private theorem smul_left' (f g : α →₂[μ] E) (r : 𝕜) : ⟪r • f, g⟫ = conj r * inner f g 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (f g : ↥(MeasureTheory.Lp E 2 μ)) (r : 𝕜)
Goal: ⟪r • f, g⟫_𝕜 = (starRingEnd 𝕜) r * ⟪f, g⟫_𝕜
-- MeasureTheory.L2.inner_indicatorConstLp_eq_setIntegral_inner
theorem inner_indicatorConstLp_eq_setIntegral_inner (f : Lp E 2 μ) (hs : MeasurableSet s) (c : E)
    (hμs : μ s ≠ ∞) : (⟪indicatorConstLp 2 hs hμs c, f⟫ : 𝕜) = ∫ x in s, ⟪c, f x⟫ ∂μ 
Arguments: {α : Type u_1} {E : Type u_2} (𝕜 : Type u_4) [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {s : Set α} (f : ↥(MeasureTheory.Lp E 2 μ)) (hs : MeasurableSet s) (c : E) (hμs : μ s ≠ ⊤)
Goal: ⟪MeasureTheory.indicatorConstLp 2 hs hμs c, f⟫_𝕜 = ∫ (x : α) in s, ⟪c, ↑↑f x⟫_𝕜 ∂μ
-- MeasureTheory.L2.inner_indicatorConstLp_eq_inner_setIntegral
theorem inner_indicatorConstLp_eq_inner_setIntegral [CompleteSpace E] [NormedSpace ℝ E]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) (f : Lp E 2 μ) :
    (⟪indicatorConstLp 2 hs hμs c, f⟫ : 𝕜) = ⟪c, ∫ x in s, f x ∂μ⟫ 
Arguments: {α : Type u_1} {E : Type u_2} (𝕜 : Type u_4) [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {s : Set α} [CompleteSpace E] [NormedSpace ℝ E] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : E) (f : ↥(MeasureTheory.Lp E 2 μ))
Goal: ⟪MeasureTheory.indicatorConstLp 2 hs hμs c, f⟫_𝕜 = ⟪c, ∫ (x : α) in s, ↑↑f x ∂μ⟫_𝕜
-- MeasureTheory.L2.inner_indicatorConstLp_one
theorem inner_indicatorConstLp_one (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (f : Lp 𝕜 2 μ) :
    ⟪indicatorConstLp 2 hs hμs (1 : 𝕜), f⟫ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {𝕜 : Type u_4} [RCLike 𝕜] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (f : ↥(MeasureTheory.Lp 𝕜 2 μ))
Goal: ⟪MeasureTheory.indicatorConstLp 2 hs hμs 1, f⟫_𝕜 = ∫ (x : α) in s, ↑↑f x ∂μ
-- MeasureTheory.BoundedContinuousFunction.inner_toLp
theorem BoundedContinuousFunction.inner_toLp (f g : α →ᵇ 𝕜) :
    ⟪BoundedContinuousFunction.toLp (E := 𝕜) 2 μ 𝕜 f,
        BoundedContinuousFunction.toLp (E := 𝕜) 2 μ 𝕜 g⟫ =
      ∫ x, conj (f x) * g x ∂μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [RCLike 𝕜] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f g : BoundedContinuousFunction α 𝕜)
Goal: ⟪(BoundedContinuousFunction.toLp 2 μ 𝕜) f, (BoundedContinuousFunction.toLp 2 μ 𝕜) g⟫_𝕜 =
  ∫ (x : α), (starRingEnd 𝕜) (f x) * g x ∂μ
-- MeasureTheory.ContinuousMap.inner_toLp
theorem ContinuousMap.inner_toLp (f g : C(α, 𝕜)) :
    ⟪ContinuousMap.toLp (E := 𝕜) 2 μ 𝕜 f, ContinuousMap.toLp (E := 𝕜) 2 μ 𝕜 g⟫ =
      ∫ x, conj (f x) * g x ∂μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [RCLike 𝕜] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [CompactSpace α] (f g : C(α, 𝕜))
Goal: ⟪(ContinuousMap.toLp 2 μ 𝕜) f, (ContinuousMap.toLp 2 μ 𝕜) g⟫_𝕜 = ∫ (x : α), (starRingEnd 𝕜) (f x) * g x ∂μ
-- MeasureTheory.eLpNorm'_trim
theorem eLpNorm'_trim (hm : m ≤ m0) {f : α → E} (hf : StronglyMeasurable[m] f) :
    eLpNorm' f q (μ.trim hm) = eLpNorm' f q μ 
Arguments: {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {q : ℝ} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hm : m ≤ m0) {f : α → E} (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.eLpNorm' f q (μ.trim hm) = MeasureTheory.eLpNorm' f q μ
-- MeasureTheory.limsup_trim
theorem limsup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
    limsup f (ae (μ.trim hm)) = limsup f (ae μ) 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ENNReal} (hf : Measurable f)
Goal: Filter.limsup f (MeasureTheory.ae (μ.trim hm)) = Filter.limsup f (MeasureTheory.ae μ)
-- MeasureTheory.essSup_trim
theorem essSup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
    essSup f (μ.trim hm) = essSup f μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ENNReal} (hf : Measurable f)
Goal: essSup f (μ.trim hm) = essSup f μ
-- MeasureTheory.eLpNormEssSup_trim
theorem eLpNormEssSup_trim (hm : m ≤ m0) {f : α → E} (hf : StronglyMeasurable[m] f) :
    eLpNormEssSup f (μ.trim hm) = eLpNormEssSup f μ 
Arguments: {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hm : m ≤ m0) {f : α → E} (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.eLpNormEssSup f (μ.trim hm) = MeasureTheory.eLpNormEssSup f μ
-- MeasureTheory.eLpNorm_trim
theorem eLpNorm_trim (hm : m ≤ m0) {f : α → E} (hf : StronglyMeasurable[m] f) :
    eLpNorm f p (μ.trim hm) = eLpNorm f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hm : m ≤ m0) {f : α → E} (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.eLpNorm f p (μ.trim hm) = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.eLpNorm_trim_ae
theorem eLpNorm_trim_ae (hm : m ≤ m0) {f : α → E} (hf : AEStronglyMeasurable f (μ.trim hm)) :
    eLpNorm f p (μ.trim hm) = eLpNorm f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hm : m ≤ m0) {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f (μ.trim hm))
Goal: MeasureTheory.eLpNorm f p (μ.trim hm) = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.memℒp_of_memℒp_trim
theorem memℒp_of_memℒp_trim (hm : m ≤ m0) {f : α → E} (hf : Memℒp f p (μ.trim hm)) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (hm : m ≤ m0) {f : α → E} (hf : MeasureTheory.Memℒp f p (μ.trim hm))
Goal: MeasureTheory.Memℒp f p μ
-- MeasureTheory.AEStronglyMeasurable.comp_measurePreserving
theorem MeasureTheory.AEStronglyMeasurable.comp_measurePreserving
    {γ : Type*} {_ : MeasurableSpace γ} {_ : MeasurableSpace α} {f : γ → α} {μ : Measure γ}
    {ν : Measure α} (hg : AEStronglyMeasurable g ν) (hf : MeasurePreserving f μ ν) :
    AEStronglyMeasurable (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {g : α → β} {γ : Type u_4}
Goal: ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ}
  {ν : MeasureTheory.Measure α},
  MeasureTheory.AEStronglyMeasurable g ν →
    MeasureTheory.MeasurePreserving f μ ν → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ
-- MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff
theorem MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff {β : Type*}
    {f : α → β} {mα : MeasurableSpace α} {μa : Measure α} {mβ : MeasurableSpace β} {μb : Measure β}
    (hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ} :
    AEStronglyMeasurable (g ∘ f) μa ↔ AEStronglyMeasurable g μb 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace γ] {β : Type u_4} {f : α → β} {mα : MeasurableSpace α} {μa : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {μb : MeasureTheory.Measure β} (hf : MeasureTheory.MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f) {g : β → γ}
Goal: MeasureTheory.AEStronglyMeasurable (g ∘ f) μa ↔ MeasureTheory.AEStronglyMeasurable g μb
-- aestronglyMeasurable_smul_const_iff
theorem aestronglyMeasurable_smul_const_iff {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
    AEStronglyMeasurable (fun x => f x • c) μ ↔ AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : α → 𝕜} {c : E} (hc : c ≠ 0)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x • c) μ ↔ MeasureTheory.AEStronglyMeasurable f μ
-- StronglyMeasurable.apply_continuousLinearMap
theorem StronglyMeasurable.apply_continuousLinearMap
    {_m : MeasurableSpace α} {φ : α → F →L[𝕜] E} (hφ : StronglyMeasurable φ) (v : F) :
    StronglyMeasurable fun a => φ a v 
Arguments: {α : Type u_1} {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_6} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {_m : MeasurableSpace α} {φ : α → F →L[𝕜] E} (hφ : MeasureTheory.StronglyMeasurable φ) (v : F)
Goal: MeasureTheory.StronglyMeasurable fun a => (φ a) v
-- MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap
@[measurability]
theorem MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap {φ : α → F →L[𝕜] E}
    (hφ : AEStronglyMeasurable φ μ) (v : F) :
    AEStronglyMeasurable (fun a => φ a v) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_6} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ : α → F →L[𝕜] E} (hφ : MeasureTheory.AEStronglyMeasurable φ μ) (v : F)
Goal: MeasureTheory.AEStronglyMeasurable (fun a => (φ a) v) μ
-- ContinuousLinearMap.aestronglyMeasurable_comp₂
theorem ContinuousLinearMap.aestronglyMeasurable_comp₂ (L : E →L[𝕜] F →L[𝕜] G) {f : α → E}
    {g : α → F} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun x => L (f x) (g x)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_6} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_7} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (L : E →L[𝕜] F →L[𝕜] G) {f : α → E} {g : α → F} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (L (f x)) (g x)) μ
-- aestronglyMeasurable_withDensity_iff
theorem aestronglyMeasurable_withDensity_iff {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
    AEStronglyMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
      AEStronglyMeasurable (fun x => (f x : ℝ) • g x) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → ℝ≥0} (hf : Measurable f) {g : α → E}
Goal: MeasureTheory.AEStronglyMeasurable g (μ.withDensity fun x => ↑(f x)) ↔
  MeasureTheory.AEStronglyMeasurable (fun x => ↑(f x) • g x) μ
-- MeasureTheory.LocallyIntegrableOn.mono_set
theorem LocallyIntegrableOn.mono_set (hf : LocallyIntegrableOn f s μ) {t : Set X}
    (hst : t ⊆ s) : LocallyIntegrableOn f t μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ) {t : Set X} (hst : t ⊆ s)
Goal: MeasureTheory.LocallyIntegrableOn f t μ
-- MeasureTheory.LocallyIntegrableOn.norm
theorem LocallyIntegrableOn.norm (hf : LocallyIntegrableOn f s μ) :
    LocallyIntegrableOn (fun x => ‖f x‖) s μ 
Arguments: {X : Type u_1} {E : Type u_3} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ)
Goal: MeasureTheory.LocallyIntegrableOn (fun x => ‖f x‖) s μ
-- MeasureTheory.LocallyIntegrableOn.mono
theorem LocallyIntegrableOn.mono (hf : LocallyIntegrableOn f s μ) {g : X → F}
    (hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
    LocallyIntegrableOn g s μ 
Arguments: {X : Type u_1} {E : Type u_3} {F : Type u_4} [MeasurableSpace X] [TopologicalSpace X] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : X → E} {μ : MeasureTheory.Measure X} {s : Set X} (hf : MeasureTheory.LocallyIntegrableOn f s μ) {g : X → F} (hg : MeasureTheory.AEStronglyMeasurable g μ) (h : ∀ᵐ (x : X) ∂μ, ‖g x‖ ≤ ‖f x‖)
Goal: MeasureTheory.LocallyIntegrableOn g s μ