ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Content.outerMeasure_eq_iInf
theorem outerMeasure_eq_iInf (A : Set G) :
    μ.outerMeasure A = ⨅ (U : Set G) (hU : IsOpen U) (_ : A ⊆ U), μ.innerContent ⟨U, hU⟩ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (A : Set G)
Goal: μ.outerMeasure A = ⨅ U, ⨅ (hU : IsOpen U), ⨅ (_ : A ⊆ U), μ.innerContent { carrier := U, is_open' := hU }
-- MeasureTheory.Content.outerMeasure_interior_compacts
theorem outerMeasure_interior_compacts (K : Compacts G) : μ.outerMeasure (interior K) ≤ μ K 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (K : TopologicalSpace.Compacts G)
Goal: μ.outerMeasure (interior ↑K) ≤ (fun s => ↑(μ.toFun s)) K
-- MeasureTheory.Content.outerMeasure_exists_compact
theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.outerMeasure U ≠ ∞) {ε : ℝ≥0}
    (hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.outerMeasure U ≤ μ.outerMeasure K + ε 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] {U : TopologicalSpace.Opens G} (hU : μ.outerMeasure ↑U ≠ ⊤) {ε : ℝ≥0} (hε : ε ≠ 0)
Goal: ∃ K, ↑K ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure ↑K + ↑ε
-- MeasureTheory.Content.outerMeasure_exists_open
theorem outerMeasure_exists_open {A : Set G} (hA : μ.outerMeasure A ≠ ∞) {ε : ℝ≥0} (hε : ε ≠ 0) :
    ∃ U : Opens G, A ⊆ U ∧ μ.outerMeasure U ≤ μ.outerMeasure A + ε 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] {A : Set G} (hA : μ.outerMeasure A ≠ ⊤) {ε : ℝ≥0} (hε : ε ≠ 0)
Goal: ∃ U, A ⊆ ↑U ∧ μ.outerMeasure ↑U ≤ μ.outerMeasure A + ↑ε
-- MeasureTheory.Content.outerMeasure_preimage
theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.continuous) = μ K)
    (A : Set G) : μ.outerMeasure (f ⁻¹' A) = μ.outerMeasure A 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (f : G ≃ₜ G) (h :
  ∀ ⦃K : TopologicalSpace.Compacts G⦄,
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map ⇑f ⋯ K) = (fun s => ↑(μ.toFun s)) K) (A : Set G)
Goal: μ.outerMeasure (⇑f ⁻¹' A) = μ.outerMeasure A
-- MeasureTheory.Content.outerMeasure_lt_top_of_isCompact
theorem outerMeasure_lt_top_of_isCompact [WeaklyLocallyCompactSpace G]
    {K : Set G} (hK : IsCompact K) :
    μ.outerMeasure K < ∞ 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [WeaklyLocallyCompactSpace G] {K : Set G} (hK : IsCompact K)
Goal: μ.outerMeasure K < ⊤
-- MeasureTheory.Content.is_add_left_invariant_outerMeasure
@[to_additive]
theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
    (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
    (A : Set G) : μ.outerMeasure ((g * ·) ⁻¹' A) = μ.outerMeasure A 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [AddGroup G] [TopologicalAddGroup G] (h :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (g : G) (A : Set G)
Goal: μ.outerMeasure ((fun x => g + x) ⁻¹' A) = μ.outerMeasure A
-- MeasureTheory.Content.is_mul_left_invariant_outerMeasure
@[to_additive]
theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
    (h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
    (A : Set G) : μ.outerMeasure ((g * ·) ⁻¹' A) = μ.outerMeasure A 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [Group G] [TopologicalGroup G] (h :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (g : G) (A : Set G)
Goal: μ.outerMeasure ((fun x => g * x) ⁻¹' A) = μ.outerMeasure A
-- MeasureTheory.Content.outerMeasure_caratheodory
theorem outerMeasure_caratheodory (A : Set G) :
    MeasurableSet[μ.outerMeasure.caratheodory] A ↔
      ∀ U : Opens G, μ.outerMeasure (U ∩ A) + μ.outerMeasure (U \ A) ≤ μ.outerMeasure U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (A : Set G)
Goal: MeasurableSet A ↔
  ∀ (U : TopologicalSpace.Opens G), μ.outerMeasure (↑U ∩ A) + μ.outerMeasure (↑U \ A) ≤ μ.outerMeasure ↑U
-- MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant
@[to_additive]
theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
    (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
    (hK : μ K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty) : 0 < μ.outerMeasure U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [Group G] [TopologicalGroup G] (h3 :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (K : TopologicalSpace.Compacts G) (hK : (fun s => ↑(μ.toFun s)) K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty)
Goal: 0 < μ.outerMeasure U
-- MeasureTheory.Content.outerMeasure_pos_of_is_add_left_invariant
@[to_additive]
theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
    (h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
    (hK : μ K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty) : 0 < μ.outerMeasure U 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [AddGroup G] [TopologicalAddGroup G] (h3 :
  ∀ (g : G) {K : TopologicalSpace.Compacts G},
    (fun s => ↑(μ.toFun s)) (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = (fun s => ↑(μ.toFun s)) K) (K : TopologicalSpace.Compacts G) (hK : (fun s => ↑(μ.toFun s)) K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty)
Goal: 0 < μ.outerMeasure U
-- MeasureTheory.Content.borel_le_caratheodory
theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [S : MeasurableSpace G] [BorelSpace G]
Goal: S ≤ μ.outerMeasure.caratheodory
-- MeasureTheory.Content.measure_apply
theorem measure_apply {s : Set G} (hs : MeasurableSet s) : μ.measure s = μ.outerMeasure s 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [S : MeasurableSpace G] [BorelSpace G] {s : Set G} (hs : MeasurableSet s)
Goal: μ.measure s = μ.outerMeasure s
-- MeasureTheory.Content.contentRegular_exists_compact
theorem contentRegular_exists_compact (H : ContentRegular μ) (K : TopologicalSpace.Compacts G)
    {ε : NNReal} (hε : ε ≠ 0) :
    ∃ K' : TopologicalSpace.Compacts G, K.carrier ⊆ interior K'.carrier ∧ μ K' ≤ μ K + ε 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (H : μ.ContentRegular) (K : TopologicalSpace.Compacts G) {ε : ℝ≥0} (hε : ε ≠ 0)
Goal: ∃ K', K.carrier ⊆ interior K'.carrier ∧ (fun s => ↑(μ.toFun s)) K' ≤ (fun s => ↑(μ.toFun s)) K + ↑ε
-- MeasureTheory.Content.measure_eq_content_of_regular
theorem measure_eq_content_of_regular (H : MeasureTheory.Content.ContentRegular μ)
    (K : TopologicalSpace.Compacts G) : μ.measure ↑K = μ K 
Arguments: {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [MeasurableSpace G] [R1Space G] [BorelSpace G] (H : μ.ContentRegular) (K : TopologicalSpace.Compacts G)
Goal: μ.measure ↑K = (fun s => ↑(μ.toFun s)) K
-- MeasureTheory.withDensityᵥ_apply
theorem withDensityᵥ_apply (hf : Integrable f μ) {s : Set α} (hs : MeasurableSet s) :
    μ.withDensityᵥ f s = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E} (hf : MeasureTheory.Integrable f μ) {s : Set α} (hs : MeasurableSet s)
Goal: ↑(μ.withDensityᵥ f) s = ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.withDensityᵥ_zero
@[simp]
theorem withDensityᵥ_zero : μ.withDensityᵥ (0 : α → E) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E]
Goal: μ.withDensityᵥ 0 = 0
-- MeasureTheory.withDensityᵥ_neg
@[simp]
theorem withDensityᵥ_neg : μ.withDensityᵥ (-f) = -μ.withDensityᵥ f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E}
Goal: μ.withDensityᵥ (-f) = -μ.withDensityᵥ f
-- MeasureTheory.withDensityᵥ_neg'
theorem withDensityᵥ_neg' : (μ.withDensityᵥ fun x => -f x) = -μ.withDensityᵥ f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E}
Goal: (μ.withDensityᵥ fun x => -f x) = -μ.withDensityᵥ f
-- MeasureTheory.withDensityᵥ_add
@[simp]
theorem withDensityᵥ_add (hf : Integrable f μ) (hg : Integrable g μ) :
    μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ.withDensityᵥ (f + g) = μ.withDensityᵥ f + μ.withDensityᵥ g
-- MeasureTheory.withDensityᵥ_add'
theorem withDensityᵥ_add' (hf : Integrable f μ) (hg : Integrable g μ) :
    (μ.withDensityᵥ fun x => f x + g x) = μ.withDensityᵥ f + μ.withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: (μ.withDensityᵥ fun x => f x + g x) = μ.withDensityᵥ f + μ.withDensityᵥ g
-- MeasureTheory.withDensityᵥ_sub
@[simp]
theorem withDensityᵥ_sub (hf : Integrable f μ) (hg : Integrable g μ) :
    μ.withDensityᵥ (f - g) = μ.withDensityᵥ f - μ.withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ.withDensityᵥ (f - g) = μ.withDensityᵥ f - μ.withDensityᵥ g
-- MeasureTheory.withDensityᵥ_sub'
theorem withDensityᵥ_sub' (hf : Integrable f μ) (hg : Integrable g μ) :
    (μ.withDensityᵥ fun x => f x - g x) = μ.withDensityᵥ f - μ.withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: (μ.withDensityᵥ fun x => f x - g x) = μ.withDensityᵥ f - μ.withDensityᵥ g
-- MeasureTheory.withDensityᵥ_smul
@[simp]
theorem withDensityᵥ_smul {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    [SMulCommClass ℝ 𝕜 E] (f : α → E) (r : 𝕜) : μ.withDensityᵥ (r • f) = r • μ.withDensityᵥ f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℝ 𝕜 E] (f : α → E) (r : 𝕜)
Goal: μ.withDensityᵥ (r • f) = r • μ.withDensityᵥ f
-- MeasureTheory.withDensityᵥ_smul'
theorem withDensityᵥ_smul' {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    [SMulCommClass ℝ 𝕜 E] (f : α → E) (r : 𝕜) :
    (μ.withDensityᵥ fun x => r • f x) = r • μ.withDensityᵥ f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_4} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℝ 𝕜 E] (f : α → E) (r : 𝕜)
Goal: (μ.withDensityᵥ fun x => r • f x) = r • μ.withDensityᵥ f
-- MeasureTheory.withDensityᵥ_smul_eq_withDensityᵥ_withDensity
theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity {f : α → ℝ≥0} {g : α → E}
    (hf : AEMeasurable f μ) (hfg : Integrable (f • g) μ) :
    μ.withDensityᵥ (f • g) = (μ.withDensity (fun x ↦ f x)).withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → ℝ≥0} {g : α → E} (hf : AEMeasurable f μ) (hfg : MeasureTheory.Integrable (f • g) μ)
Goal: μ.withDensityᵥ (f • g) = (μ.withDensity fun x => ↑(f x)).withDensityᵥ g
-- MeasureTheory.withDensityᵥ_smul_eq_withDensityᵥ_withDensity'
theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity' {f : α → ℝ≥0∞} {g : α → E}
    (hf : AEMeasurable f μ) (hflt : ∀ᵐ x ∂μ, f x < ∞)
    (hfg : Integrable (fun x ↦ (f x).toReal • g x) μ) :
    μ.withDensityᵥ (fun x ↦ (f x).toReal • g x) = (μ.withDensity f).withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → ℝ≥0∞} {g : α → E} (hf : AEMeasurable f μ) (hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤) (hfg : MeasureTheory.Integrable (fun x => (f x).toReal • g x) μ)
Goal: (μ.withDensityᵥ fun x => (f x).toReal • g x) = (μ.withDensity f).withDensityᵥ g
-- MeasureTheory.Measure.withDensityᵥ_absolutelyContinuous
theorem Measure.withDensityᵥ_absolutelyContinuous (μ : Measure α) (f : α → ℝ) :
    μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ)
Goal: μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure
-- MeasureTheory.Integrable.ae_eq_of_withDensityᵥ_eq
theorem Integrable.ae_eq_of_withDensityᵥ_eq {f g : α → E} (hf : Integrable f μ)
    (hg : Integrable g μ) (hfg : μ.withDensityᵥ f = μ.withDensityᵥ g) : f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : μ.withDensityᵥ f = μ.withDensityᵥ g)
Goal: f =ᵐ[μ] g
-- MeasureTheory.WithDensityᵥEq.congr_ae
theorem WithDensityᵥEq.congr_ae {f g : α → E} (h : f =ᵐ[μ] g) :
    μ.withDensityᵥ f = μ.withDensityᵥ g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : α → E} (h : f =ᵐ[μ] g)
Goal: μ.withDensityᵥ f = μ.withDensityᵥ g
-- MeasureTheory.Integrable.withDensityᵥ_eq_iff
theorem Integrable.withDensityᵥ_eq_iff {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) :
    μ.withDensityᵥ f = μ.withDensityᵥ g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ.withDensityᵥ f = μ.withDensityᵥ g ↔ f =ᵐ[μ] g
-- MeasureTheory.withDensityᵥ_toReal
theorem withDensityᵥ_toReal {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hf : (∫⁻ x, f x ∂μ) ≠ ∞) :
    (μ.withDensityᵥ fun x => (f x).toReal) =
      @toSignedMeasure α _ (μ.withDensity f) (isFiniteMeasure_withDensity hf) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: (μ.withDensityᵥ fun x => (f x).toReal) = (μ.withDensity f).toSignedMeasure
-- MeasureTheory.withDensityᵥ_eq_withDensity_pos_part_sub_withDensity_neg_part
theorem withDensityᵥ_eq_withDensity_pos_part_sub_withDensity_neg_part {f : α → ℝ}
    (hfi : Integrable f μ) :
    μ.withDensityᵥ f =
      @toSignedMeasure α _ (μ.withDensity fun x => ENNReal.ofReal <| f x)
          (isFiniteMeasure_withDensity_ofReal hfi.2) -
        @toSignedMeasure α _ (μ.withDensity fun x => ENNReal.ofReal <| -f x)
          (isFiniteMeasure_withDensity_ofReal hfi.neg.2) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hfi : MeasureTheory.Integrable f μ)
Goal: μ.withDensityᵥ f =
  (μ.withDensity fun x => ENNReal.ofReal (f x)).toSignedMeasure -
    (μ.withDensity fun x => ENNReal.ofReal (-f x)).toSignedMeasure
-- MeasureTheory.Integrable.withDensityᵥ_trim_eq_integral
theorem Integrable.withDensityᵥ_trim_eq_integral {m m0 : MeasurableSpace α} {μ : Measure α}
    (hm : m ≤ m0) {f : α → ℝ} (hf : Integrable f μ) {i : Set α} (hi : MeasurableSet[m] i) :
    (μ.withDensityᵥ f).trim hm i = ∫ x in i, f x ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) {f : α → ℝ} (hf : MeasureTheory.Integrable f μ) {i : Set α} (hi : MeasurableSet i)
Goal: ↑((μ.withDensityᵥ f).trim hm) i = ∫ (x : α) in i, f x ∂μ
-- MeasureTheory.Integrable.withDensityᵥ_trim_absolutelyContinuous
theorem Integrable.withDensityᵥ_trim_absolutelyContinuous {m m0 : MeasurableSpace α} {μ : Measure α}
    (hm : m ≤ m0) (hfi : Integrable f μ) :
    (μ.withDensityᵥ f).trim hm ≪ᵥ (μ.trim hm).toENNRealVectorMeasure 
Arguments: {α : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α → E} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (hfi : MeasureTheory.Integrable f μ)
Goal: (μ.withDensityᵥ f).trim hm ≪ᵥ (μ.trim hm).toENNRealVectorMeasure
-- MeasureTheory.FiniteMeasure.toMeasure_prod
@[simp] lemma toMeasure_prod : (μ.prod ν).toMeasure = μ.toMeasure.prod ν.toMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β)
Goal: ↑(μ.prod ν) = (↑μ).prod ↑ν
-- MeasureTheory.FiniteMeasure.prod_apply
lemma prod_apply (s : Set (α × β)) (s_mble : MeasurableSet s) :
    μ.prod ν s = ENNReal.toNNReal (∫⁻ x, ν.toMeasure (Prod.mk x ⁻¹' s) ∂μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s)
Goal: (μ.prod ν) s = (∫⁻ (x : α), ↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal
-- MeasureTheory.FiniteMeasure.prod_apply_symm
lemma prod_apply_symm (s : Set (α × β)) (s_mble : MeasurableSet s) :
    μ.prod ν s = ENNReal.toNNReal (∫⁻ y, μ.toMeasure ((fun x ↦ ⟨x, y⟩) ⁻¹' s) ∂ν) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s)
Goal: (μ.prod ν) s = (∫⁻ (y : β), ↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal
-- MeasureTheory.FiniteMeasure.prod_prod
lemma prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set α) (t : Set β)
Goal: (μ.prod ν) (s ×ˢ t) = μ s * ν t
-- MeasureTheory.FiniteMeasure.mass_prod
@[simp] lemma mass_prod : (μ.prod ν).mass = μ.mass * ν.mass 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β)
Goal: (μ.prod ν).mass = μ.mass * ν.mass
-- MeasureTheory.FiniteMeasure.zero_prod
@[simp] lemma zero_prod : (0 : FiniteMeasure α).prod ν = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (ν : MeasureTheory.FiniteMeasure β)
Goal: MeasureTheory.FiniteMeasure.prod 0 ν = 0
-- MeasureTheory.FiniteMeasure.prod_zero
@[simp] lemma prod_zero : μ.prod (0 : FiniteMeasure β) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α)
Goal: μ.prod 0 = 0
-- MeasureTheory.FiniteMeasure.map_fst_prod
@[simp] lemma map_fst_prod : (μ.prod ν).map Prod.fst = ν univ • μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β)
Goal: (μ.prod ν).map Prod.fst = ν Set.univ • μ
-- MeasureTheory.FiniteMeasure.map_snd_prod
@[simp] lemma map_snd_prod : (μ.prod ν).map Prod.snd = μ univ • ν 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β)
Goal: (μ.prod ν).map Prod.snd = μ Set.univ • ν
-- MeasureTheory.FiniteMeasure.map_prod_map
lemma map_prod_map {α' : Type*} [MeasurableSpace α'] {β' : Type*} [MeasurableSpace β']
    {f : α → α'} {g : β → β'}  (f_mble : Measurable f) (g_mble : Measurable g) :
    (μ.map f).prod (ν.map g) = (μ.prod ν).map (Prod.map f g) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g)
Goal: (μ.map f).prod (ν.map g) = (μ.prod ν).map (Prod.map f g)
-- MeasureTheory.FiniteMeasure.prod_swap
lemma prod_swap : (μ.prod ν).map Prod.swap = ν.prod μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β)
Goal: (μ.prod ν).map Prod.swap = ν.prod μ
-- MeasureTheory.ProbabilityMeasure.toMeasure_prod
@[simp] lemma toMeasure_prod : (μ.prod ν).toMeasure = μ.toMeasure.prod ν.toMeasure 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β)
Goal: ↑(μ.prod ν) = (↑μ).prod ↑ν
-- MeasureTheory.ProbabilityMeasure.prod_apply
lemma prod_apply (s : Set (α × β)) (s_mble : MeasurableSet s) :
    μ.prod ν s = ENNReal.toNNReal (∫⁻ x, ν.toMeasure (Prod.mk x ⁻¹' s) ∂μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s)
Goal: (μ.prod ν) s = (∫⁻ (x : α), ↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal
-- MeasureTheory.ProbabilityMeasure.prod_apply_symm
lemma prod_apply_symm (s : Set (α × β)) (s_mble : MeasurableSet s) :
    μ.prod ν s = ENNReal.toNNReal (∫⁻ y, μ.toMeasure ((fun x ↦ ⟨x, y⟩) ⁻¹' s) ∂ν) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s)
Goal: (μ.prod ν) s = (∫⁻ (y : β), ↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal
-- MeasureTheory.ProbabilityMeasure.prod_prod
lemma prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set α) (t : Set β)
Goal: (μ.prod ν) (s ×ˢ t) = μ s * ν t
-- MeasureTheory.ProbabilityMeasure.map_fst_prod
@[simp] lemma map_fst_prod : (μ.prod ν).map measurable_fst.aemeasurable = μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β)
Goal: (μ.prod ν).map ⋯ = μ
-- MeasureTheory.ProbabilityMeasure.map_snd_prod
@[simp] lemma map_snd_prod : (μ.prod ν).map measurable_snd.aemeasurable = ν 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β)
Goal: (μ.prod ν).map ⋯ = ν
-- MeasureTheory.ProbabilityMeasure.map_prod_map
lemma map_prod_map {α' : Type*} [MeasurableSpace α'] {β' : Type*} [MeasurableSpace β']
    {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) :
    (μ.map f_mble.aemeasurable).prod (ν.map g_mble.aemeasurable)
      = (μ.prod ν).map (f_mble.prod_map g_mble).aemeasurable 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g)
Goal: (μ.map ⋯).prod (ν.map ⋯) = (μ.prod ν).map ⋯
-- MeasureTheory.ProbabilityMeasure.prod_swap
lemma prod_swap : (μ.prod ν).map measurable_swap.aemeasurable = ν.prod μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β)
Goal: (μ.prod ν).map ⋯ = ν.prod μ
-- MeasureTheory.Measure.sub_def
theorem sub_def : μ - ν = sInf { d | μ ≤ d + ν } 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α}
Goal: μ - ν = sInf {d | μ ≤ d + ν}