ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.measure_lintegral_le_pos
theorem measure_lintegral_le_pos (hint : ∫⁻ a, f a ∂μ ≠ ∞) : 0 < μ {x | ∫⁻ a, f a ∂μ ≤ f x} 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} [MeasureTheory.IsProbabilityMeasure μ] (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤)
Goal: 0 < μ {x | ∫⁻ (a : α), f a ∂μ ≤ f x}
-- MeasureTheory.exists_le_lintegral
theorem exists_le_lintegral (hf : AEMeasurable f μ) : ∃ x, f x ≤ ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} [MeasureTheory.IsProbabilityMeasure μ] (hf : AEMeasurable f μ)
Goal: ∃ x, f x ≤ ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.exists_lintegral_le
theorem exists_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) : ∃ x, ∫⁻ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} [MeasureTheory.IsProbabilityMeasure μ] (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤)
Goal: ∃ x, ∫⁻ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.exists_not_mem_null_le_lintegral
theorem exists_not_mem_null_le_lintegral (hf : AEMeasurable f μ) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ f x ≤ ∫⁻ a, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ≥0∞} [MeasureTheory.IsProbabilityMeasure μ] (hf : AEMeasurable f μ) (hN : μ N = 0)
Goal: ∃ x ∉ N, f x ≤ ∫⁻ (a : α), f a ∂μ
-- MeasureTheory.exists_not_mem_null_lintegral_le
theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
    ∃ x, x ∉ N ∧ ∫⁻ a, f a ∂μ ≤ f x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {N : Set α} {f : α → ℝ≥0∞} [MeasureTheory.IsProbabilityMeasure μ] (hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤) (hN : μ N = 0)
Goal: ∃ x ∉ N, ∫⁻ (a : α), f a ∂μ ≤ f x
-- MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub
theorem tendsto_integral_smul_of_tendsto_average_norm_sub
    {ι : Type*} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ)
    (hf : Tendsto (fun i ↦ ⨍ y in a i, ‖f y - c‖ ∂μ) l (𝓝 0))
    (f_int : ∀ᶠ i in l, IntegrableOn f (a i) μ)
    (hg : Tendsto (fun i ↦ ∫ y, g i y ∂μ) l (𝓝 1))
    (g_supp : ∀ᶠ i in l, Function.support (g i) ⊆ a i)
    (g_bound : ∀ᶠ i in l, ∀ x, |g i x| ≤ K / (μ (a i)).toReal) :
    Tendsto (fun i ↦ ∫ y, g i y • f y ∂μ) l (𝓝 c) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {μ : MeasureTheory.Measure α} {ι : Type u_4} {a : ι → Set α} {l : Filter ι} {f : α → E} {c : E} {g : ι → α → ℝ} (K : ℝ) (hf : Filter.Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)) (f_int : ∀ᶠ (i : ι) in l, MeasureTheory.IntegrableOn f (a i) μ) (hg : Filter.Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)) (g_supp : ∀ᶠ (i : ι) in l, Function.support (g i) ⊆ a i) (g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal)
Goal: Filter.Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)
-- IsPiSystem.prod
theorem IsPiSystem.prod {C : Set (Set α)} {D : Set (Set β)} (hC : IsPiSystem C)
    (hD : IsPiSystem D) : IsPiSystem (image2 (· ×ˢ ·) C D) 
Arguments: {α : Type u_1} {β : Type u_3} {C : Set (Set α)} {D : Set (Set β)} (hC : IsPiSystem C) (hD : IsPiSystem D)
Goal: IsPiSystem (Set.image2 (fun x x_1 => x ×ˢ x_1) C D)
-- IsCountablySpanning.prod
theorem IsCountablySpanning.prod {C : Set (Set α)} {D : Set (Set β)} (hC : IsCountablySpanning C)
    (hD : IsCountablySpanning D) : IsCountablySpanning (image2 (· ×ˢ ·) C D) 
Arguments: {α : Type u_1} {β : Type u_3} {C : Set (Set α)} {D : Set (Set β)} (hC : IsCountablySpanning C) (hD : IsCountablySpanning D)
Goal: IsCountablySpanning (Set.image2 (fun x x_1 => x ×ˢ x_1) C D)
-- generateFrom_prod_eq
theorem generateFrom_prod_eq {α β} {C : Set (Set α)} {D : Set (Set β)} (hC : IsCountablySpanning C)
    (hD : IsCountablySpanning D) :
    @Prod.instMeasurableSpace _ _ (generateFrom C) (generateFrom D) =
      generateFrom (image2 (· ×ˢ ·) C D) 
Arguments: {α : Type u_7} {β : Type u_8} {C : Set (Set α)} {D : Set (Set β)} (hC : IsCountablySpanning C) (hD : IsCountablySpanning D)
Goal: Prod.instMeasurableSpace = MeasurableSpace.generateFrom (Set.image2 (fun x x_1 => x ×ˢ x_1) C D)
-- generateFrom_eq_prod
theorem generateFrom_eq_prod {C : Set (Set α)} {D : Set (Set β)} (hC : generateFrom C = ‹_›)
    (hD : generateFrom D = ‹_›) (h2C : IsCountablySpanning C) (h2D : IsCountablySpanning D) :
    generateFrom (image2 (· ×ˢ ·) C D) = Prod.instMeasurableSpace 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {C : Set (Set α)} {D : Set (Set β)} (hC : MeasurableSpace.generateFrom C = inst✝) (hD : MeasurableSpace.generateFrom D = inst✝¹) (h2C : IsCountablySpanning C) (h2D : IsCountablySpanning D)
Goal: MeasurableSpace.generateFrom (Set.image2 (fun x x_1 => x ×ˢ x_1) C D) = Prod.instMeasurableSpace
-- generateFrom_prod
theorem generateFrom_prod :
    generateFrom (image2 (· ×ˢ ·) { s : Set α | MeasurableSet s } { t : Set β | MeasurableSet t }) =
      Prod.instMeasurableSpace 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β]
Goal: MeasurableSpace.generateFrom (Set.image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t}) =
  Prod.instMeasurableSpace
-- isPiSystem_prod
theorem isPiSystem_prod :
    IsPiSystem (image2 (· ×ˢ ·) { s : Set α | MeasurableSet s } { t : Set β | MeasurableSet t }) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β]
Goal: IsPiSystem (Set.image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t})
-- measurable_measure_prod_mk_left_finite
theorem measurable_measure_prod_mk_left_finite [IsFiniteMeasure ν] {s : Set (α × β)}
    (hs : MeasurableSet s) : Measurable fun x => ν (Prod.mk x ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [MeasureTheory.IsFiniteMeasure ν] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: Measurable fun x => ν (Prod.mk x ⁻¹' s)
-- measurable_measure_prod_mk_left
theorem measurable_measure_prod_mk_left [SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s) :
    Measurable fun x => ν (Prod.mk x ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: Measurable fun x => ν (Prod.mk x ⁻¹' s)
-- measurable_measure_prod_mk_right
theorem measurable_measure_prod_mk_right {μ : Measure α} [SFinite μ] {s : Set (α × β)}
    (hs : MeasurableSet s) : Measurable fun y => μ ((fun x => (x, y)) ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: Measurable fun y => μ ((fun x => (x, y)) ⁻¹' s)
-- Measurable.map_prod_mk_left
theorem Measurable.map_prod_mk_left [SFinite ν] :
    Measurable fun x : α => map (Prod.mk x) ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν]
Goal: Measurable fun x => MeasureTheory.Measure.map (Prod.mk x) ν
-- Measurable.map_prod_mk_right
theorem Measurable.map_prod_mk_right {μ : Measure α} [SFinite μ] :
    Measurable fun y : β => map (fun x : α => (x, y)) μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ]
Goal: Measurable fun y => MeasureTheory.Measure.map (fun x => (x, y)) μ
-- MeasurableEmbedding.prod_mk
theorem MeasurableEmbedding.prod_mk {α β γ δ : Type*} {mα : MeasurableSpace α}
    {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ} {f : α → β}
    {g : γ → δ} (hg : MeasurableEmbedding g) (hf : MeasurableEmbedding f) :
    MeasurableEmbedding fun x : γ × α => (g x.1, f x.2) 
Arguments: {α : Type u_7} {β : Type u_8} {γ : Type u_9} {δ : Type u_10} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ} {f : α → β} {g : γ → δ} (hg : MeasurableEmbedding g) (hf : MeasurableEmbedding f)
Goal: MeasurableEmbedding fun x => (g x.1, f x.2)
-- MeasurableEmbedding.prod_mk_left
lemma MeasurableEmbedding.prod_mk_left {β γ : Type*} [MeasurableSingletonClass α]
    {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
    (x : α) {f : γ → β} (hf : MeasurableEmbedding f) :
    MeasurableEmbedding (fun y ↦ (x, f y)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_7} {γ : Type u_8} [MeasurableSingletonClass α] {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (x : α) {f : γ → β} (hf : MeasurableEmbedding f)
Goal: MeasurableEmbedding fun y => (x, f y)
-- measurableEmbedding_prod_mk_left
lemma measurableEmbedding_prod_mk_left [MeasurableSingletonClass α] (x : α) :
    MeasurableEmbedding (Prod.mk x : β → α × β) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] (x : α)
Goal: MeasurableEmbedding (Prod.mk x)
-- MeasurableEmbedding.prod_mk_right
lemma MeasurableEmbedding.prod_mk_right {β γ : Type*} [MeasurableSingletonClass α]
    {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
    {f : γ → β} (hf : MeasurableEmbedding f) (x : α) :
    MeasurableEmbedding (fun y ↦ (f y, x)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_7} {γ : Type u_8} [MeasurableSingletonClass α] {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {f : γ → β} (hf : MeasurableEmbedding f) (x : α)
Goal: MeasurableEmbedding fun y => (f y, x)
-- measurableEmbedding_prod_mk_right
lemma measurableEmbedding_prod_mk_right [MeasurableSingletonClass α] (x : α) :
    MeasurableEmbedding (fun y ↦ (y, x) : β → β × α) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass α] (x : α)
Goal: MeasurableEmbedding fun y => (y, x)
-- Measurable.lintegral_prod_right'
theorem Measurable.lintegral_prod_right' [SFinite ν] :
    ∀ {f : α × β → ℝ≥0∞}, Measurable f → Measurable fun x => ∫⁻ y, f (x, y) ∂ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {f : α × β → ℝ≥0∞}
Goal: Measurable f → Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν
-- Measurable.lintegral_prod_right
theorem Measurable.lintegral_prod_right [SFinite ν] {f : α → β → ℝ≥0∞}
    (hf : Measurable (uncurry f)) : Measurable fun x => ∫⁻ y, f x y ∂ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {f : α → β → ℝ≥0∞} (hf : Measurable (Function.uncurry f))
Goal: Measurable fun x => ∫⁻ (y : β), f x y ∂ν
-- Measurable.lintegral_prod_left'
theorem Measurable.lintegral_prod_left' [SFinite μ] {f : α × β → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun y => ∫⁻ x, f (x, y) ∂μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {f : α × β → ℝ≥0∞} (hf : Measurable f)
Goal: Measurable fun y => ∫⁻ (x : α), f (x, y) ∂μ
-- Measurable.lintegral_prod_left
theorem Measurable.lintegral_prod_left [SFinite μ] {f : α → β → ℝ≥0∞}
    (hf : Measurable (uncurry f)) : Measurable fun y => ∫⁻ x, f x y ∂μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {f : α → β → ℝ≥0∞} (hf : Measurable (Function.uncurry f))
Goal: Measurable fun y => ∫⁻ (x : α), f x y ∂μ
-- MeasureTheory.Measure.prod_def
protected irreducible_def prod (μ : Measure α) (ν : Measure β) : Measure (α × β) 
Arguments: {α : Type u_7} {β : Type u_8} [MeasurableSpace α] [MeasurableSpace β] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure β)
Goal: μ.prod ν = μ.bind fun x => MeasureTheory.Measure.map (Prod.mk x) ν
-- MeasureTheory.Measure.volume_eq_prod
theorem volume_eq_prod (α β) [MeasureSpace α] [MeasureSpace β] :
    (volume : Measure (α × β)) = (volume : Measure α).prod (volume : Measure β) 
Arguments: (α : Type u_7) (β : Type u_8) [MeasureTheory.MeasureSpace α] [MeasureTheory.MeasureSpace β]
Goal: MeasureTheory.volume = MeasureTheory.volume.prod MeasureTheory.volume
-- MeasureTheory.Measure.prod_apply
theorem prod_apply {s : Set (α × β)} (hs : MeasurableSet s) :
    μ.prod ν s = ∫⁻ x, ν (Prod.mk x ⁻¹' s) ∂μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: (μ.prod ν) s = ∫⁻ (x : α), ν (Prod.mk x ⁻¹' s) ∂μ
-- MeasureTheory.Measure.prod_prod
@[simp]
theorem prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] (s : Set α) (t : Set β)
Goal: (μ.prod ν) (s ×ˢ t) = μ s * ν t
-- MeasureTheory.Measure.map_fst_prod
@[simp] lemma map_fst_prod : Measure.map Prod.fst (μ.prod ν) = (ν univ) • μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν]
Goal: MeasureTheory.Measure.map Prod.fst (μ.prod ν) = ν Set.univ • μ
-- MeasureTheory.Measure.map_snd_prod
@[simp] lemma map_snd_prod : Measure.map Prod.snd (μ.prod ν) = (μ univ) • ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν]
Goal: MeasureTheory.Measure.map Prod.snd (μ.prod ν) = μ Set.univ • ν
-- MeasureTheory.Measure.ae_measure_lt_top
theorem ae_measure_lt_top {s : Set (α × β)} (hs : MeasurableSet s) (h2s : (μ.prod ν) s ≠ ∞) :
    ∀ᵐ x ∂μ, ν (Prod.mk x ⁻¹' s) < ∞ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s) (h2s : (μ.prod ν) s ≠ ⊤)
Goal: ∀ᵐ (x : α) ∂μ, ν (Prod.mk x ⁻¹' s) < ⊤
-- MeasureTheory.Measure.measure_prod_null
theorem measure_prod_null {s : Set (α × β)} (hs : MeasurableSet s) :
    μ.prod ν s = 0 ↔ (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: (μ.prod ν) s = 0 ↔ (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0
-- MeasureTheory.Measure.measure_ae_null_of_prod_null
theorem measure_ae_null_of_prod_null {s : Set (α × β)} (h : μ.prod ν s = 0) :
    (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (h : (μ.prod ν) s = 0)
Goal: (fun x => ν (Prod.mk x ⁻¹' s)) =ᵐ[μ] 0
-- MeasureTheory.Measure.AbsolutelyContinuous.prod
theorem AbsolutelyContinuous.prod [SFinite ν'] (h1 : μ ≪ μ') (h2 : ν ≪ ν') :
    μ.prod ν ≪ μ'.prod ν' 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ μ' : MeasureTheory.Measure α} {ν ν' : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [MeasureTheory.SFinite ν'] (h1 : μ ≪ μ') (h2 : ν ≪ ν')
Goal: μ.prod ν ≪ μ'.prod ν'
-- MeasureTheory.Measure.ae_ae_of_ae_prod
theorem ae_ae_of_ae_prod {p : α × β → Prop} (h : ∀ᵐ z ∂μ.prod ν, p z) :
    ∀ᵐ x ∂μ, ∀ᵐ y ∂ν, p (x, y) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {p : α × β → Prop} (h : ∀ᵐ (z : α × β) ∂μ.prod ν, p z)
Goal: ∀ᵐ (x : α) ∂μ, ∀ᵐ (y : β) ∂ν, p (x, y)
-- MeasureTheory.Measure.ae_ae_eq_curry_of_prod
theorem ae_ae_eq_curry_of_prod {f g : α × β → γ} (h : f =ᵐ[μ.prod ν] g) :
    ∀ᵐ x ∂μ, curry f x =ᵐ[ν] curry g x 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {f g : α × β → γ} (h : f =ᵐ[μ.prod ν] g)
Goal: ∀ᵐ (x : α) ∂μ, Function.curry f x =ᵐ[ν] Function.curry g x
-- MeasureTheory.Measure.ae_ae_eq_of_ae_eq_uncurry
theorem ae_ae_eq_of_ae_eq_uncurry {f g : α → β → γ} (h : uncurry f =ᵐ[μ.prod ν] uncurry g) :
    ∀ᵐ x ∂μ, f x =ᵐ[ν] g x 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {f g : α → β → γ} (h : Function.uncurry f =ᵐ[μ.prod ν] Function.uncurry g)
Goal: ∀ᵐ (x : α) ∂μ, f x =ᵐ[ν] g x
-- MeasureTheory.Measure.ae_prod_mem_iff_ae_ae_mem
theorem ae_prod_mem_iff_ae_ae_mem {s : Set (α × β)} (hs : MeasurableSet s) :
    (∀ᵐ z ∂μ.prod ν, z ∈ s) ↔ ∀ᵐ x ∂μ, ∀ᵐ y ∂ν, (x, y) ∈ s 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: (∀ᵐ (z : α × β) ∂μ.prod ν, z ∈ s) ↔ ∀ᵐ (x : α) ∂μ, ∀ᵐ (y : β) ∂ν, (x, y) ∈ s
-- MeasureTheory.Measure.quasiMeasurePreserving_fst
theorem quasiMeasurePreserving_fst : QuasiMeasurePreserving Prod.fst (μ.prod ν) μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving Prod.fst (μ.prod ν) μ
-- MeasureTheory.Measure.quasiMeasurePreserving_snd
theorem quasiMeasurePreserving_snd : QuasiMeasurePreserving Prod.snd (μ.prod ν) ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν]
Goal: MeasureTheory.Measure.QuasiMeasurePreserving Prod.snd (μ.prod ν) ν
-- MeasureTheory.Measure.set_prod_ae_eq
lemma set_prod_ae_eq {s s' : Set α} {t t' : Set β} (hs : s =ᵐ[μ] s') (ht : t =ᵐ[ν] t') :
    (s ×ˢ t : Set (α × β)) =ᵐ[μ.prod ν] (s' ×ˢ t' : Set (α × β)) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s s' : Set α} {t t' : Set β} (hs : s =ᵐ[μ] s') (ht : t =ᵐ[ν] t')
Goal: s ×ˢ t =ᵐ[μ.prod ν] s' ×ˢ t'
-- MeasureTheory.Measure.measure_prod_compl_eq_zero
lemma measure_prod_compl_eq_zero {s : Set α} {t : Set β}
    (s_ae_univ : μ sᶜ = 0) (t_ae_univ : ν tᶜ = 0) :
    μ.prod ν (s ×ˢ t)ᶜ = 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set α} {t : Set β} (s_ae_univ : μ sᶜ = 0) (t_ae_univ : ν tᶜ = 0)
Goal: (μ.prod ν) (s ×ˢ t)ᶜ = 0
-- MeasureTheory.NullMeasurableSet.prod
lemma _root_.MeasureTheory.NullMeasurableSet.prod {s : Set α} {t : Set β}
    (s_mble : NullMeasurableSet s μ) (t_mble : NullMeasurableSet t ν) :
    NullMeasurableSet (s ×ˢ t) (μ.prod ν) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set α} {t : Set β} (s_mble : MeasureTheory.NullMeasurableSet s μ) (t_mble : MeasureTheory.NullMeasurableSet t ν)
Goal: MeasureTheory.NullMeasurableSet (s ×ˢ t) (μ.prod ν)
-- MeasureTheory.NullMeasurableSet.right_of_prod
lemma _root_.MeasureTheory.NullMeasurableSet.right_of_prod {s : Set α} {t : Set β}
    (h : NullMeasurableSet (s ×ˢ t) (μ.prod ν)) (hs : μ s ≠ 0) : NullMeasurableSet t ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set α} {t : Set β} (h : MeasureTheory.NullMeasurableSet (s ×ˢ t) (μ.prod ν)) (hs : μ s ≠ 0)
Goal: MeasureTheory.NullMeasurableSet t ν
-- MeasureTheory.NullMeasurableSet.of_preimage_snd
lemma _root_.MeasureTheory.NullMeasurableSet.of_preimage_snd [NeZero μ] {t : Set β}
    (h : NullMeasurableSet (Prod.snd ⁻¹' t) (μ.prod ν)) : NullMeasurableSet t ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [NeZero μ] {t : Set β} (h : MeasureTheory.NullMeasurableSet (Prod.snd ⁻¹' t) (μ.prod ν))
Goal: MeasureTheory.NullMeasurableSet t ν
-- MeasureTheory.Measure.nullMeasurableSet_preimage_snd
lemma nullMeasurableSet_preimage_snd [NeZero μ] {t : Set β} :
    NullMeasurableSet (Prod.snd ⁻¹' t) (μ.prod ν) ↔ NullMeasurableSet t ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [NeZero μ] {t : Set β}
Goal: MeasureTheory.NullMeasurableSet (Prod.snd ⁻¹' t) (μ.prod ν) ↔ MeasureTheory.NullMeasurableSet t ν
-- MeasureTheory.Measure.nullMeasurable_comp_snd
lemma nullMeasurable_comp_snd [NeZero μ] {f : β → γ} :
    NullMeasurable (f ∘ Prod.snd) (μ.prod ν) ↔ NullMeasurable f ν 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [NeZero μ] {f : β → γ}
Goal: MeasureTheory.NullMeasurable (f ∘ Prod.snd) (μ.prod ν) ↔ MeasureTheory.NullMeasurable f ν
-- MeasureTheory.Measure.prod_sum_left
lemma prod_sum_left {ι : Type*} (m : ι → Measure α) (μ : Measure β) [SFinite μ] :
    (Measure.sum m).prod μ = Measure.sum (fun i ↦ (m i).prod μ) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ι : Type u_7} (m : ι → MeasureTheory.Measure α) (μ : MeasureTheory.Measure β) [MeasureTheory.SFinite μ]
Goal: (MeasureTheory.Measure.sum m).prod μ = MeasureTheory.Measure.sum fun i => (m i).prod μ
-- MeasureTheory.Measure.prod_sum_right
lemma prod_sum_right {ι' : Type*} [Countable ι'] (m : Measure α) (m' : ι' → Measure β)
    [∀ n, SFinite (m' n)] :
    m.prod (Measure.sum m') = Measure.sum (fun p ↦ m.prod (m' p)) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ι' : Type u_7} [Countable ι'] (m : MeasureTheory.Measure α) (m' : ι' → MeasureTheory.Measure β) [∀ (n : ι'), MeasureTheory.SFinite (m' n)]
Goal: m.prod (MeasureTheory.Measure.sum m') = MeasureTheory.Measure.sum fun p => m.prod (m' p)
-- MeasureTheory.Measure.prod_sum
lemma prod_sum {ι ι' : Type*} [Countable ι'] (m : ι → Measure α) (m' : ι' → Measure β)
    [∀ n, SFinite (m' n)] :
    (Measure.sum m).prod (Measure.sum m') =
      Measure.sum (fun (p : ι × ι') ↦ (m p.1).prod (m' p.2)) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ι : Type u_7} {ι' : Type u_8} [Countable ι'] (m : ι → MeasureTheory.Measure α) (m' : ι' → MeasureTheory.Measure β) [∀ (n : ι'), MeasureTheory.SFinite (m' n)]
Goal: (MeasureTheory.Measure.sum m).prod (MeasureTheory.Measure.sum m') =
  MeasureTheory.Measure.sum fun p => (m p.1).prod (m' p.2)
-- MeasureTheory.Measure.prod_eq_generateFrom
theorem prod_eq_generateFrom {μ : Measure α} {ν : Measure β} {C : Set (Set α)} {D : Set (Set β)}
    (hC : generateFrom C = ‹_›) (hD : generateFrom D = ‹_›) (h2C : IsPiSystem C)
    (h2D : IsPiSystem D) (h3C : μ.FiniteSpanningSetsIn C) (h3D : ν.FiniteSpanningSetsIn D)
    {μν : Measure (α × β)} (h₁ : ∀ s ∈ C, ∀ t ∈ D, μν (s ×ˢ t) = μ s * ν t) : μ.prod ν = μν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {C : Set (Set α)} {D : Set (Set β)} (hC : MeasurableSpace.generateFrom C = inst✝) (hD : MeasurableSpace.generateFrom D = inst✝¹) (h2C : IsPiSystem C) (h2D : IsPiSystem D) (h3C : μ.FiniteSpanningSetsIn C) (h3D : ν.FiniteSpanningSetsIn D) {μν : MeasureTheory.Measure (α × β)} (h₁ : ∀ s ∈ C, ∀ t ∈ D, μν (s ×ˢ t) = μ s * ν t)
Goal: μ.prod ν = μν