ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.prod_eq
theorem prod_eq {μ : Measure α} [SigmaFinite μ] {ν : Measure β} [SigmaFinite ν]
    {μν : Measure (α × β)}
    (h : ∀ s t, MeasurableSet s → MeasurableSet t → 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} [MeasureTheory.SigmaFinite μ] {ν : MeasureTheory.Measure β} [MeasureTheory.SigmaFinite ν] {μν : MeasureTheory.Measure (α × β)} (h : ∀ (s : Set α) (t : Set β), MeasurableSet s → MeasurableSet t → μν (s ×ˢ t) = μ s * ν t)
Goal: μ.prod ν = μν
-- MeasureTheory.Measure.prod_swap
theorem prod_swap : map Prod.swap (μ.prod ν) = ν.prod 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] [MeasureTheory.SFinite μ]
Goal: MeasureTheory.Measure.map Prod.swap (μ.prod ν) = ν.prod μ
-- MeasureTheory.Measure.measurePreserving_swap
theorem measurePreserving_swap : MeasurePreserving Prod.swap (μ.prod ν) (ν.prod 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] [MeasureTheory.SFinite μ]
Goal: MeasureTheory.MeasurePreserving Prod.swap (μ.prod ν) (ν.prod μ)
-- MeasureTheory.Measure.prod_apply_symm
theorem prod_apply_symm {s : Set (α × β)} (hs : MeasurableSet s) :
    μ.prod ν s = ∫⁻ y, μ ((fun x => (x, y)) ⁻¹' s) ∂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] [MeasureTheory.SFinite μ] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: (μ.prod ν) s = ∫⁻ (y : β), μ ((fun x => (x, y)) ⁻¹' s) ∂ν
-- MeasureTheory.NullMeasurableSet.left_of_prod
lemma _root_.MeasureTheory.NullMeasurableSet.left_of_prod {s : Set α} {t : Set β}
    (h : NullMeasurableSet (s ×ˢ t) (μ.prod ν)) (ht : ν t ≠ 0) : NullMeasurableSet 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] [MeasureTheory.SFinite μ] {s : Set α} {t : Set β} (h : MeasureTheory.NullMeasurableSet (s ×ˢ t) (μ.prod ν)) (ht : ν t ≠ 0)
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.NullMeasurableSet.of_preimage_fst
lemma _root_.MeasureTheory.NullMeasurableSet.of_preimage_fst [NeZero ν] {s : Set α}
    (h : NullMeasurableSet (Prod.fst ⁻¹' s) (μ.prod ν)) : NullMeasurableSet 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] [MeasureTheory.SFinite μ] [NeZero ν] {s : Set α} (h : MeasureTheory.NullMeasurableSet (Prod.fst ⁻¹' s) (μ.prod ν))
Goal: MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.Measure.nullMeasurableSet_preimage_fst
lemma nullMeasurableSet_preimage_fst [NeZero ν] {s : Set α} :
    NullMeasurableSet (Prod.fst ⁻¹' s) (μ.prod ν) ↔ NullMeasurableSet 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] [MeasureTheory.SFinite μ] [NeZero ν] {s : Set α}
Goal: MeasureTheory.NullMeasurableSet (Prod.fst ⁻¹' s) (μ.prod ν) ↔ MeasureTheory.NullMeasurableSet s μ
-- MeasureTheory.Measure.nullMeasurable_comp_fst
lemma nullMeasurable_comp_fst [NeZero ν] {f : α → γ} :
    NullMeasurable (f ∘ Prod.fst) (μ.prod ν) ↔ NullMeasurable f μ 
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] [MeasureTheory.SFinite μ] [NeZero ν] {f : α → γ}
Goal: MeasureTheory.NullMeasurable (f ∘ Prod.fst) (μ.prod ν) ↔ MeasureTheory.NullMeasurable f μ
-- MeasureTheory.Measure.nullMeasurableSet_prod_of_ne_zero
lemma nullMeasurableSet_prod_of_ne_zero {s : Set α} {t : Set β} (hs : μ s ≠ 0) (ht : ν t ≠ 0) :
    NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ 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] [MeasureTheory.SFinite μ] {s : Set α} {t : Set β} (hs : μ s ≠ 0) (ht : ν t ≠ 0)
Goal: MeasureTheory.NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔
  MeasureTheory.NullMeasurableSet s μ ∧ MeasureTheory.NullMeasurableSet t ν
-- MeasureTheory.Measure.nullMeasurableSet_prod
lemma nullMeasurableSet_prod {s : Set α} {t : Set β} :
    NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔
      NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 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] [MeasureTheory.SFinite μ] {s : Set α} {t : Set β}
Goal: MeasureTheory.NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔
  MeasureTheory.NullMeasurableSet s μ ∧ MeasureTheory.NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0
-- MeasureTheory.Measure.prodAssoc_prod
theorem prodAssoc_prod [SFinite τ] :
    map MeasurableEquiv.prodAssoc ((μ.prod ν).prod τ) = μ.prod (ν.prod τ) 
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} {τ : MeasureTheory.Measure γ} [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite τ]
Goal: MeasureTheory.Measure.map (⇑MeasurableEquiv.prodAssoc) ((μ.prod ν).prod τ) = μ.prod (ν.prod τ)
-- MeasureTheory.Measure.prod_restrict
theorem prod_restrict (s : Set α) (t : Set β) :
    (μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s ×ˢ 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] [MeasureTheory.SFinite μ] (s : Set α) (t : Set β)
Goal: (μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s ×ˢ t)
-- MeasureTheory.Measure.restrict_prod_eq_prod_univ
theorem restrict_prod_eq_prod_univ (s : Set α) :
    (μ.restrict s).prod ν = (μ.prod ν).restrict (s ×ˢ univ) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] (s : Set α)
Goal: (μ.restrict s).prod ν = (μ.prod ν).restrict (s ×ˢ Set.univ)
-- MeasureTheory.Measure.prod_dirac
theorem prod_dirac (y : β) : μ.prod (dirac y) = map (fun x => (x, y)) μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (y : β)
Goal: μ.prod (MeasureTheory.Measure.dirac y) = MeasureTheory.Measure.map (fun x => (x, y)) μ
-- MeasureTheory.Measure.dirac_prod
theorem dirac_prod (x : α) : (dirac x).prod ν = 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] (x : α)
Goal: (MeasureTheory.Measure.dirac x).prod ν = MeasureTheory.Measure.map (Prod.mk x) ν
-- MeasureTheory.Measure.dirac_prod_dirac
theorem dirac_prod_dirac {x : α} {y : β} : (dirac x).prod (dirac y) = dirac (x, y) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {x : α} {y : β}
Goal: (MeasureTheory.Measure.dirac x).prod (MeasureTheory.Measure.dirac y) = MeasureTheory.Measure.dirac (x, y)
-- MeasureTheory.Measure.prod_add
theorem prod_add (ν' : Measure β) [SFinite ν'] : μ.prod (ν + ν') = μ.prod ν + 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] [MeasureTheory.SFinite μ] (ν' : MeasureTheory.Measure β) [MeasureTheory.SFinite ν']
Goal: μ.prod (ν + ν') = μ.prod ν + μ.prod ν'
-- MeasureTheory.Measure.add_prod
theorem add_prod (μ' : Measure α) [SFinite μ'] : (μ + μ').prod ν = μ.prod ν + 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] [MeasureTheory.SFinite μ] (μ' : MeasureTheory.Measure α) [MeasureTheory.SFinite μ']
Goal: (μ + μ').prod ν = μ.prod ν + μ'.prod ν
-- MeasureTheory.Measure.zero_prod
@[simp]
theorem zero_prod (ν : Measure β) : (0 : Measure α).prod ν = 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (ν : MeasureTheory.Measure β)
Goal: MeasureTheory.Measure.prod 0 ν = 0
-- MeasureTheory.Measure.prod_zero
@[simp]
theorem prod_zero (μ : Measure α) : μ.prod (0 : Measure β) = 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] (μ : MeasureTheory.Measure α)
Goal: μ.prod 0 = 0
-- MeasureTheory.Measure.map_prod_map
theorem map_prod_map {δ} [MeasurableSpace δ] {f : α → β} {g : γ → δ} (μa : Measure α)
    (μc : Measure γ) [SFinite μa] [SFinite μc] (hf : Measurable f) (hg : Measurable g) :
    (map f μa).prod (map g μc) = map (Prod.map f g) (μa.prod μc) 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {δ : Type u_7} [MeasurableSpace δ] {f : α → β} {g : γ → δ} (μa : MeasureTheory.Measure α) (μc : MeasureTheory.Measure γ) [MeasureTheory.SFinite μa] [MeasureTheory.SFinite μc] (hf : Measurable f) (hg : Measurable g)
Goal: (MeasureTheory.Measure.map f μa).prod (MeasureTheory.Measure.map g μc) =
  MeasureTheory.Measure.map (Prod.map f g) (μa.prod μc)
-- MeasureTheory.MeasurePreserving.skew_product
theorem skew_product [SFinite μa] [SFinite μc] {f : α → β} (hf : MeasurePreserving f μa μb)
    {g : α → γ → δ} (hgm : Measurable (uncurry g)) (hg : ∀ᵐ x ∂μa, map (g x) μc = μd) :
    MeasurePreserving (fun p : α × γ => (f p.1, g p.1 p.2)) (μa.prod μc) (μb.prod μd) 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {δ : Type u_7} [MeasurableSpace δ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {μd : MeasureTheory.Measure δ} [MeasureTheory.SFinite μa] [MeasureTheory.SFinite μc] {f : α → β} (hf : MeasureTheory.MeasurePreserving f μa μb) {g : α → γ → δ} (hgm : Measurable (Function.uncurry g)) (hg : ∀ᵐ (x : α) ∂μa, MeasureTheory.Measure.map (g x) μc = μd)
Goal: MeasureTheory.MeasurePreserving (fun p => (f p.1, g p.1 p.2)) (μa.prod μc) (μb.prod μd)
-- MeasureTheory.MeasurePreserving.prod
protected theorem prod [SFinite μa] [SFinite μc] {f : α → β} {g : γ → δ}
    (hf : MeasurePreserving f μa μb) (hg : MeasurePreserving g μc μd) :
    MeasurePreserving (Prod.map f g) (μa.prod μc) (μb.prod μd) 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {δ : Type u_7} [MeasurableSpace δ] {μa : MeasureTheory.Measure α} {μb : MeasureTheory.Measure β} {μc : MeasureTheory.Measure γ} {μd : MeasureTheory.Measure δ} [MeasureTheory.SFinite μa] [MeasureTheory.SFinite μc] {f : α → β} {g : γ → δ} (hf : MeasureTheory.MeasurePreserving f μa μb) (hg : MeasureTheory.MeasurePreserving g μc μd)
Goal: MeasureTheory.MeasurePreserving (Prod.map f g) (μa.prod μc) (μb.prod μd)
-- MeasureTheory.QuasiMeasurePreserving.prod_of_right
theorem prod_of_right {f : α × β → γ} {μ : Measure α} {ν : Measure β} {τ : Measure γ}
    (hf : Measurable f) [SFinite ν]
    (h2f : ∀ᵐ x ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ) :
    QuasiMeasurePreserving f (μ.prod ν) τ 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : α × β → γ} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {τ : MeasureTheory.Measure γ} (hf : Measurable f) [MeasureTheory.SFinite ν] (h2f : ∀ᵐ (x : α) ∂μ, MeasureTheory.Measure.QuasiMeasurePreserving (fun y => f (x, y)) ν τ)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f (μ.prod ν) τ
-- MeasureTheory.QuasiMeasurePreserving.prod_of_left
theorem prod_of_left {α β γ} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
    {f : α × β → γ} {μ : Measure α} {ν : Measure β} {τ : Measure γ} (hf : Measurable f)
    [SFinite μ] [SFinite ν]
    (h2f : ∀ᵐ y ∂ν, QuasiMeasurePreserving (fun x => f (x, y)) μ τ) :
    QuasiMeasurePreserving f (μ.prod ν) τ 
Arguments: {α : Type u_7} {β : Type u_8} {γ : Type u_9} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {f : α × β → γ} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {τ : MeasureTheory.Measure γ} (hf : Measurable f) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] (h2f : ∀ᵐ (y : β) ∂ν, MeasureTheory.Measure.QuasiMeasurePreserving (fun x => f (x, y)) μ τ)
Goal: MeasureTheory.Measure.QuasiMeasurePreserving f (μ.prod ν) τ
-- AEMeasurable.prod_swap
theorem AEMeasurable.prod_swap [SFinite μ] [SFinite ν] {f : β × α → γ}
    (hf : AEMeasurable f (ν.prod μ)) : AEMeasurable (fun z : α × β => f z.swap) (μ.prod 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 μ] [MeasureTheory.SFinite ν] {f : β × α → γ} (hf : AEMeasurable f (ν.prod μ))
Goal: AEMeasurable (fun z => f z.swap) (μ.prod ν)
-- AEMeasurable.fst
theorem AEMeasurable.fst [SFinite ν] {f : α → γ} (hf : AEMeasurable f μ) :
    AEMeasurable (fun z : α × β => f z.1) (μ.prod 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] {f : α → γ} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun z => f z.1) (μ.prod ν)
-- AEMeasurable.snd
theorem AEMeasurable.snd [SFinite ν] {f : β → γ} (hf : AEMeasurable f ν) :
    AEMeasurable (fun z : α × β => f z.2) (μ.prod 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] {f : β → γ} (hf : AEMeasurable f ν)
Goal: AEMeasurable (fun z => f z.2) (μ.prod ν)
-- MeasureTheory.lintegral_prod_swap
theorem lintegral_prod_swap [SFinite μ] (f : α × β → ℝ≥0∞) :
    ∫⁻ z, f z.swap ∂ν.prod μ = ∫⁻ z, f z ∂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] [MeasureTheory.SFinite μ] (f : α × β → ℝ≥0∞)
Goal: ∫⁻ (z : β × α), f z.swap ∂ν.prod μ = ∫⁻ (z : α × β), f z ∂μ.prod ν
-- MeasureTheory.lintegral_prod_of_measurable
theorem lintegral_prod_of_measurable :
    ∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ z, f z ∂μ.prod ν = ∫⁻ x, ∫⁻ y, f (x, y) ∂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] (f : α × β → ℝ≥0∞)
Goal: Measurable f → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ
-- MeasureTheory.lintegral_prod
theorem lintegral_prod (f : α × β → ℝ≥0∞) (hf : AEMeasurable f (μ.prod ν)) :
    ∫⁻ z, f z ∂μ.prod ν = ∫⁻ x, ∫⁻ y, f (x, y) ∂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] (f : α × β → ℝ≥0∞) (hf : AEMeasurable f (μ.prod ν))
Goal: ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ
-- MeasureTheory.lintegral_prod_symm
theorem lintegral_prod_symm [SFinite μ] (f : α × β → ℝ≥0∞) (hf : AEMeasurable f (μ.prod ν)) :
    ∫⁻ z, f z ∂μ.prod ν = ∫⁻ y, ∫⁻ x, f (x, y) ∂μ ∂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] [MeasureTheory.SFinite μ] (f : α × β → ℝ≥0∞) (hf : AEMeasurable f (μ.prod ν))
Goal: ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν
-- MeasureTheory.lintegral_prod_symm'
theorem lintegral_prod_symm' [SFinite μ] (f : α × β → ℝ≥0∞) (hf : Measurable f) :
    ∫⁻ z, f z ∂μ.prod ν = ∫⁻ y, ∫⁻ x, f (x, y) ∂μ ∂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] [MeasureTheory.SFinite μ] (f : α × β → ℝ≥0∞) (hf : Measurable f)
Goal: ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν
-- MeasureTheory.lintegral_lintegral
theorem lintegral_lintegral ⦃f : α → β → ℝ≥0∞⦄ (hf : AEMeasurable (uncurry f) (μ.prod ν)) :
    ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ = ∫⁻ z, f z.1 z.2 ∂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] ⦃f : α → β → ℝ≥0∞⦄ (hf : AEMeasurable (Function.uncurry f) (μ.prod ν))
Goal: ∫⁻ (x : α), ∫⁻ (y : β), f x y ∂ν ∂μ = ∫⁻ (z : α × β), f z.1 z.2 ∂μ.prod ν
-- MeasureTheory.lintegral_lintegral_symm
theorem lintegral_lintegral_symm [SFinite μ] ⦃f : α → β → ℝ≥0∞⦄
    (hf : AEMeasurable (uncurry f) (μ.prod ν)) :
    ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ = ∫⁻ z, f z.2 z.1 ∂ν.prod 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] [MeasureTheory.SFinite μ] ⦃f : α → β → ℝ≥0∞⦄ (hf : AEMeasurable (Function.uncurry f) (μ.prod ν))
Goal: ∫⁻ (x : α), ∫⁻ (y : β), f x y ∂ν ∂μ = ∫⁻ (z : β × α), f z.2 z.1 ∂ν.prod μ
-- MeasureTheory.lintegral_lintegral_swap
theorem lintegral_lintegral_swap [SFinite μ] ⦃f : α → β → ℝ≥0∞⦄
    (hf : AEMeasurable (uncurry f) (μ.prod ν)) :
    ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ = ∫⁻ y, ∫⁻ x, f x y ∂μ ∂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] [MeasureTheory.SFinite μ] ⦃f : α → β → ℝ≥0∞⦄ (hf : AEMeasurable (Function.uncurry f) (μ.prod ν))
Goal: ∫⁻ (x : α), ∫⁻ (y : β), f x y ∂ν ∂μ = ∫⁻ (y : β), ∫⁻ (x : α), f x y ∂μ ∂ν
-- MeasureTheory.lintegral_prod_mul
theorem lintegral_prod_mul {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g ν) : ∫⁻ z, f z.1 * g z.2 ∂μ.prod ν = (∫⁻ x, f x ∂μ) * ∫⁻ y, g y ∂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] {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g ν)
Goal: ∫⁻ (z : α × β), f z.1 * g z.2 ∂μ.prod ν = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν
-- MeasureTheory.Measure.fst_apply
theorem fst_apply {s : Set α} (hs : MeasurableSet s) : ρ.fst s = ρ (Prod.fst ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)} {s : Set α} (hs : MeasurableSet s)
Goal: ρ.fst s = ρ (Prod.fst ⁻¹' s)
-- MeasureTheory.Measure.fst_univ
theorem fst_univ : ρ.fst univ = ρ univ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)}
Goal: ρ.fst Set.univ = ρ Set.univ
-- MeasureTheory.Measure.fst_zero
@[simp] theorem fst_zero : fst (0 : Measure (α × β)) = 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β]
Goal: MeasureTheory.Measure.fst 0 = 0
-- MeasureTheory.Measure.fst_prod
@[simp]
lemma fst_prod [IsProbabilityMeasure ν] : (μ.prod ν).fst = 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] [MeasureTheory.IsProbabilityMeasure ν]
Goal: (μ.prod ν).fst = μ
-- MeasureTheory.Measure.fst_map_prod_mk₀
theorem fst_map_prod_mk₀ {X : α → β} {Y : α → γ} {μ : Measure α}
    (hY : AEMeasurable Y μ) : (μ.map fun a => (X a, Y a)).fst = μ.map X 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {X : α → β} {Y : α → γ} {μ : MeasureTheory.Measure α} (hY : AEMeasurable Y μ)
Goal: (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).fst = MeasureTheory.Measure.map X μ
-- MeasureTheory.Measure.fst_map_prod_mk
theorem fst_map_prod_mk {X : α → β} {Y : α → γ} {μ : Measure α}
    (hY : Measurable Y) : (μ.map fun a => (X a, Y a)).fst = μ.map X 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {X : α → β} {Y : α → γ} {μ : MeasureTheory.Measure α} (hY : Measurable Y)
Goal: (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).fst = MeasureTheory.Measure.map X μ
-- MeasureTheory.Measure.snd_apply
theorem snd_apply {s : Set β} (hs : MeasurableSet s) : ρ.snd s = ρ (Prod.snd ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)} {s : Set β} (hs : MeasurableSet s)
Goal: ρ.snd s = ρ (Prod.snd ⁻¹' s)
-- MeasureTheory.Measure.snd_univ
theorem snd_univ : ρ.snd univ = ρ univ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)}
Goal: ρ.snd Set.univ = ρ Set.univ
-- MeasureTheory.Measure.snd_zero
@[simp] theorem snd_zero : snd (0 : Measure (α × β)) = 0 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β]
Goal: MeasureTheory.Measure.snd 0 = 0
-- MeasureTheory.Measure.snd_prod
@[simp]
lemma snd_prod [IsProbabilityMeasure μ] : (μ.prod ν).snd = 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] [MeasureTheory.IsProbabilityMeasure μ]
Goal: (μ.prod ν).snd = ν
-- MeasureTheory.Measure.snd_map_prod_mk₀
theorem snd_map_prod_mk₀ {X : α → β} {Y : α → γ} {μ : Measure α} (hX : AEMeasurable X μ) :
    (μ.map fun a => (X a, Y a)).snd = μ.map Y 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {X : α → β} {Y : α → γ} {μ : MeasureTheory.Measure α} (hX : AEMeasurable X μ)
Goal: (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).snd = MeasureTheory.Measure.map Y μ
-- MeasureTheory.Measure.snd_map_prod_mk
theorem snd_map_prod_mk {X : α → β} {Y : α → γ} {μ : Measure α} (hX : Measurable X) :
    (μ.map fun a => (X a, Y a)).snd = μ.map Y 
Arguments: {α : Type u_1} {β : Type u_3} {γ : Type u_5} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {X : α → β} {Y : α → γ} {μ : MeasureTheory.Measure α} (hX : Measurable X)
Goal: (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).snd = MeasureTheory.Measure.map Y μ
-- MeasureTheory.Measure.fst_map_swap
@[simp] lemma fst_map_swap : (ρ.map Prod.swap).fst = ρ.snd 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)}
Goal: (MeasureTheory.Measure.map Prod.swap ρ).fst = ρ.snd
-- MeasureTheory.Measure.snd_map_swap
@[simp] lemma snd_map_swap : (ρ.map Prod.swap).snd = ρ.fst 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)}
Goal: (MeasureTheory.Measure.map Prod.swap ρ).snd = ρ.fst
-- RCLike.measurable_re
@[measurability]
theorem measurable_re : Measurable (re : 𝕜 → ℝ) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜]
Goal: Measurable ⇑RCLike.re
-- RCLike.measurable_im
@[measurability]
theorem measurable_im : Measurable (im : 𝕜 → ℝ) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜]
Goal: Measurable ⇑RCLike.im
-- Measurable.re
@[measurability]
theorem Measurable.re (hf : Measurable f) : Measurable fun x => RCLike.re (f x) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {m : MeasurableSpace α} {f : α → 𝕜} (hf : Measurable f)
Goal: Measurable fun x => RCLike.re (f x)
-- AEMeasurable.re
@[measurability]
theorem AEMeasurable.re (hf : AEMeasurable f μ) : AEMeasurable (fun x => RCLike.re (f x)) μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] {m : MeasurableSpace α} {f : α → 𝕜} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => RCLike.re (f x)) μ