ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.defaultRatCDF_le_one
lemma defaultRatCDF_le_one (q : ℚ) : defaultRatCDF q ≤ 1 
Arguments: (q : ℚ)
Goal: ProbabilityTheory.defaultRatCDF q ≤ 1
-- ProbabilityTheory.tendsto_defaultRatCDF_atTop
lemma tendsto_defaultRatCDF_atTop : Tendsto defaultRatCDF atTop (𝓝 1) 
Goal: Filter.Tendsto ProbabilityTheory.defaultRatCDF Filter.atTop (𝓝 1)
-- ProbabilityTheory.tendsto_defaultRatCDF_atBot
lemma tendsto_defaultRatCDF_atBot : Tendsto defaultRatCDF atBot (𝓝 0) 
Goal: Filter.Tendsto ProbabilityTheory.defaultRatCDF Filter.atBot (𝓝 0)
-- ProbabilityTheory.iInf_rat_gt_defaultRatCDF
lemma iInf_rat_gt_defaultRatCDF (t : ℚ) :
    ⨅ r : Ioi t, defaultRatCDF r = defaultRatCDF t 
Arguments: (t : ℚ)
Goal: ⨅ r, ProbabilityTheory.defaultRatCDF ↑r = ProbabilityTheory.defaultRatCDF t
-- ProbabilityTheory.isRatStieltjesPoint_defaultRatCDF
lemma isRatStieltjesPoint_defaultRatCDF (a : α) :
    IsRatStieltjesPoint (fun (_ : α) ↦ defaultRatCDF) a 
Arguments: {α : Type u_1} (a : α)
Goal: ProbabilityTheory.IsRatStieltjesPoint (fun x => ProbabilityTheory.defaultRatCDF) a
-- ProbabilityTheory.IsMeasurableRatCDF_defaultRatCDF
lemma IsMeasurableRatCDF_defaultRatCDF (α : Type*) [MeasurableSpace α] :
    IsMeasurableRatCDF (fun (_ : α) (q : ℚ) ↦ defaultRatCDF q) 
Arguments: (α : Type u_4) [MeasurableSpace α]
Goal: ProbabilityTheory.IsMeasurableRatCDF fun x q => ProbabilityTheory.defaultRatCDF q
-- ProbabilityTheory.toRatCDF_of_isRatStieltjesPoint
lemma toRatCDF_of_isRatStieltjesPoint {a : α} (h : IsRatStieltjesPoint f a) (q : ℚ) :
    toRatCDF f a q = f a q 
Arguments: {α : Type u_1} {f : α → ℚ → ℝ} {a : α} (h : ProbabilityTheory.IsRatStieltjesPoint f a) (q : ℚ)
Goal: ProbabilityTheory.toRatCDF f a q = f a q
-- ProbabilityTheory.measurable_toRatCDF
lemma measurable_toRatCDF (hf : Measurable f) : Measurable (toRatCDF f) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f)
Goal: Measurable (ProbabilityTheory.toRatCDF f)
-- ProbabilityTheory.isMeasurableRatCDF_toRatCDF
lemma isMeasurableRatCDF_toRatCDF (hf : Measurable f) :
    IsMeasurableRatCDF (toRatCDF f) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f)
Goal: ProbabilityTheory.IsMeasurableRatCDF (ProbabilityTheory.toRatCDF f)
-- ProbabilityTheory.toRatCDF_unit_prod
lemma toRatCDF_unit_prod (a : α) :
    toRatCDF (fun (p : Unit × α) ↦ f p.2) ((), a) = toRatCDF f a 
Arguments: {α : Type u_1} {f : α → ℚ → ℝ} (a : α)
Goal: ProbabilityTheory.toRatCDF (fun p => f p.2) ((), a) = ProbabilityTheory.toRatCDF f a
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def
noncomputable irreducible_def IsMeasurableRatCDF.stieltjesFunctionAux (f : α → ℚ → ℝ) :
    α → ℝ → ℝ 
Arguments: {α : Type u_4} (f : α → ℚ → ℝ) (a : α) (x : ℝ)
Goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a x = ⨅ q, f a ↑q
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def'
lemma IsMeasurableRatCDF.stieltjesFunctionAux_def' (f : α → ℚ → ℝ) (a : α) :
    IsMeasurableRatCDF.stieltjesFunctionAux f a
      = fun (t : ℝ) ↦ ⨅ r : { r' : ℚ // t < r' }, f a r 
Arguments: {α : Type u_1} (f : α → ℚ → ℝ) (a : α)
Goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a = fun t => ⨅ r, f a ↑r
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod
lemma IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod {f : α → ℚ → ℝ} (a : α) :
    IsMeasurableRatCDF.stieltjesFunctionAux (fun (p : Unit × α) ↦ f p.2) ((), a)
      = IsMeasurableRatCDF.stieltjesFunctionAux f a 
Arguments: {α : Type u_1} {f : α → ℚ → ℝ} (a : α)
Goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux (fun p => f p.2) ((), a) =
  ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_eq
lemma IsMeasurableRatCDF.stieltjesFunctionAux_eq (a : α) (r : ℚ) :
    IsMeasurableRatCDF.stieltjesFunctionAux f a r = f a r 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (r : ℚ)
Goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a ↑r = f a r
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_nonneg
lemma IsMeasurableRatCDF.stieltjesFunctionAux_nonneg (a : α) (r : ℝ) :
    0 ≤ IsMeasurableRatCDF.stieltjesFunctionAux f a r 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (r : ℝ)
Goal: 0 ≤ ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a r
-- ProbabilityTheory.IsMeasurableRatCDF.monotone_stieltjesFunctionAux
lemma IsMeasurableRatCDF.monotone_stieltjesFunctionAux (a : α) :
    Monotone (IsMeasurableRatCDF.stieltjesFunctionAux f a) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α)
Goal: Monotone (ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a)
-- ProbabilityTheory.IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici
lemma IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici (a : α) (x : ℝ) :
    ContinuousWithinAt (IsMeasurableRatCDF.stieltjesFunctionAux f a) (Ici x) x 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (x : ℝ)
Goal: ContinuousWithinAt (ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a) (Set.Ici x) x
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_eq
lemma IsMeasurableRatCDF.stieltjesFunction_eq (a : α) (r : ℚ) : hf.stieltjesFunction a r = f a r 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (r : ℚ)
Goal: ↑(hf.stieltjesFunction a) ↑r = f a r
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_nonneg
lemma IsMeasurableRatCDF.stieltjesFunction_nonneg (a : α) (r : ℝ) : 0 ≤ hf.stieltjesFunction a r 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (r : ℝ)
Goal: 0 ≤ ↑(hf.stieltjesFunction a) r
-- ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_le_one
lemma IsMeasurableRatCDF.stieltjesFunction_le_one (a : α) (x : ℝ) :
    hf.stieltjesFunction a x ≤ 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (x : ℝ)
Goal: ↑(hf.stieltjesFunction a) x ≤ 1
-- ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot
lemma IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot (a : α) :
    Tendsto (hf.stieltjesFunction a) atBot (𝓝 0) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α)
Goal: Filter.Tendsto (↑(hf.stieltjesFunction a)) Filter.atBot (𝓝 0)
-- ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop
lemma IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop (a : α) :
    Tendsto (hf.stieltjesFunction a) atTop (𝓝 1) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α)
Goal: Filter.Tendsto (↑(hf.stieltjesFunction a)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.IsMeasurableRatCDF.measurable_stieltjesFunction
lemma IsMeasurableRatCDF.measurable_stieltjesFunction (x : ℝ) :
    Measurable fun a ↦ hf.stieltjesFunction a x 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (x : ℝ)
Goal: Measurable fun a => ↑(hf.stieltjesFunction a) x
-- ProbabilityTheory.IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction
lemma IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction (x : ℝ) :
    StronglyMeasurable fun a ↦ hf.stieltjesFunction a x 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (x : ℝ)
Goal: MeasureTheory.StronglyMeasurable fun a => ↑(hf.stieltjesFunction a) x
-- ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_Iic
lemma IsMeasurableRatCDF.measure_stieltjesFunction_Iic (a : α) (x : ℝ) :
    (hf.stieltjesFunction a).measure (Iic x) = ENNReal.ofReal (hf.stieltjesFunction a x) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α) (x : ℝ)
Goal: (hf.stieltjesFunction a).measure (Set.Iic x) = ENNReal.ofReal (↑(hf.stieltjesFunction a) x)
-- ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_univ
lemma IsMeasurableRatCDF.measure_stieltjesFunction_univ (a : α) :
    (hf.stieltjesFunction a).measure univ = 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f) (a : α)
Goal: (hf.stieltjesFunction a).measure Set.univ = 1
-- ProbabilityTheory.IsMeasurableRatCDF.measurable_measure_stieltjesFunction
lemma IsMeasurableRatCDF.measurable_measure_stieltjesFunction :
    Measurable fun a ↦ (hf.stieltjesFunction a).measure 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : ProbabilityTheory.IsMeasurableRatCDF f)
Goal: Measurable fun a => (hf.stieltjesFunction a).measure
-- ProbabilityTheory.stieltjesOfMeasurableRat_eq
lemma stieltjesOfMeasurableRat_eq (hf : Measurable f) (a : α) (r : ℚ) :
    stieltjesOfMeasurableRat f hf a r = toRatCDF f a r 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α) (r : ℚ)
Goal: ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) ↑r = ProbabilityTheory.toRatCDF f a r
-- ProbabilityTheory.stieltjesOfMeasurableRat_unit_prod
lemma stieltjesOfMeasurableRat_unit_prod (hf : Measurable f) (a : α) :
    stieltjesOfMeasurableRat (fun (p : Unit × α) ↦ f p.2) (hf.comp measurable_snd) ((), a)
      = stieltjesOfMeasurableRat f hf a 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α)
Goal: ProbabilityTheory.stieltjesOfMeasurableRat (fun p => f p.2) ⋯ ((), a) =
  ProbabilityTheory.stieltjesOfMeasurableRat f hf a
-- ProbabilityTheory.stieltjesOfMeasurableRat_nonneg
lemma stieltjesOfMeasurableRat_nonneg (hf : Measurable f) (a : α) (r : ℝ) :
    0 ≤ stieltjesOfMeasurableRat f hf a r 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α) (r : ℝ)
Goal: 0 ≤ ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) r
-- ProbabilityTheory.stieltjesOfMeasurableRat_le_one
lemma stieltjesOfMeasurableRat_le_one (hf : Measurable f) (a : α) (x : ℝ) :
    stieltjesOfMeasurableRat f hf a x ≤ 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α) (x : ℝ)
Goal: ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x ≤ 1
-- ProbabilityTheory.tendsto_stieltjesOfMeasurableRat_atBot
lemma tendsto_stieltjesOfMeasurableRat_atBot (hf : Measurable f) (a : α) :
    Tendsto (stieltjesOfMeasurableRat f hf a) atBot (𝓝 0) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α)
Goal: Filter.Tendsto (↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a)) Filter.atBot (𝓝 0)
-- ProbabilityTheory.tendsto_stieltjesOfMeasurableRat_atTop
lemma tendsto_stieltjesOfMeasurableRat_atTop (hf : Measurable f) (a : α) :
    Tendsto (stieltjesOfMeasurableRat f hf a) atTop (𝓝 1) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α)
Goal: Filter.Tendsto (↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.measurable_stieltjesOfMeasurableRat
lemma measurable_stieltjesOfMeasurableRat (hf : Measurable f) (x : ℝ) :
    Measurable fun a ↦ stieltjesOfMeasurableRat f hf a x 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (x : ℝ)
Goal: Measurable fun a => ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x
-- ProbabilityTheory.stronglyMeasurable_stieltjesOfMeasurableRat
lemma stronglyMeasurable_stieltjesOfMeasurableRat (hf : Measurable f) (x : ℝ) :
    StronglyMeasurable fun a ↦ stieltjesOfMeasurableRat f hf a x 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (x : ℝ)
Goal: MeasureTheory.StronglyMeasurable fun a => ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x
-- ProbabilityTheory.measure_stieltjesOfMeasurableRat_Iic
lemma measure_stieltjesOfMeasurableRat_Iic (hf : Measurable f) (a : α) (x : ℝ) :
    (stieltjesOfMeasurableRat f hf a).measure (Iic x)
      = ENNReal.ofReal (stieltjesOfMeasurableRat f hf a x) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α) (x : ℝ)
Goal: (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure (Set.Iic x) =
  ENNReal.ofReal (↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x)
-- ProbabilityTheory.measure_stieltjesOfMeasurableRat_univ
lemma measure_stieltjesOfMeasurableRat_univ (hf : Measurable f) (a : α) :
    (stieltjesOfMeasurableRat f hf a).measure univ = 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f) (a : α)
Goal: (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure Set.univ = 1
-- ProbabilityTheory.measurable_measure_stieltjesOfMeasurableRat
lemma measurable_measure_stieltjesOfMeasurableRat (hf : Measurable f) :
    Measurable fun a ↦ (stieltjesOfMeasurableRat f hf a).measure 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℚ → ℝ} (hf : Measurable f)
Goal: Measurable fun a => (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure
-- ProbabilityTheory.eq_condKernel_of_measure_eq_compProd'
theorem eq_condKernel_of_measure_eq_compProd' (κ : Kernel α Ω) [IsSFiniteKernel κ]
    (hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s) :
    ∀ᵐ x ∂ρ.fst, κ x s = ρ.condKernel x s 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {ρ : MeasureTheory.Measure (α × Ω)} [MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α Ω) [ProbabilityTheory.IsSFiniteKernel κ] (hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s)
Goal: ∀ᵐ (x : α) ∂ρ.fst, (κ x) s = (ρ.condKernel x) s
-- ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
lemma eq_condKernel_of_measure_eq_compProd_real {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ]
    (κ : Kernel α ℝ) [IsFiniteKernel κ] (hκ : ρ = ρ.fst ⊗ₘ κ) :
    ∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {ρ : MeasureTheory.Measure (α × ℝ)} [MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α ℝ) [ProbabilityTheory.IsFiniteKernel κ] (hκ : ρ = ρ.fst ⊗ₘ κ)
Goal: ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
-- ProbabilityTheory.eq_condKernel_of_measure_eq_compProd
theorem eq_condKernel_of_measure_eq_compProd (κ : Kernel α Ω) [IsFiniteKernel κ]
    (hκ : ρ = ρ.fst ⊗ₘ κ) :
    ∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x 
Arguments: {α : Type u_1} {Ω : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {ρ : MeasureTheory.Measure (α × Ω)} [MeasureTheory.IsFiniteMeasure ρ] (κ : ProbabilityTheory.Kernel α Ω) [ProbabilityTheory.IsFiniteKernel κ] (hκ : ρ = ρ.fst ⊗ₘ κ)
Goal: ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
-- ProbabilityTheory.Kernel.apply_eq_measure_condKernel_of_compProd_eq
lemma Kernel.apply_eq_measure_condKernel_of_compProd_eq
    {ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
    (hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
    (fun b ↦ κ (a, b)) =ᵐ[Kernel.fst ρ a] (ρ a).condKernel 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {ρ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel ρ] {κ : ProbabilityTheory.Kernel (α × β) Ω} [ProbabilityTheory.IsFiniteKernel κ] (hκ : ρ.fst ⊗ₖ κ = ρ) (a : α)
Goal: (fun b => κ (a, b)) =ᵐ[ρ.fst a] ⇑(ρ a).condKernel
-- ProbabilityTheory.Kernel.condKernel_apply_eq_condKernel
lemma Kernel.condKernel_apply_eq_condKernel [CountableOrCountablyGenerated α β]
    (κ : Kernel α (β × Ω)) [IsFiniteKernel κ] (a : α) :
    (fun b ↦ Kernel.condKernel κ (a, b)) =ᵐ[Kernel.fst κ a] (κ a).condKernel 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] (κ : ProbabilityTheory.Kernel α (β × Ω)) [ProbabilityTheory.IsFiniteKernel κ] (a : α)
Goal: (fun b => κ.condKernel (a, b)) =ᵐ[κ.fst a] ⇑(κ a).condKernel
-- ProbabilityTheory.condKernel_const
lemma condKernel_const [CountableOrCountablyGenerated α β] (ρ : Measure (β × Ω)) [IsFiniteMeasure ρ]
    (a : α) :
    (fun b ↦ Kernel.condKernel (Kernel.const α ρ) (a, b)) =ᵐ[ρ.fst] ρ.condKernel 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] (ρ : MeasureTheory.Measure (β × Ω)) [MeasureTheory.IsFiniteMeasure ρ] (a : α)
Goal: (fun b => (ProbabilityTheory.Kernel.const α ρ).condKernel (a, b)) =ᵐ[ρ.fst] ⇑ρ.condKernel
-- ProbabilityTheory.eq_condKernel_of_kernel_eq_compProd
theorem eq_condKernel_of_kernel_eq_compProd [CountableOrCountablyGenerated α β]
    {ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
    (hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
    ∀ᵐ x ∂(Kernel.fst ρ a), κ (a, x) = Kernel.condKernel ρ (a, x) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {ρ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel ρ] {κ : ProbabilityTheory.Kernel (α × β) Ω} [ProbabilityTheory.IsFiniteKernel κ] (hκ : ρ.fst ⊗ₖ κ = ρ) (a : α)
Goal: ∀ᵐ (x : β) ∂ρ.fst a, κ (a, x) = ρ.condKernel (a, x)
-- MeasureTheory.Measure.compProd_of_not_sfinite
lemma compProd_of_not_sfinite (μ : Measure α) (κ : Kernel α β) (h : ¬ SFinite μ) :
    μ ⊗ₘ κ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (h : ¬MeasureTheory.SFinite μ)
Goal: μ ⊗ₘ κ = 0
-- MeasureTheory.Measure.compProd_of_not_isSFiniteKernel
lemma compProd_of_not_isSFiniteKernel (μ : Measure α) (κ : Kernel α β) (h : ¬ IsSFiniteKernel κ) :
    μ ⊗ₘ κ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (h : ¬ProbabilityTheory.IsSFiniteKernel κ)
Goal: μ ⊗ₘ κ = 0
-- MeasureTheory.Measure.compProd_zero_left
@[simp] lemma compProd_zero_left (κ : Kernel α β) : (0 : Measure α) ⊗ₘ κ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β)
Goal: 0 ⊗ₘ κ = 0
-- MeasureTheory.Measure.compProd_zero_right
@[simp] lemma compProd_zero_right (μ : Measure α) : μ ⊗ₘ (0 : Kernel α β) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α)
Goal: μ ⊗ₘ 0 = 0
-- MeasureTheory.Measure.compProd_apply
lemma compProd_apply [SFinite μ] [IsSFiniteKernel κ] {s : Set (α × β)} (hs : MeasurableSet s) :
    (μ ⊗ₘ κ) s = ∫⁻ a, κ a (Prod.mk a ⁻¹' s) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: (μ ⊗ₘ κ) s = ∫⁻ (a : α), (κ a) (Prod.mk a ⁻¹' s) ∂μ
-- MeasureTheory.Measure.compProd_apply_prod
lemma compProd_apply_prod [SFinite μ] [IsSFiniteKernel κ]
    {s : Set α} {t : Set β} (hs : MeasurableSet s) (ht : MeasurableSet t) :
    (μ ⊗ₘ κ) (s ×ˢ t) = ∫⁻ a in s, κ a t ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {s : Set α} {t : Set β} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: (μ ⊗ₘ κ) (s ×ˢ t) = ∫⁻ (a : α) in s, (κ a) t ∂μ
-- MeasureTheory.Measure.compProd_congr
lemma compProd_congr [SFinite μ] [IsSFiniteKernel κ] [IsSFiniteKernel η]
    (h : κ =ᵐ[μ] η) : μ ⊗ₘ κ = μ ⊗ₘ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] (h : ⇑κ =ᵐ[μ] ⇑η)
Goal: μ ⊗ₘ κ = μ ⊗ₘ η
-- MeasureTheory.Measure.ae_compProd_of_ae_ae
lemma ae_compProd_of_ae_ae [SFinite μ] [IsSFiniteKernel κ] {p : α × β → Prop}
    (hp : MeasurableSet {x | p x}) (h : ∀ᵐ a ∂μ, ∀ᵐ b ∂(κ a), p (a, b)) :
    ∀ᵐ x ∂(μ ⊗ₘ κ), p x 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {p : α × β → Prop} (hp : MeasurableSet {x | p x}) (h : ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b))
Goal: ∀ᵐ (x : α × β) ∂μ ⊗ₘ κ, p x
-- MeasureTheory.Measure.ae_ae_of_ae_compProd
lemma ae_ae_of_ae_compProd [SFinite μ] [IsSFiniteKernel κ] {p : α × β → Prop}
    (h : ∀ᵐ x ∂(μ ⊗ₘ κ), p x) :
    ∀ᵐ a ∂μ, ∀ᵐ b ∂κ a, p (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {p : α × β → Prop} (h : ∀ᵐ (x : α × β) ∂μ ⊗ₘ κ, p x)
Goal: ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b)
-- MeasureTheory.Measure.ae_compProd_iff
lemma ae_compProd_iff [SFinite μ] [IsSFiniteKernel κ] {p : α × β → Prop}
    (hp : MeasurableSet {x | p x}) :
    (∀ᵐ x ∂(μ ⊗ₘ κ), p x) ↔ ∀ᵐ a ∂μ, ∀ᵐ b ∂(κ a), p (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {p : α × β → Prop} (hp : MeasurableSet {x | p x})
Goal: (∀ᵐ (x : α × β) ∂μ ⊗ₘ κ, p x) ↔ ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b)
-- MeasureTheory.Measure.compProd_add_left
lemma compProd_add_left (μ ν : Measure α) [SFinite μ] [SFinite ν] (κ : Kernel α β)
    [IsSFiniteKernel κ] :
    (μ + ν) ⊗ₘ κ = μ ⊗ₘ κ + ν ⊗ₘ κ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ]
Goal: (μ + ν) ⊗ₘ κ = μ ⊗ₘ κ + ν ⊗ₘ κ
-- MeasureTheory.Measure.compProd_add_right
lemma compProd_add_right (μ : Measure α) [SFinite μ] (κ η : Kernel α β)
    [IsSFiniteKernel κ] [IsSFiniteKernel η] :
    μ ⊗ₘ (κ + η) = μ ⊗ₘ κ + μ ⊗ₘ η 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (κ η : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η]
Goal: μ ⊗ₘ (κ + η) = μ ⊗ₘ κ + μ ⊗ₘ η
-- MeasureTheory.Measure.lintegral_compProd
lemma lintegral_compProd [SFinite μ] [IsSFiniteKernel κ]
    {f : α × β → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ x, f x ∂(μ ⊗ₘ κ) = ∫⁻ a, ∫⁻ b, f (a, b) ∂(κ a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {f : α × β → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (x : α × β), f x ∂μ ⊗ₘ κ = ∫⁻ (a : α), ∫⁻ (b : β), f (a, b) ∂κ a ∂μ
-- MeasureTheory.Measure.setLIntegral_compProd
lemma setLIntegral_compProd [SFinite μ] [IsSFiniteKernel κ]
    {f : α × β → ℝ≥0∞} (hf : Measurable f)
    {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) :
    ∫⁻ x in s ×ˢ t, f x ∂(μ ⊗ₘ κ) = ∫⁻ a in s, ∫⁻ b in t, f (a, b) ∂(κ a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {f : α × β → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t)
Goal: ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ ⊗ₘ κ = ∫⁻ (a : α) in s, ∫⁻ (b : β) in t, f (a, b) ∂κ a ∂μ