ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.Kernel.condexp_densityProcess
lemma condexp_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    {i j : ℕ} (hij : i ≤ j) (a : α) {s : Set β} (hs : MeasurableSet s) :
    (ν a)[fun x ↦ densityProcess κ ν j a x s | countableFiltration γ i]
      =ᵐ[ν a] fun x ↦ densityProcess κ ν i a x s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] {i j : ℕ} (hij : i ≤ j) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ν a[fun x => κ.densityProcess ν j a x s|↑(ProbabilityTheory.countableFiltration γ) i] =ᵐ[ν a] fun x =>
  κ.densityProcess ν i a x s
-- ProbabilityTheory.Kernel.martingale_densityProcess
lemma martingale_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    Martingale (fun n x ↦ densityProcess κ ν n a x s) (countableFiltration γ) (ν a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Martingale (fun n x => κ.densityProcess ν n a x s) (ProbabilityTheory.countableFiltration γ) (ν a)
-- ProbabilityTheory.Kernel.densityProcess_mono_set
lemma densityProcess_mono_set (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ)
    {s s' : Set β} (h : s ⊆ s') :
    densityProcess κ ν n a x s ≤ densityProcess κ ν n a x s' 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (n : ℕ) (a : α) (x : γ) {s s' : Set β} (h : s ⊆ s')
Goal: κ.densityProcess ν n a x s ≤ κ.densityProcess ν n a x s'
-- ProbabilityTheory.Kernel.densityProcess_mono_kernel_left
lemma densityProcess_mono_kernel_left {κ' : Kernel α (γ × β)} (hκκ' : κ ≤ κ')
    (hκ'ν : fst κ' ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β) :
    densityProcess κ ν n a x s ≤ densityProcess κ' ν n a x s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} {κ' : ProbabilityTheory.Kernel α (γ × β)} (hκκ' : κ ≤ κ') (hκ'ν : κ'.fst ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β)
Goal: κ.densityProcess ν n a x s ≤ κ'.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.densityProcess_antitone_kernel_right
lemma densityProcess_antitone_kernel_right {ν' : Kernel α γ}
    (hνν' : ν ≤ ν') (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β) :
    densityProcess κ ν' n a x s ≤ densityProcess κ ν n a x s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν ν' : ProbabilityTheory.Kernel α γ} (hνν' : ν ≤ ν') (hκν : κ.fst ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β)
Goal: κ.densityProcess ν' n a x s ≤ κ.densityProcess ν n a x s
-- ProbabilityTheory.Kernel.densityProcess_empty
@[simp]
lemma densityProcess_empty (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ) (a : α) (x : γ) :
    densityProcess κ ν n a x ∅ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (n : ℕ) (a : α) (x : γ)
Goal: κ.densityProcess ν n a x ∅ = 0
-- ProbabilityTheory.Kernel.tendsto_densityProcess_atTop_empty_of_antitone
lemma tendsto_densityProcess_atTop_empty_of_antitone (κ : Kernel α (γ × β)) (ν : Kernel α γ)
    [IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ)
    (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
    (hseq_meas : ∀ m, MeasurableSet (seq m)) :
    Tendsto (fun m ↦ densityProcess κ ν n a x (seq m)) atTop
      (𝓝 (densityProcess κ ν n a x ∅)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ) (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅) (hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m))
Goal: Filter.Tendsto (fun m => κ.densityProcess ν n a x (seq m)) Filter.atTop (𝓝 (κ.densityProcess ν n a x ∅))
-- ProbabilityTheory.Kernel.tendsto_densityProcess_atTop_of_antitone
lemma tendsto_densityProcess_atTop_of_antitone (κ : Kernel α (γ × β)) (ν : Kernel α γ)
    [IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ)
    (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
    (hseq_meas : ∀ m, MeasurableSet (seq m)) :
    Tendsto (fun m ↦ densityProcess κ ν n a x (seq m)) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ) (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅) (hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m))
Goal: Filter.Tendsto (fun m => κ.densityProcess ν n a x (seq m)) Filter.atTop (𝓝 0)
-- ProbabilityTheory.Kernel.tendsto_densityProcess_limitProcess
lemma tendsto_densityProcess_limitProcess (hκν : fst κ ≤ ν)
    [IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) :
    ∀ᵐ x ∂(ν a), Tendsto (fun n ↦ densityProcess κ ν n a x s) atTop
      (𝓝 ((countableFiltration γ).limitProcess
      (fun n x ↦ densityProcess κ ν n a x s) (ν a) x)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ∀ᵐ (x : γ) ∂ν a,
  Filter.Tendsto (fun n => κ.densityProcess ν n a x s) Filter.atTop
    (𝓝
      (MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
        (ProbabilityTheory.countableFiltration γ) (ν a) x))
-- ProbabilityTheory.Kernel.memL1_limitProcess_densityProcess
lemma memL1_limitProcess_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    Memℒp ((countableFiltration γ).limitProcess
      (fun n x ↦ densityProcess κ ν n a x s) (ν a)) 1 (ν a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Memℒp
  (MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
    (ProbabilityTheory.countableFiltration γ) (ν a))
  1 (ν a)
-- ProbabilityTheory.Kernel.tendsto_eLpNorm_one_densityProcess_limitProcess
lemma tendsto_eLpNorm_one_densityProcess_limitProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    Tendsto (fun n ↦ eLpNorm ((fun x ↦ densityProcess κ ν n a x s)
      - (countableFiltration γ).limitProcess (fun n x ↦ densityProcess κ ν n a x s) (ν a))
      1 (ν a)) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: Filter.Tendsto
  (fun n =>
    MeasureTheory.eLpNorm
      ((fun x => κ.densityProcess ν n a x s) -
        MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
          (ProbabilityTheory.countableFiltration γ) (ν a))
      1 (ν a))
  Filter.atTop (𝓝 0)
-- ProbabilityTheory.Kernel.tendsto_eLpNorm_one_restrict_densityProcess_limitProcess
lemma tendsto_eLpNorm_one_restrict_densityProcess_limitProcess [IsFiniteKernel ν]
    (hκν : fst κ ≤ ν) (a : α) {s : Set β} (hs : MeasurableSet s) (A : Set γ) :
    Tendsto (fun n ↦ eLpNorm ((fun x ↦ densityProcess κ ν n a x s)
      - (countableFiltration γ).limitProcess (fun n x ↦ densityProcess κ ν n a x s) (ν a))
      1 ((ν a).restrict A)) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} [ProbabilityTheory.IsFiniteKernel ν] (hκν : κ.fst ≤ ν) (a : α) {s : Set β} (hs : MeasurableSet s) (A : Set γ)
Goal: Filter.Tendsto
  (fun n =>
    MeasureTheory.eLpNorm
      ((fun x => κ.densityProcess ν n a x s) -
        MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
          (ProbabilityTheory.countableFiltration γ) (ν a))
      1 ((ν a).restrict A))
  Filter.atTop (𝓝 0)
-- ProbabilityTheory.Kernel.density_ae_eq_limitProcess
lemma density_ae_eq_limitProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    (fun x ↦ density κ ν a x s)
      =ᵐ[ν a] (countableFiltration γ).limitProcess
        (fun n x ↦ densityProcess κ ν n a x s) (ν a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: (fun x => κ.density ν a x s) =ᵐ[ν a]
  MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
    (ProbabilityTheory.countableFiltration γ) (ν a)
-- ProbabilityTheory.Kernel.tendsto_m_density
lemma tendsto_m_density (hκν : fst κ ≤ ν) (a : α) [IsFiniteKernel ν]
    {s : Set β} (hs : MeasurableSet s) :
    ∀ᵐ x ∂(ν a),
      Tendsto (fun n ↦ densityProcess κ ν n a x s) atTop (𝓝 (density κ ν a x s)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (a : α) [ProbabilityTheory.IsFiniteKernel ν] {s : Set β} (hs : MeasurableSet s)
Goal: ∀ᵐ (x : γ) ∂ν a, Filter.Tendsto (fun n => κ.densityProcess ν n a x s) Filter.atTop (𝓝 (κ.density ν a x s))
-- ProbabilityTheory.Kernel.measurable_density
lemma measurable_density (κ : Kernel α (γ × β)) (ν : Kernel α γ)
    {s : Set β} (hs : MeasurableSet s) :
    Measurable (fun (p : α × γ) ↦ density κ ν p.1 p.2 s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun p => κ.density ν p.1 p.2 s
-- ProbabilityTheory.Kernel.measurable_density_left
lemma measurable_density_left (κ : Kernel α (γ × β)) (ν : Kernel α γ) (x : γ)
    {s : Set β} (hs : MeasurableSet s) :
    Measurable (fun a ↦ density κ ν a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) (x : γ) {s : Set β} (hs : MeasurableSet s)
Goal: Measurable fun a => κ.density ν a x s
-- ProbabilityTheory.Kernel.measurable_density_right
lemma measurable_density_right (κ : Kernel α (γ × β)) (ν : Kernel α γ)
    {s : Set β} (hs : MeasurableSet s) (a : α) :
    Measurable (fun x ↦ density κ ν a x s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) (ν : ProbabilityTheory.Kernel α γ) {s : Set β} (hs : MeasurableSet s) (a : α)
Goal: Measurable fun x => κ.density ν a x s
-- ProbabilityTheory.Kernel.density_mono_set
lemma density_mono_set (hκν : fst κ ≤ ν) (a : α) (x : γ) {s s' : Set β} (h : s ⊆ s') :
    density κ ν a x s ≤ density κ ν a x s' 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (a : α) (x : γ) {s s' : Set β} (h : s ⊆ s')
Goal: κ.density ν a x s ≤ κ.density ν a x s'
-- ProbabilityTheory.Kernel.density_nonneg
lemma density_nonneg (hκν : fst κ ≤ ν) (a : α) (x : γ) (s : Set β) :
    0 ≤ density κ ν a x s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (a : α) (x : γ) (s : Set β)
Goal: 0 ≤ κ.density ν a x s
-- ProbabilityTheory.Kernel.density_le_one
lemma density_le_one (hκν : fst κ ≤ ν) (a : α) (x : γ) (s : Set β) :
    density κ ν a x s ≤ 1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (a : α) (x : γ) (s : Set β)
Goal: κ.density ν a x s ≤ 1
-- ProbabilityTheory.Kernel.eLpNorm_density_le
lemma eLpNorm_density_le (hκν : fst κ ≤ ν) (a : α) (s : Set β) :
    eLpNorm (fun x ↦ density κ ν a x s) 1 (ν a) ≤ ν a univ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) (a : α) (s : Set β)
Goal: MeasureTheory.eLpNorm (fun x => κ.density ν a x s) 1 (ν a) ≤ (ν a) Set.univ
-- ProbabilityTheory.Kernel.integrable_density
lemma integrable_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    Integrable (fun x ↦ density κ ν a x s) (ν a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (fun x => κ.density ν a x s) (ν a)
-- ProbabilityTheory.Kernel.tendsto_setIntegral_densityProcess
lemma tendsto_setIntegral_densityProcess (hκν : fst κ ≤ ν)
    [IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) (A : Set γ) :
    Tendsto (fun i ↦ ∫ x in A, densityProcess κ ν i a x s ∂(ν a)) atTop
      (𝓝 (∫ x in A, density κ ν a x s ∂(ν a))) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) (A : Set γ)
Goal: Filter.Tendsto (fun i => ∫ (x : γ) in A, κ.densityProcess ν i a x s ∂ν a) Filter.atTop
  (𝓝 (∫ (x : γ) in A, κ.density ν a x s ∂ν a))
-- ProbabilityTheory.Kernel.setIntegral_density_of_measurableSet
lemma setIntegral_density_of_measurableSet (hκν : fst κ ≤ ν)
    [IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
    (hA : MeasurableSet[countableFiltration γ n] A) :
    ∫ x in A, density κ ν a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A)
Goal: ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
-- ProbabilityTheory.Kernel.integral_density
lemma integral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    ∫ x, density κ ν a x s ∂(ν a) = (κ a (univ ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ∫ (x : γ), κ.density ν a x s ∂ν a = ((κ a) (Set.univ ×ˢ s)).toReal
-- ProbabilityTheory.Kernel.setIntegral_density
lemma setIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A) :
    ∫ x in A, density κ ν a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A)
Goal: ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
-- ProbabilityTheory.Kernel.setLIntegral_density
lemma setLIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A) :
    ∫⁻ x in A, ENNReal.ofReal (density κ ν a x s) ∂(ν a) = κ a (A ×ˢ s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A)
Goal: ∫⁻ (x : γ) in A, ENNReal.ofReal (κ.density ν a x s) ∂ν a = (κ a) (A ×ˢ s)
-- ProbabilityTheory.Kernel.lintegral_density
lemma lintegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ x, ENNReal.ofReal (density κ ν a x s) ∂(ν a) = κ a (univ ×ˢ s) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (x : γ), ENNReal.ofReal (κ.density ν a x s) ∂ν a = (κ a) (Set.univ ×ˢ s)
-- ProbabilityTheory.Kernel.tendsto_integral_density_of_monotone
lemma tendsto_integral_density_of_monotone (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ)
    (hseq_meas : ∀ m, MeasurableSet (seq m)) :
    Tendsto (fun m ↦ ∫ x, density κ ν a x (seq m) ∂(ν a)) atTop (𝓝 (κ a univ).toReal) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = Set.univ) (hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m))
Goal: Filter.Tendsto (fun m => ∫ (x : γ), κ.density ν a x (seq m) ∂ν a) Filter.atTop (𝓝 ((κ a) Set.univ).toReal)
-- ProbabilityTheory.Kernel.tendsto_integral_density_of_antitone
lemma tendsto_integral_density_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (a : α)
    (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
    (hseq_meas : ∀ m, MeasurableSet (seq m)) :
    Tendsto (fun m ↦ ∫ x, density κ ν a x (seq m) ∂(ν a)) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅) (hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m))
Goal: Filter.Tendsto (fun m => ∫ (x : γ), κ.density ν a x (seq m) ∂ν a) Filter.atTop (𝓝 0)
-- ProbabilityTheory.Kernel.tendsto_density_atTop_ae_of_antitone
lemma tendsto_density_atTop_ae_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (a : α)
    (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
    (hseq_meas : ∀ m, MeasurableSet (seq m)) :
    ∀ᵐ x ∂(ν a), Tendsto (fun m ↦ density κ ν a x (seq m)) atTop (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} {ν : ProbabilityTheory.Kernel α γ} (hκν : κ.fst ≤ ν) [ProbabilityTheory.IsFiniteKernel ν] (a : α) (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅) (hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m))
Goal: ∀ᵐ (x : γ) ∂ν a, Filter.Tendsto (fun m => κ.density ν a x (seq m)) Filter.atTop (𝓝 0)
-- ProbabilityTheory.Kernel.densityProcess_fst_univ
lemma densityProcess_fst_univ [IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ) :
    densityProcess κ (fst κ) n a x univ
      = if fst κ a (countablePartitionSet n x) = 0 then 0 else 1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)} [ProbabilityTheory.IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ)
Goal: κ.densityProcess κ.fst n a x Set.univ = if (κ.fst a) (MeasurableSpace.countablePartitionSet n x) = 0 then 0 else 1
-- ProbabilityTheory.Kernel.densityProcess_fst_univ_ae
lemma densityProcess_fst_univ_ae (κ : Kernel α (γ × β)) [IsFiniteKernel κ] (n : ℕ) (a : α) :
    ∀ᵐ x ∂(fst κ a), densityProcess κ (fst κ) n a x univ = 1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × β)) [ProbabilityTheory.IsFiniteKernel κ] (n : ℕ) (a : α)
Goal: ∀ᵐ (x : γ) ∂κ.fst a, κ.densityProcess κ.fst n a x Set.univ = 1