ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.upcrossings_eq_top_of_frequently_lt
theorem upcrossings_eq_top_of_frequently_lt (hab : a < b) (h₁ : ∃ᶠ n in atTop, f n ω < a)
    (h₂ : ∃ᶠ n in atTop, b < f n ω) : upcrossings a b f ω = ∞ 
Arguments: {Ω : Type u_1} {a b : ℝ} {f : ℕ → Ω → ℝ} {ω : Ω} (hab : a < b) (h₁ : ∃ᶠ (n : ℕ) in Filter.atTop, f n ω < a) (h₂ : ∃ᶠ (n : ℕ) in Filter.atTop, b < f n ω)
Goal: MeasureTheory.upcrossings a b f ω = ⊤
-- MeasureTheory.tendsto_of_uncrossing_lt_top
theorem tendsto_of_uncrossing_lt_top (hf₁ : liminf (fun n => (‖f n ω‖₊ : ℝ≥0∞)) atTop < ∞)
    (hf₂ : ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞) :
    ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) 
Arguments: {Ω : Type u_1} {f : ℕ → Ω → ℝ} {ω : Ω} (hf₁ : Filter.liminf (fun n => ↑‖f n ω‖₊) Filter.atTop < ⊤) (hf₂ : ∀ (a b : ℚ), a < b → MeasureTheory.upcrossings (↑a) (↑b) f ω < ⊤)
Goal: ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 c)
-- MeasureTheory.Submartingale.upcrossings_ae_lt_top'
theorem Submartingale.upcrossings_ae_lt_top' [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) (hab : a < b) : ∀ᵐ ω ∂μ, upcrossings a b f ω < ∞ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {a b : ℝ} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R) (hab : a < b)
Goal: ∀ᵐ (ω : Ω) ∂μ, MeasureTheory.upcrossings a b f ω < ⊤
-- MeasureTheory.Submartingale.upcrossings_ae_lt_top
theorem Submartingale.upcrossings_ae_lt_top [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, ∀ (a b : ℚ), a < b → MeasureTheory.upcrossings (↑a) (↑b) f ω < ⊤
-- MeasureTheory.Submartingale.exists_ae_tendsto_of_bdd
theorem Submartingale.exists_ae_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 c)
-- MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd
theorem Submartingale.exists_ae_trim_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) :
    ∀ᵐ ω ∂μ.trim (sSup_le fun m ⟨n, hn⟩ => hn ▸ ℱ.le _ : ⨆ n, ℱ n ≤ m0),
      ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ.trim ⋯, ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 c)
-- MeasureTheory.Submartingale.ae_tendsto_limitProcess
theorem Submartingale.ae_tendsto_limitProcess [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) :
    ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (ℱ.limitProcess f μ ω)) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) 1 μ ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 (MeasureTheory.Filtration.limitProcess f ℱ μ ω))
-- MeasureTheory.Submartingale.memℒp_limitProcess
theorem Submartingale.memℒp_limitProcess {p : ℝ≥0∞} (hf : Submartingale f ℱ μ)
    (hbdd : ∀ n, eLpNorm (f n) p μ ≤ R) : Memℒp (ℱ.limitProcess f μ) p μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} {p : ℝ≥0∞} (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ (n : ℕ), MeasureTheory.eLpNorm (f n) p μ ≤ ↑R)
Goal: MeasureTheory.Memℒp (MeasureTheory.Filtration.limitProcess f ℱ μ) p μ
-- MeasureTheory.Submartingale.tendsto_eLpNorm_one_limitProcess
theorem Submartingale.tendsto_eLpNorm_one_limitProcess (hf : Submartingale f ℱ μ)
    (hunif : UniformIntegrable f 1 μ) :
    Tendsto (fun n => eLpNorm (f n - ℱ.limitProcess f μ) 1 μ) atTop (𝓝 0) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hunif : MeasureTheory.UniformIntegrable f 1 μ)
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - MeasureTheory.Filtration.limitProcess f ℱ μ) 1 μ) Filter.atTop
  (𝓝 0)
-- MeasureTheory.Submartingale.ae_tendsto_limitProcess_of_uniformIntegrable
theorem Submartingale.ae_tendsto_limitProcess_of_uniformIntegrable (hf : Submartingale f ℱ μ)
    (hunif : UniformIntegrable f 1 μ) :
    ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (ℱ.limitProcess f μ ω)) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hunif : MeasureTheory.UniformIntegrable f 1 μ)
Goal: ∀ᵐ (ω : Ω) ∂μ, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 (MeasureTheory.Filtration.limitProcess f ℱ μ ω))
-- MeasureTheory.Martingale.eq_condexp_of_tendsto_eLpNorm
theorem Martingale.eq_condexp_of_tendsto_eLpNorm {μ : Measure Ω} (hf : Martingale f ℱ μ)
    (hg : Integrable g μ) (hgtends : Tendsto (fun n => eLpNorm (f n - g) 1 μ) atTop (𝓝 0)) (n : ℕ) :
    f n =ᵐ[μ] μ[g|ℱ n] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {g : Ω → ℝ} {μ : MeasureTheory.Measure Ω} (hf : MeasureTheory.Martingale f ℱ μ) (hg : MeasureTheory.Integrable g μ) (hgtends : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) 1 μ) Filter.atTop (𝓝 0)) (n : ℕ)
Goal: f n =ᵐ[μ] μ[g|↑ℱ n]
-- MeasureTheory.Martingale.ae_eq_condexp_limitProcess
theorem Martingale.ae_eq_condexp_limitProcess (hf : Martingale f ℱ μ)
    (hbdd : UniformIntegrable f 1 μ) (n : ℕ) : f n =ᵐ[μ] μ[ℱ.limitProcess f μ|ℱ n] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Martingale f ℱ μ) (hbdd : MeasureTheory.UniformIntegrable f 1 μ) (n : ℕ)
Goal: f n =ᵐ[μ] μ[MeasureTheory.Filtration.limitProcess f ℱ μ|↑ℱ n]
-- MeasureTheory.Integrable.tendsto_ae_condexp
theorem Integrable.tendsto_ae_condexp (hg : Integrable g μ)
    (hgmeas : StronglyMeasurable[⨆ n, ℱ n] g) :
    ∀ᵐ x ∂μ, Tendsto (fun n => (μ[g|ℱ n]) x) atTop (𝓝 (g x)) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {g : Ω → ℝ} (hg : MeasureTheory.Integrable g μ) (hgmeas : MeasureTheory.StronglyMeasurable g)
Goal: ∀ᵐ (x : Ω) ∂μ, Filter.Tendsto (fun n => (μ[g|↑ℱ n]) x) Filter.atTop (𝓝 (g x))
-- MeasureTheory.Integrable.tendsto_eLpNorm_condexp
theorem Integrable.tendsto_eLpNorm_condexp (hg : Integrable g μ)
    (hgmeas : StronglyMeasurable[⨆ n, ℱ n] g) :
    Tendsto (fun n => eLpNorm (μ[g|ℱ n] - g) 1 μ) atTop (𝓝 0) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {g : Ω → ℝ} (hg : MeasureTheory.Integrable g μ) (hgmeas : MeasureTheory.StronglyMeasurable g)
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (μ[g|↑ℱ n] - g) 1 μ) Filter.atTop (𝓝 0)
-- MeasureTheory.tendsto_ae_condexp
theorem tendsto_ae_condexp (g : Ω → ℝ) :
    ∀ᵐ x ∂μ, Tendsto (fun n => (μ[g|ℱ n]) x) atTop (𝓝 ((μ[g|⨆ n, ℱ n]) x)) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (g : Ω → ℝ)
Goal: ∀ᵐ (x : Ω) ∂μ, Filter.Tendsto (fun n => (μ[g|↑ℱ n]) x) Filter.atTop (𝓝 ((μ[g|⨆ n, ↑ℱ n]) x))
-- MeasureTheory.tendsto_eLpNorm_condexp
theorem tendsto_eLpNorm_condexp (g : Ω → ℝ) :
    Tendsto (fun n => eLpNorm (μ[g|ℱ n] - μ[g|⨆ n, ℱ n]) 1 μ) atTop (𝓝 0) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] (g : Ω → ℝ)
Goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (μ[g|↑ℱ n] - μ[g|⨆ n, ↑ℱ n]) 1 μ) Filter.atTop (𝓝 0)
-- Ideal.iSup_iInf_eq_top_iff_pairwise
theorem iSup_iInf_eq_top_iff_pairwise {t : Finset ι} (h : t.Nonempty) (I : ι → Ideal R) :
    (⨆ i ∈ t, ⨅ (j) (_ : j ∈ t) (_ : j ≠ i), I j) = ⊤ ↔
      (t : Set ι).Pairwise fun i j => I i ⊔ I j = ⊤ 
Arguments: {ι : Type u_1} {R : Type u_2} [CommSemiring R] {t : Finset ι} (h : t.Nonempty) (I : ι → Ideal R)
Goal: ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤
-- ProbabilityTheory.Kernel.compProdFun_empty
theorem compProdFun_empty (κ : Kernel α β) (η : Kernel (α × β) γ) (a : α) :
    compProdFun κ η a ∅ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) (a : α)
Goal: κ.compProdFun η a ∅ = 0
-- ProbabilityTheory.Kernel.compProdFun_iUnion
theorem compProdFun_iUnion (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel η] (a : α)
    (f : ℕ → Set (β × γ)) (hf_meas : ∀ i, MeasurableSet (f i))
    (hf_disj : Pairwise (Disjoint on f)) :
    compProdFun κ η a (⋃ i, f i) = ∑' i, compProdFun κ η a (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (f : ℕ → Set (β × γ)) (hf_meas : ∀ (i : ℕ), MeasurableSet (f i)) (hf_disj : Pairwise (Disjoint on f))
Goal: κ.compProdFun η a (⋃ i, f i) = ∑' (i : ℕ), κ.compProdFun η a (f i)
-- ProbabilityTheory.Kernel.compProdFun_tsum_right
theorem compProdFun_tsum_right (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel η] (a : α)
    (hs : MeasurableSet s) : compProdFun κ η a s = ∑' n, compProdFun κ (seq η n) a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (hs : MeasurableSet s)
Goal: κ.compProdFun η a s = ∑' (n : ℕ), κ.compProdFun (η.seq n) a s
-- ProbabilityTheory.Kernel.compProdFun_tsum_left
theorem compProdFun_tsum_left (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel κ] (a : α)
    (s : Set (β × γ)) : compProdFun κ η a s = ∑' n, compProdFun (seq κ n) η a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel κ] (a : α) (s : Set (β × γ))
Goal: κ.compProdFun η a s = ∑' (n : ℕ), (κ.seq n).compProdFun η a s
-- ProbabilityTheory.Kernel.compProdFun_eq_tsum
theorem compProdFun_eq_tsum (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
    compProdFun κ η a s = ∑' (n) (m), compProdFun (seq κ n) (seq η m) a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (hs : MeasurableSet s)
Goal: κ.compProdFun η a s = ∑' (n : ℕ) (m : ℕ), (κ.seq n).compProdFun (η.seq m) a s
-- ProbabilityTheory.Kernel.measurable_compProdFun_of_finite
theorem measurable_compProdFun_of_finite (κ : Kernel α β) [IsFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsFiniteKernel η] (hs : MeasurableSet s) : Measurable fun a => compProdFun κ η a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsFiniteKernel η] (hs : MeasurableSet s)
Goal: Measurable fun a => κ.compProdFun η a s
-- ProbabilityTheory.Kernel.measurable_compProdFun
theorem measurable_compProdFun (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (hs : MeasurableSet s) : Measurable fun a => compProdFun κ η a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (hs : MeasurableSet s)
Goal: Measurable fun a => κ.compProdFun η a s
-- ProbabilityTheory.Kernel.compProd_apply_eq_compProdFun
theorem compProd_apply_eq_compProdFun (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
    (κ ⊗ₖ η) a s = compProdFun κ η a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (hs : MeasurableSet s)
Goal: ((κ.compProd η) a) s = κ.compProdFun η a s
-- ProbabilityTheory.Kernel.compProd_of_not_isSFiniteKernel_left
theorem compProd_of_not_isSFiniteKernel_left (κ : Kernel α β) (η : Kernel (α × β) γ)
    (h : ¬ IsSFiniteKernel κ) :
    κ ⊗ₖ η = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) (h : ¬ProbabilityTheory.IsSFiniteKernel κ)
Goal: κ.compProd η = 0
-- ProbabilityTheory.Kernel.compProd_of_not_isSFiniteKernel_right
theorem compProd_of_not_isSFiniteKernel_right (κ : Kernel α β) (η : Kernel (α × β) γ)
    (h : ¬ IsSFiniteKernel η) :
    κ ⊗ₖ η = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel (α × β) γ) (h : ¬ProbabilityTheory.IsSFiniteKernel η)
Goal: κ.compProd η = 0
-- ProbabilityTheory.Kernel.compProd_apply
theorem compProd_apply (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
    (κ ⊗ₖ η) a s = ∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (hs : MeasurableSet s)
Goal: ((κ.compProd η) a) s = ∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂κ a
-- ProbabilityTheory.Kernel.le_compProd_apply
theorem le_compProd_apply (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) (s : Set (β × γ)) :
    ∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂κ a ≤ (κ ⊗ₖ η) a s 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) (s : Set (β × γ))
Goal: ∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂κ a ≤ ((κ.compProd η) a) s
-- ProbabilityTheory.Kernel.compProd_zero_left
@[simp]
lemma compProd_zero_left (κ : Kernel (α × β) γ) :
    (0 : Kernel α β) ⊗ₖ κ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel (α × β) γ)
Goal: ProbabilityTheory.Kernel.compProd 0 κ = 0
-- ProbabilityTheory.Kernel.compProd_zero_right
@[simp]
lemma compProd_zero_right (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ] :
    κ ⊗ₖ (0 : Kernel (α × β) γ) = 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (γ : Type u_5) [MeasurableSpace γ]
Goal: κ.compProd 0 = 0
-- ProbabilityTheory.Kernel.ae_kernel_lt_top
theorem ae_kernel_lt_top (a : α) (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
    ∀ᵐ b ∂κ a, η (a, b) (Prod.mk b ⁻¹' s) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] (a : α) (h2s : ((κ.compProd η) a) s ≠ ⊤)
Goal: ∀ᵐ (b : β) ∂κ a, (η (a, b)) (Prod.mk b ⁻¹' s) < ⊤
-- ProbabilityTheory.Kernel.compProd_null
theorem compProd_null (a : α) (hs : MeasurableSet s) :
    (κ ⊗ₖ η) a s = 0 ↔ (fun b => η (a, b) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] (a : α) (hs : MeasurableSet s)
Goal: ((κ.compProd η) a) s = 0 ↔ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0
-- ProbabilityTheory.Kernel.ae_null_of_compProd_null
theorem ae_null_of_compProd_null (h : (κ ⊗ₖ η) a s = 0) :
    (fun b => η (a, b) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {s : Set (β × γ)} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} (h : ((κ.compProd η) a) s = 0)
Goal: (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0
-- ProbabilityTheory.Kernel.ae_ae_of_ae_compProd
theorem ae_ae_of_ae_compProd {p : β × γ → Prop} (h : ∀ᵐ bc ∂(κ ⊗ₖ η) a, p bc) :
    ∀ᵐ b ∂κ a, ∀ᵐ c ∂η (a, b), p (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} {p : β × γ → Prop} (h : ∀ᵐ (bc : β × γ) ∂(κ.compProd η) a, p bc)
Goal: ∀ᵐ (b : β) ∂κ a, ∀ᵐ (c : γ) ∂η (a, b), p (b, c)
-- ProbabilityTheory.Kernel.ae_compProd_of_ae_ae
lemma ae_compProd_of_ae_ae {p : β × γ → Prop} (hp : MeasurableSet {x | p x})
    (h : ∀ᵐ b ∂κ a, ∀ᵐ c ∂η (a, b), p (b, c)) :
    ∀ᵐ bc ∂(κ ⊗ₖ η) a, p bc 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} {p : β × γ → Prop} (hp : MeasurableSet {x | p x}) (h : ∀ᵐ (b : β) ∂κ a, ∀ᵐ (c : γ) ∂η (a, b), p (b, c))
Goal: ∀ᵐ (bc : β × γ) ∂(κ.compProd η) a, p bc
-- ProbabilityTheory.Kernel.ae_compProd_iff
lemma ae_compProd_iff {p : β × γ → Prop} (hp : MeasurableSet {x | p x}) :
    (∀ᵐ bc ∂(κ ⊗ₖ η) a, p bc) ↔ ∀ᵐ b ∂κ a, ∀ᵐ c ∂η (a, b), p (b, c) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} {p : β × γ → Prop} (hp : MeasurableSet {x | p x})
Goal: (∀ᵐ (bc : β × γ) ∂(κ.compProd η) a, p bc) ↔ ∀ᵐ (b : β) ∂κ a, ∀ᵐ (c : γ) ∂η (a, b), p (b, c)
-- ProbabilityTheory.Kernel.compProd_restrict
theorem compProd_restrict {s : Set β} {t : Set γ} (hs : MeasurableSet s) (ht : MeasurableSet t) :
    Kernel.restrict κ hs ⊗ₖ Kernel.restrict η ht = Kernel.restrict (κ ⊗ₖ η) (hs.prod ht) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {s : Set β} {t : Set γ} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: (κ.restrict hs).compProd (η.restrict ht) = (κ.compProd η).restrict ⋯
-- ProbabilityTheory.Kernel.compProd_restrict_left
theorem compProd_restrict_left {s : Set β} (hs : MeasurableSet s) :
    Kernel.restrict κ hs ⊗ₖ η = Kernel.restrict (κ ⊗ₖ η) (hs.prod MeasurableSet.univ) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {s : Set β} (hs : MeasurableSet s)
Goal: (κ.restrict hs).compProd η = (κ.compProd η).restrict ⋯
-- ProbabilityTheory.Kernel.compProd_restrict_right
theorem compProd_restrict_right {t : Set γ} (ht : MeasurableSet t) :
    κ ⊗ₖ Kernel.restrict η ht = Kernel.restrict (κ ⊗ₖ η) (MeasurableSet.univ.prod ht) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {t : Set γ} (ht : MeasurableSet t)
Goal: κ.compProd (η.restrict ht) = (κ.compProd η).restrict ⋯
-- ProbabilityTheory.Kernel.lintegral_compProd'
theorem lintegral_compProd' (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β → γ → ℝ≥0∞} (hf : Measurable (Function.uncurry f)) :
    ∫⁻ bc, f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ b, ∫⁻ c, f b c ∂η (a, b) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β → γ → ℝ≥0∞} (hf : Measurable (Function.uncurry f))
Goal: ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂(κ.compProd η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂η (a, b) ∂κ a
-- ProbabilityTheory.Kernel.lintegral_compProd
theorem lintegral_compProd (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ bc, f bc ∂(κ ⊗ₖ η) a = ∫⁻ b, ∫⁻ c, f (b, c) ∂η (a, b) ∂κ a 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_4} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) γ) [ProbabilityTheory.IsSFiniteKernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (bc : β × γ), f bc ∂(κ.compProd η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f (b, c) ∂η (a, b) ∂κ a