ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.iIndepFun.condexp_natural_ae_eq_of_lt
theorem iIndepFun.condexp_natural_ae_eq_of_lt [SecondCountableTopology β] [CompleteSpace β]
    [NormedSpace ℝ β] (hf : ∀ i, StronglyMeasurable (f i)) (hfi : iIndepFun (fun _ => mβ) f μ)
    (hij : i < j) : μ[f j|Filtration.natural f hf i] =ᵐ[μ] fun _ => μ[f j] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} {β : Type u_3} [LinearOrder ι] [mβ : MeasurableSpace β] [NormedAddCommGroup β] [BorelSpace β] {f : ι → Ω → β} {i j : ι} [SecondCountableTopology β] [CompleteSpace β] [NormedSpace ℝ β] (hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (hfi : ProbabilityTheory.iIndepFun (fun x => mβ) f μ) (hij : i < j)
Goal: μ[f j|↑(MeasureTheory.Filtration.natural f hf) i] =ᵐ[μ] fun x => ∫ (x : Ω), f j x ∂μ
-- ProbabilityTheory.iIndepSet.condexp_indicator_filtrationOfSet_ae_eq
theorem iIndepSet.condexp_indicator_filtrationOfSet_ae_eq (hsm : ∀ n, MeasurableSet (s n))
    (hs : iIndepSet s μ) (hij : i < j) :
    μ[(s j).indicator (fun _ => 1 : Ω → ℝ)|filtrationOfSet hsm i] =ᵐ[μ]
    fun _ => (μ (s j)).toReal 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_2} [LinearOrder ι] {i j : ι} {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (hij : i < j)
Goal: μ[(s j).indicator fun x => 1|↑(MeasureTheory.filtrationOfSet hsm) i] =ᵐ[μ] fun x => (μ (s j)).toReal
-- ProbabilityTheory.measure_limsup_eq_one
theorem measure_limsup_eq_one {s : ℕ → Set Ω} (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ)
    (hs' : (∑' n, μ (s n)) = ∞) : μ (limsup s atTop) = 1 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {s : ℕ → Set Ω} (hsm : ∀ (n : ℕ), MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (hs' : ∑' (n : ℕ), μ (s n) = ⊤)
Goal: μ (Filter.limsup s Filter.atTop) = 1
-- PMF.binomial_apply
theorem binomial_apply (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) (i : Fin (n + 1)) :
    binomial p h n i = p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ) 
Arguments: (p : ENNReal) (h : p ≤ 1) (n : ℕ) (i : Fin (n + 1))
Goal: (PMF.binomial p h n) i = p ^ ↑i * (1 - p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)
-- PMF.binomial_apply_zero
@[simp]
theorem binomial_apply_zero (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
    binomial p h n 0 = (1-p)^n 
Arguments: (p : ENNReal) (h : p ≤ 1) (n : ℕ)
Goal: (PMF.binomial p h n) 0 = (1 - p) ^ n
-- PMF.binomial_apply_last
@[simp]
theorem binomial_apply_last (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
    binomial p h n (.last n) = p^n 
Arguments: (p : ENNReal) (h : p ≤ 1) (n : ℕ)
Goal: (PMF.binomial p h n) (Fin.last n) = p ^ n
-- PMF.binomial_apply_self
theorem binomial_apply_self (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
    binomial p h n n = p^n 
Arguments: (p : ENNReal) (h : p ≤ 1) (n : ℕ)
Goal: (PMF.binomial p h n) ↑n = p ^ n
-- PMF.binomial_one_eq_bernoulli
theorem binomial_one_eq_bernoulli (p : ℝ≥0∞) (h : p ≤ 1) :
    binomial p h 1 = (bernoulli p h).map (cond · 1 0) 
Arguments: (p : ENNReal) (h : p ≤ 1)
Goal: PMF.binomial p h 1 = PMF.map (fun x => bif x then 1 else 0) (PMF.bernoulli p h)
-- PMF.pure_apply
@[simp]
theorem pure_apply : pure a a' = if a' = a then 1 else 0 
Arguments: {α : Type u_1} (a a' : α)
Goal: (PMF.pure a) a' = if a' = a then 1 else 0
-- PMF.support_pure
@[simp]
theorem support_pure : (pure a).support = {a} 
Arguments: {α : Type u_1} (a : α)
Goal: (PMF.pure a).support = {a}
-- PMF.mem_support_pure_iff
theorem mem_support_pure_iff : a' ∈ (pure a).support ↔ a' = a 
Arguments: {α : Type u_1} (a a' : α)
Goal: a' ∈ (PMF.pure a).support ↔ a' = a
-- PMF.pure_apply_self
theorem pure_apply_self : pure a a = 1 
Arguments: {α : Type u_1} (a : α)
Goal: (PMF.pure a) a = 1
-- PMF.pure_apply_of_ne
theorem pure_apply_of_ne (h : a' ≠ a) : pure a a' = 0 
Arguments: {α : Type u_1} (a a' : α) (h : a' ≠ a)
Goal: (PMF.pure a) a' = 0
-- PMF.toOuterMeasure_pure_apply
@[simp]
theorem toOuterMeasure_pure_apply : (pure a).toOuterMeasure s = if a ∈ s then 1 else 0 
Arguments: {α : Type u_1} (a : α) (s : Set α)
Goal: (PMF.pure a).toOuterMeasure s = if a ∈ s then 1 else 0
-- PMF.toMeasure_pure_apply
@[simp]
theorem toMeasure_pure_apply (hs : MeasurableSet s) :
    (pure a).toMeasure s = if a ∈ s then 1 else 0 
Arguments: {α : Type u_1} (a : α) (s : Set α) [MeasurableSpace α] (hs : MeasurableSet s)
Goal: (PMF.pure a).toMeasure s = if a ∈ s then 1 else 0
-- PMF.toMeasure_pure
theorem toMeasure_pure : (pure a).toMeasure = Measure.dirac a 
Arguments: {α : Type u_1} (a : α) [MeasurableSpace α]
Goal: (PMF.pure a).toMeasure = MeasureTheory.Measure.dirac a
-- PMF.toPMF_dirac
@[simp]
theorem toPMF_dirac [Countable α] [h : MeasurableSingletonClass α] :
    (Measure.dirac a).toPMF = pure a 
Arguments: {α : Type u_1} (a : α) [MeasurableSpace α] [Countable α] [h : MeasurableSingletonClass α]
Goal: (MeasureTheory.Measure.dirac a).toPMF = PMF.pure a
-- PMF.bind_apply
@[simp]
theorem bind_apply (b : β) : p.bind f b = ∑' a, p a * f a b 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β) (b : β)
Goal: (p.bind f) b = ∑' (a : α), p a * (f a) b
-- PMF.support_bind
@[simp]
theorem support_bind : (p.bind f).support = ⋃ a ∈ p.support, (f a).support 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β)
Goal: (p.bind f).support = ⋃ a ∈ p.support, (f a).support
-- PMF.mem_support_bind_iff
theorem mem_support_bind_iff (b : β) :
    b ∈ (p.bind f).support ↔ ∃ a ∈ p.support, b ∈ (f a).support 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β) (b : β)
Goal: b ∈ (p.bind f).support ↔ ∃ a ∈ p.support, b ∈ (f a).support
-- PMF.pure_bind
@[simp]
theorem pure_bind (a : α) (f : α → PMF β) : (pure a).bind f = f a 
Arguments: {α : Type u_1} {β : Type u_2} (a : α) (f : α → PMF β)
Goal: (PMF.pure a).bind f = f a
-- PMF.bind_pure
@[simp]
theorem bind_pure : p.bind pure = p 
Arguments: {α : Type u_1} (p : PMF α)
Goal: p.bind PMF.pure = p
-- PMF.bind_const
@[simp]
theorem bind_const (p : PMF α) (q : PMF β) : (p.bind fun _ => q) = q 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (q : PMF β)
Goal: (p.bind fun x => q) = q
-- PMF.bind_bind
@[simp]
theorem bind_bind : (p.bind f).bind g = p.bind fun a => (f a).bind g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : PMF α) (f : α → PMF β) (g : β → PMF γ)
Goal: (p.bind f).bind g = p.bind fun a => (f a).bind g
-- PMF.bind_comm
theorem bind_comm (p : PMF α) (q : PMF β) (f : α → β → PMF γ) :
    (p.bind fun a => q.bind (f a)) = q.bind fun b => p.bind fun a => f a b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : PMF α) (q : PMF β) (f : α → β → PMF γ)
Goal: (p.bind fun a => q.bind (f a)) = q.bind fun b => p.bind fun a => f a b
-- PMF.toOuterMeasure_bind_apply
@[simp]
theorem toOuterMeasure_bind_apply :
    (p.bind f).toOuterMeasure s = ∑' a, p a * (f a).toOuterMeasure s 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β) (s : Set β)
Goal: (p.bind f).toOuterMeasure s = ∑' (a : α), p a * (f a).toOuterMeasure s
-- PMF.toMeasure_bind_apply
@[simp]
theorem toMeasure_bind_apply [MeasurableSpace β] (hs : MeasurableSet s) :
    (p.bind f).toMeasure s = ∑' a, p a * (f a).toMeasure s 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β) (s : Set β) [MeasurableSpace β] (hs : MeasurableSet s)
Goal: (p.bind f).toMeasure s = ∑' (a : α), p a * (f a).toMeasure s
-- PMF.bindOnSupport_apply
@[simp]
theorem bindOnSupport_apply (b : β) :
    p.bindOnSupport f b = ∑' a, p a * if h : p a = 0 then 0 else f a h b 
Arguments: {α : Type u_1} {β : Type u_2} {p : PMF α} (f : (a : α) → a ∈ p.support → PMF β) (b : β)
Goal: (p.bindOnSupport f) b = ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h) b
-- PMF.support_bindOnSupport
@[simp]
theorem support_bindOnSupport :
    (p.bindOnSupport f).support = ⋃ (a : α) (h : a ∈ p.support), (f a h).support 
Arguments: {α : Type u_1} {β : Type u_2} {p : PMF α} (f : (a : α) → a ∈ p.support → PMF β)
Goal: (p.bindOnSupport f).support = ⋃ a, ⋃ (h : a ∈ p.support), (f a h).support
-- PMF.mem_support_bindOnSupport_iff
theorem mem_support_bindOnSupport_iff (b : β) :
    b ∈ (p.bindOnSupport f).support ↔ ∃ (a : α) (h : a ∈ p.support), b ∈ (f a h).support 
Arguments: {α : Type u_1} {β : Type u_2} {p : PMF α} (f : (a : α) → a ∈ p.support → PMF β) (b : β)
Goal: b ∈ (p.bindOnSupport f).support ↔ ∃ a, ∃ (h : a ∈ p.support), b ∈ (f a h).support
-- PMF.bindOnSupport_eq_bind
@[simp]
theorem bindOnSupport_eq_bind (p : PMF α) (f : α → PMF β) :
    (p.bindOnSupport fun a _ => f a) = p.bind f 
Arguments: {α : Type u_1} {β : Type u_2} (p : PMF α) (f : α → PMF β)
Goal: (p.bindOnSupport fun a x => f a) = p.bind f
-- PMF.bindOnSupport_eq_zero_iff
theorem bindOnSupport_eq_zero_iff (b : β) :
    p.bindOnSupport f b = 0 ↔ ∀ (a) (ha : p a ≠ 0), f a ha b = 0 
Arguments: {α : Type u_1} {β : Type u_2} {p : PMF α} (f : (a : α) → a ∈ p.support → PMF β) (b : β)
Goal: (p.bindOnSupport f) b = 0 ↔ ∀ (a : α) (ha : p a ≠ 0), (f a ha) b = 0
-- PMF.pure_bindOnSupport
@[simp]
theorem pure_bindOnSupport (a : α) (f : ∀ (a' : α) (_ : a' ∈ (pure a).support), PMF β) :
    (pure a).bindOnSupport f = f a ((mem_support_pure_iff a a).mpr rfl) 
Arguments: {α : Type u_1} {β : Type u_2} (a : α) (f : (a' : α) → a' ∈ (PMF.pure a).support → PMF β)
Goal: (PMF.pure a).bindOnSupport f = f a ⋯
-- PMF.bindOnSupport_pure
theorem bindOnSupport_pure (p : PMF α) : (p.bindOnSupport fun a _ => pure a) = p 
Arguments: {α : Type u_1} (p : PMF α)
Goal: (p.bindOnSupport fun a x => PMF.pure a) = p
-- PMF.bindOnSupport_bindOnSupport
@[simp]
theorem bindOnSupport_bindOnSupport (p : PMF α) (f : ∀ a ∈ p.support, PMF β)
    (g : ∀ b ∈ (p.bindOnSupport f).support, PMF γ) :
    (p.bindOnSupport f).bindOnSupport g =
      p.bindOnSupport fun a ha =>
        (f a ha).bindOnSupport fun b hb =>
          g b ((mem_support_bindOnSupport_iff f b).mpr ⟨a, ha, hb⟩) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : PMF α) (f : (a : α) → a ∈ p.support → PMF β) (g : (b : β) → b ∈ (p.bindOnSupport f).support → PMF γ)
Goal: (p.bindOnSupport f).bindOnSupport g = p.bindOnSupport fun a ha => (f a ha).bindOnSupport fun b hb => g b ⋯
-- PMF.bindOnSupport_comm
theorem bindOnSupport_comm (p : PMF α) (q : PMF β) (f : ∀ a ∈ p.support, ∀ b ∈ q.support, PMF γ) :
    (p.bindOnSupport fun a ha => q.bindOnSupport (f a ha)) =
      q.bindOnSupport fun b hb => p.bindOnSupport fun a ha => f a ha b hb 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : PMF α) (q : PMF β) (f : (a : α) → a ∈ p.support → (b : β) → b ∈ q.support → PMF γ)
Goal: (p.bindOnSupport fun a ha => q.bindOnSupport (f a ha)) =
  q.bindOnSupport fun b hb => p.bindOnSupport fun a ha => f a ha b hb
-- PMF.toOuterMeasure_bindOnSupport_apply
@[simp]
theorem toOuterMeasure_bindOnSupport_apply :
    (p.bindOnSupport f).toOuterMeasure s =
      ∑' a, p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s 
Arguments: {α : Type u_1} {β : Type u_2} {p : PMF α} (f : (a : α) → a ∈ p.support → PMF β) (s : Set β)
Goal: (p.bindOnSupport f).toOuterMeasure s = ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s
-- PMF.toMeasure_bindOnSupport_apply
@[simp]
theorem toMeasure_bindOnSupport_apply [MeasurableSpace β] (hs : MeasurableSet s) :
    (p.bindOnSupport f).toMeasure s =
      ∑' a, p a * if h : p a = 0 then 0 else (f a h).toMeasure s 
Arguments: {α : Type u_1} {β : Type u_2} {p : PMF α} (f : (a : α) → a ∈ p.support → PMF β) (s : Set β) [MeasurableSpace β] (hs : MeasurableSet s)
Goal: (p.bindOnSupport f).toMeasure s = ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h).toMeasure s
-- ProbabilityTheory.IsRatCondKernelCDF.measurable
structure IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) : Prop :=
  measurable : Measurable f
  isRatStieltjesPoint_ae (a : α) : ∀ᵐ b ∂(ν a), IsRatStieltjesPoint f (a, b)
  integrable (a : α) (q : ℚ) : Integrable (fun b ↦ f (a, b) q) (ν a)
  setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (q : ℚ) :
    ∫ b in s, f (a, b) q ∂(ν a) = (κ a (s ×ˢ Iic (q : ℝ))).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDF f κ ν)
Goal: Measurable f
-- ProbabilityTheory.IsRatCondKernelCDF.setIntegral
structure IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) : Prop :=
  measurable : Measurable f
  isRatStieltjesPoint_ae (a : α) : ∀ᵐ b ∂(ν a), IsRatStieltjesPoint f (a, b)
  integrable (a : α) (q : ℚ) : Integrable (fun b ↦ f (a, b) q) (ν a)
  setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (q : ℚ) :
    ∫ b in s, f (a, b) q ∂(ν a) = (κ a (s ×ˢ Iic (q : ℝ))).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) {s : Set β} (_hs : MeasurableSet s) (q : ℚ)
Goal: ∫ (b : β) in s, f (a, b) q ∂ν a = ((κ a) (s ×ˢ Set.Iic ↑q)).toReal
-- ProbabilityTheory.IsRatCondKernelCDF.integrable
structure IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) : Prop :=
  measurable : Measurable f
  isRatStieltjesPoint_ae (a : α) : ∀ᵐ b ∂(ν a), IsRatStieltjesPoint f (a, b)
  integrable (a : α) (q : ℚ) : Integrable (fun b ↦ f (a, b) q) (ν a)
  setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (q : ℚ) :
    ∫ b in s, f (a, b) q ∂(ν a) = (κ a (s ×ˢ Iic (q : ℝ))).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ)
Goal: MeasureTheory.Integrable (fun b => f (a, b) q) (ν a)
-- ProbabilityTheory.IsRatCondKernelCDF.isRatStieltjesPoint_ae
structure IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) : Prop :=
  measurable : Measurable f
  isRatStieltjesPoint_ae (a : α) : ∀ᵐ b ∂(ν a), IsRatStieltjesPoint f (a, b)
  integrable (a : α) (q : ℚ) : Integrable (fun b ↦ f (a, b) q) (ν a)
  setIntegral (a : α) {s : Set β} (_hs : MeasurableSet s) (q : ℚ) :
    ∫ b in s, f (a, b) q ∂(ν a) = (κ a (s ×ˢ Iic (q : ℝ))).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α)
Goal: ∀ᵐ (b : β) ∂ν a, ProbabilityTheory.IsRatStieltjesPoint f (a, b)
-- ProbabilityTheory.IsRatCondKernelCDF.mono
lemma IsRatCondKernelCDF.mono (hf : IsRatCondKernelCDF f κ ν) (a : α) :
    ∀ᵐ b ∂(ν a), Monotone (f (a, b)) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α)
Goal: ∀ᵐ (b : β) ∂ν a, Monotone (f (a, b))
-- ProbabilityTheory.IsRatCondKernelCDF.tendsto_atTop_one
lemma IsRatCondKernelCDF.tendsto_atTop_one (hf : IsRatCondKernelCDF f κ ν) (a : α) :
    ∀ᵐ b ∂(ν a), Tendsto (f (a, b)) atTop (𝓝 1) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α)
Goal: ∀ᵐ (b : β) ∂ν a, Filter.Tendsto (f (a, b)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.IsRatCondKernelCDF.tendsto_atBot_zero
lemma IsRatCondKernelCDF.tendsto_atBot_zero (hf : IsRatCondKernelCDF f κ ν) (a : α) :
    ∀ᵐ b ∂(ν a), Tendsto (f (a, b)) atBot (𝓝 0) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α)
Goal: ∀ᵐ (b : β) ∂ν a, Filter.Tendsto (f (a, b)) Filter.atBot (𝓝 0)
-- ProbabilityTheory.IsRatCondKernelCDF.iInf_rat_gt_eq
lemma IsRatCondKernelCDF.iInf_rat_gt_eq (hf : IsRatCondKernelCDF f κ ν) (a : α) :
    ∀ᵐ b ∂(ν a), ∀ q, ⨅ r : Ioi q, f (a, b) r = f (a, b) q 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α)
Goal: ∀ᵐ (b : β) ∂ν a, ∀ (q : ℚ), ⨅ r, f (a, b) ↑r = f (a, b) q
-- ProbabilityTheory.stieltjesOfMeasurableRat_ae_eq
lemma stieltjesOfMeasurableRat_ae_eq (hf : IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ) :
    (fun b ↦ stieltjesOfMeasurableRat f hf.measurable (a, b) q) =ᵐ[ν a] fun b ↦ f (a, b) q 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ)
Goal: (fun b => ↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q) =ᵐ[ν a] fun b => f (a, b) q
-- ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat_rat
lemma setIntegral_stieltjesOfMeasurableRat_rat (hf : IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ)
    {s : Set β} (hs : MeasurableSet s) :
    ∫ b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) q ∂(ν a)
      = (κ a (s ×ˢ Iic (q : ℝ))).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ) {s : Set β} (hs : MeasurableSet s)
Goal: ∫ (b : β) in s, ↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q ∂ν a = ((κ a) (s ×ˢ Set.Iic ↑q)).toReal
-- ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat_rat
lemma setLIntegral_stieltjesOfMeasurableRat_rat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (q : ℚ) {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) q) ∂(ν a)
      = κ a (s ×ˢ Iic (q : ℝ)) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (b : β) in s, ENNReal.ofReal (↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q) ∂ν a =
  (κ a) (s ×ˢ Set.Iic ↑q)
-- ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat
lemma setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
    ∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
      = κ a (s ×ˢ Iic x) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (b : β) in s, ENNReal.ofReal (↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
  (κ a) (s ×ˢ Set.Iic x)
-- ProbabilityTheory.lintegral_stieltjesOfMeasurableRat
lemma lintegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (x : ℝ) :
    ∫⁻ b, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
      = κ a (univ ×ˢ Iic x) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (x : ℝ)
Goal: ∫⁻ (b : β), ENNReal.ofReal (↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
  (κ a) (Set.univ ×ˢ Set.Iic x)
-- ProbabilityTheory.integrable_stieltjesOfMeasurableRat
lemma integrable_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (x : ℝ) :
    Integrable (fun b ↦ stieltjesOfMeasurableRat f hf.measurable (a, b) x) (ν a) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (x : ℝ)
Goal: MeasureTheory.Integrable (fun b => ↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) x) (ν a)
-- ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat
lemma setIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
    ∫ b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) x ∂(ν a)
      = (κ a (s ×ˢ Iic x)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s)
Goal: ∫ (b : β) in s, ↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = ((κ a) (s ×ˢ Set.Iic x)).toReal
-- ProbabilityTheory.integral_stieltjesOfMeasurableRat
lemma integral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
    (a : α) (x : ℝ) :
    ∫ b, stieltjesOfMeasurableRat f hf.measurable (a, b) x ∂(ν a)
      = (κ a (univ ×ˢ Iic x)).toReal 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → ℚ → ℝ} [ProbabilityTheory.IsFiniteKernel κ] (hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν) (a : α) (x : ℝ)
Goal: ∫ (b : β), ↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = ((κ a) (Set.univ ×ˢ Set.Iic x)).toReal
-- ProbabilityTheory.IsRatCondKernelCDFAux.measurable
structure IsRatCondKernelCDFAux (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
    Prop :=
  measurable : Measurable f
  mono' (a : α) {q r : ℚ} (_hqr : q ≤ r) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ f (a, c) r
  nonneg' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), 0 ≤ f (a, c) q
  le_one' (a : α) (q : ℚ) : ∀ᵐ c ∂(ν a), f (a, c) q ≤ 1
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atBot (𝓝 0)` but slightly easier
  to prove in the current applications of this definition (some integral convergence lemmas
  currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_antitone (a : α) (seq : ℕ → ℚ) (_hs : Antitone seq)
    (_hs_tendsto : Tendsto seq atTop atBot) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 0)
  /- Same as `Tendsto (fun q : ℚ ↦ ∫ c, f (a, c) q ∂(ν a)) atTop (𝓝 (ν a univ).toReal)` but
  slightly easier to prove in the current applications of this definition (some integral convergence
  lemmas currently apply only to `ℕ`, not `ℚ`) -/
  tendsto_integral_of_monotone (a : α) (seq : ℕ → ℚ) (_hs : Monotone seq)
    (_hs_tendsto : Tendsto seq atTop atTop) :
    Tendsto (fun m ↦ ∫ c, f (a, c) (seq m) ∂(ν a)) atTop (𝓝 (ν a univ).toReal)
  integrable (a : α) (q : ℚ) : Integrable (fun c ↦ f (a, c) q) (ν a)
  setIntegral (a : α) {A : Set β} (_hA : MeasurableSet A) (q : ℚ) :
    ∫ c in A, f (a, c) q ∂(ν a) = (κ a (A ×ˢ Iic ↑q)).toReal
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α × β → ℚ → ℝ} {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} (self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν)
Goal: Measurable f