ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.gaussianReal_const_mul
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
    Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) 
Arguments: {μ : ℝ} {v : ℝ≥0} {Ω : Type} [MeasureTheory.MeasureSpace Ω] {X : Ω → ℝ} (hX : MeasureTheory.Measure.map X MeasureTheory.volume = ProbabilityTheory.gaussianReal μ v) (c : ℝ)
Goal: MeasureTheory.Measure.map (fun ω => c * X ω) MeasureTheory.volume =
  ProbabilityTheory.gaussianReal (c * μ) (⟨c ^ 2, ⋯⟩ * v)
-- ProbabilityTheory.gaussianReal_mul_const
lemma gaussianReal_mul_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
    Measure.map (fun ω ↦ X ω * c) ℙ = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) 
Arguments: {μ : ℝ} {v : ℝ≥0} {Ω : Type} [MeasureTheory.MeasureSpace Ω] {X : Ω → ℝ} (hX : MeasureTheory.Measure.map X MeasureTheory.volume = ProbabilityTheory.gaussianReal μ v) (c : ℝ)
Goal: MeasureTheory.Measure.map (fun ω => X ω * c) MeasureTheory.volume =
  ProbabilityTheory.gaussianReal (c * μ) (⟨c ^ 2, ⋯⟩ * v)
-- MeasureTheory.Measure.IicSnd_apply
theorem IicSnd_apply (r : ℝ) {s : Set α} (hs : MeasurableSet s) :
    ρ.IicSnd r s = ρ (s ×ˢ Iic r) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℝ) {s : Set α} (hs : MeasurableSet s)
Goal: (ρ.IicSnd r) s = ρ (s ×ˢ Set.Iic r)
-- MeasureTheory.Measure.IicSnd_univ
theorem IicSnd_univ (r : ℝ) : ρ.IicSnd r univ = ρ (univ ×ˢ Iic r) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℝ)
Goal: (ρ.IicSnd r) Set.univ = ρ (Set.univ ×ˢ Set.Iic r)
-- MeasureTheory.Measure.IicSnd_mono
theorem IicSnd_mono {r r' : ℝ} (h_le : r ≤ r') : ρ.IicSnd r ≤ ρ.IicSnd r' 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) {r r' : ℝ} (h_le : r ≤ r')
Goal: ρ.IicSnd r ≤ ρ.IicSnd r'
-- MeasureTheory.Measure.IicSnd_le_fst
theorem IicSnd_le_fst (r : ℝ) : ρ.IicSnd r ≤ ρ.fst 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℝ)
Goal: ρ.IicSnd r ≤ ρ.fst
-- MeasureTheory.Measure.IicSnd_ac_fst
theorem IicSnd_ac_fst (r : ℝ) : ρ.IicSnd r ≪ ρ.fst 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℝ)
Goal: ρ.IicSnd r ≪ ρ.fst
-- MeasureTheory.Measure.IsFiniteMeasure.IicSnd
theorem IsFiniteMeasure.IicSnd {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ] (r : ℝ) :
    IsFiniteMeasure (ρ.IicSnd r) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {ρ : MeasureTheory.Measure (α × ℝ)} [MeasureTheory.IsFiniteMeasure ρ] (r : ℝ)
Goal: MeasureTheory.IsFiniteMeasure (ρ.IicSnd r)
-- MeasureTheory.Measure.iInf_IicSnd_gt
theorem iInf_IicSnd_gt (t : ℚ) {s : Set α} (hs : MeasurableSet s) [IsFiniteMeasure ρ] :
    ⨅ r : { r' : ℚ // t < r' }, ρ.IicSnd r s = ρ.IicSnd t s 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (t : ℚ) {s : Set α} (hs : MeasurableSet s) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ⨅ r, (ρ.IicSnd ↑↑r) s = (ρ.IicSnd ↑t) s
-- MeasureTheory.Measure.tendsto_IicSnd_atTop
theorem tendsto_IicSnd_atTop {s : Set α} (hs : MeasurableSet s) :
    Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atTop (𝓝 (ρ.fst s)) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) {s : Set α} (hs : MeasurableSet s)
Goal: Filter.Tendsto (fun r => (ρ.IicSnd ↑r) s) Filter.atTop (𝓝 (ρ.fst s))
-- MeasureTheory.Measure.tendsto_IicSnd_atBot
theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
    Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s)
Goal: Filter.Tendsto (fun r => (ρ.IicSnd ↑r) s) Filter.atBot (𝓝 0)
-- ProbabilityTheory.measurable_preCDF
theorem measurable_preCDF {ρ : Measure (α × ℝ)} {r : ℚ} : Measurable (preCDF ρ r) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {ρ : MeasureTheory.Measure (α × ℝ)} {r : ℚ}
Goal: Measurable (ProbabilityTheory.preCDF ρ r)
-- ProbabilityTheory.measurable_preCDF'
lemma measurable_preCDF' {ρ : Measure (α × ℝ)} :
    Measurable fun a r ↦ (preCDF ρ r a).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {ρ : MeasureTheory.Measure (α × ℝ)}
Goal: Measurable fun a r => (ProbabilityTheory.preCDF ρ r a).toReal
-- ProbabilityTheory.withDensity_preCDF
theorem withDensity_preCDF (ρ : Measure (α × ℝ)) (r : ℚ) [IsFiniteMeasure ρ] :
    ρ.fst.withDensity (preCDF ρ r) = ρ.IicSnd r 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℚ) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ρ.fst.withDensity (ProbabilityTheory.preCDF ρ r) = ρ.IicSnd ↑r
-- ProbabilityTheory.setLIntegral_preCDF_fst
theorem setLIntegral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) {s : Set α} (hs : MeasurableSet s)
    [IsFiniteMeasure ρ] : ∫⁻ x in s, preCDF ρ r x ∂ρ.fst = ρ.IicSnd r s 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℚ) {s : Set α} (hs : MeasurableSet s) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ∫⁻ (x : α) in s, ProbabilityTheory.preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) s
-- ProbabilityTheory.lintegral_preCDF_fst
lemma lintegral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) [IsFiniteMeasure ρ] :
    ∫⁻ x, preCDF ρ r x ∂ρ.fst = ρ.IicSnd r univ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℚ) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ∫⁻ (x : α), ProbabilityTheory.preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) Set.univ
-- ProbabilityTheory.monotone_preCDF
theorem monotone_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
    ∀ᵐ a ∂ρ.fst, Monotone fun r ↦ preCDF ρ r a 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ∀ᵐ (a : α) ∂ρ.fst, Monotone fun r => ProbabilityTheory.preCDF ρ r a
-- ProbabilityTheory.preCDF_le_one
theorem preCDF_le_one (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
    ∀ᵐ a ∂ρ.fst, ∀ r, preCDF ρ r a ≤ 1 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ∀ᵐ (a : α) ∂ρ.fst, ∀ (r : ℚ), ProbabilityTheory.preCDF ρ r a ≤ 1
-- ProbabilityTheory.setIntegral_preCDF_fst
lemma setIntegral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) {s : Set α} (hs : MeasurableSet s)
    [IsFiniteMeasure ρ] :
    ∫ x in s, (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd r s).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℚ) {s : Set α} (hs : MeasurableSet s) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ∫ (x : α) in s, (ProbabilityTheory.preCDF ρ r x).toReal ∂ρ.fst = ((ρ.IicSnd ↑r) s).toReal
-- ProbabilityTheory.integral_preCDF_fst
lemma integral_preCDF_fst (ρ : Measure (α × ℝ)) (r : ℚ) [IsFiniteMeasure ρ] :
    ∫ x, (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd r univ).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (r : ℚ) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ∫ (x : α), (ProbabilityTheory.preCDF ρ r x).toReal ∂ρ.fst = ((ρ.IicSnd ↑r) Set.univ).toReal
-- ProbabilityTheory.integrable_preCDF
lemma integrable_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℚ) :
    Integrable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (x : ℚ)
Goal: MeasureTheory.Integrable (fun a => (ProbabilityTheory.preCDF ρ x a).toReal) ρ.fst
-- ProbabilityTheory.isRatCondKernelCDFAux_preCDF
lemma isRatCondKernelCDFAux_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
    IsRatCondKernelCDFAux (fun p r ↦ (preCDF ρ r p.2).toReal)
      (Kernel.const Unit ρ) (Kernel.const Unit ρ.fst) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ProbabilityTheory.IsRatCondKernelCDFAux (fun p r => (ProbabilityTheory.preCDF ρ r p.2).toReal)
  (ProbabilityTheory.Kernel.const Unit ρ) (ProbabilityTheory.Kernel.const Unit ρ.fst)
-- ProbabilityTheory.isRatCondKernelCDF_preCDF
lemma isRatCondKernelCDF_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
    IsRatCondKernelCDF (fun p r ↦ (preCDF ρ r p.2).toReal)
      (Kernel.const Unit ρ) (Kernel.const Unit ρ.fst) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ProbabilityTheory.IsRatCondKernelCDF (fun p r => (ProbabilityTheory.preCDF ρ r p.2).toReal)
  (ProbabilityTheory.Kernel.const Unit ρ) (ProbabilityTheory.Kernel.const Unit ρ.fst)
-- ProbabilityTheory.condCDF_eq_stieltjesOfMeasurableRat_unit_prod
lemma condCDF_eq_stieltjesOfMeasurableRat_unit_prod (ρ : Measure (α × ℝ)) (a : α) :
    condCDF ρ a = stieltjesOfMeasurableRat (fun (p : Unit × α) r ↦ (preCDF ρ r p.2).toReal)
      (measurable_preCDF'.comp measurable_snd) ((), a) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α)
Goal: ProbabilityTheory.condCDF ρ a =
  ProbabilityTheory.stieltjesOfMeasurableRat (fun p r => (ProbabilityTheory.preCDF ρ r p.2).toReal) ⋯ ((), a)
-- ProbabilityTheory.isCondKernelCDF_condCDF
lemma isCondKernelCDF_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] :
    IsCondKernelCDF (fun p : Unit × α ↦ condCDF ρ p.2) (Kernel.const Unit ρ)
      (Kernel.const Unit ρ.fst) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ProbabilityTheory.IsCondKernelCDF (fun p => ProbabilityTheory.condCDF ρ p.2) (ProbabilityTheory.Kernel.const Unit ρ)
  (ProbabilityTheory.Kernel.const Unit ρ.fst)
-- ProbabilityTheory.condCDF_nonneg
theorem condCDF_nonneg (ρ : Measure (α × ℝ)) (a : α) (r : ℝ) : 0 ≤ condCDF ρ a r 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α) (r : ℝ)
Goal: 0 ≤ ↑(ProbabilityTheory.condCDF ρ a) r
-- ProbabilityTheory.condCDF_le_one
theorem condCDF_le_one (ρ : Measure (α × ℝ)) (a : α) (x : ℝ) : condCDF ρ a x ≤ 1 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α) (x : ℝ)
Goal: ↑(ProbabilityTheory.condCDF ρ a) x ≤ 1
-- ProbabilityTheory.tendsto_condCDF_atBot
theorem tendsto_condCDF_atBot (ρ : Measure (α × ℝ)) (a : α) :
    Tendsto (condCDF ρ a) atBot (𝓝 0) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α)
Goal: Filter.Tendsto (↑(ProbabilityTheory.condCDF ρ a)) Filter.atBot (𝓝 0)
-- ProbabilityTheory.tendsto_condCDF_atTop
theorem tendsto_condCDF_atTop (ρ : Measure (α × ℝ)) (a : α) :
    Tendsto (condCDF ρ a) atTop (𝓝 1) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α)
Goal: Filter.Tendsto (↑(ProbabilityTheory.condCDF ρ a)) Filter.atTop (𝓝 1)
-- ProbabilityTheory.condCDF_ae_eq
theorem condCDF_ae_eq (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (r : ℚ) :
    (fun a ↦ condCDF ρ a r) =ᵐ[ρ.fst] fun a ↦ (preCDF ρ r a).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (r : ℚ)
Goal: (fun a => ↑(ProbabilityTheory.condCDF ρ a) ↑r) =ᵐ[ρ.fst] fun a => (ProbabilityTheory.preCDF ρ r a).toReal
-- ProbabilityTheory.ofReal_condCDF_ae_eq
theorem ofReal_condCDF_ae_eq (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (r : ℚ) :
    (fun a ↦ ENNReal.ofReal (condCDF ρ a r)) =ᵐ[ρ.fst] preCDF ρ r 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (r : ℚ)
Goal: (fun a => ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) ↑r)) =ᵐ[ρ.fst] ProbabilityTheory.preCDF ρ r
-- ProbabilityTheory.measurable_condCDF
theorem measurable_condCDF (ρ : Measure (α × ℝ)) (x : ℝ) : Measurable fun a ↦ condCDF ρ a x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (x : ℝ)
Goal: Measurable fun a => ↑(ProbabilityTheory.condCDF ρ a) x
-- ProbabilityTheory.stronglyMeasurable_condCDF
theorem stronglyMeasurable_condCDF (ρ : Measure (α × ℝ)) (x : ℝ) :
    StronglyMeasurable fun a ↦ condCDF ρ a x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (x : ℝ)
Goal: MeasureTheory.StronglyMeasurable fun a => ↑(ProbabilityTheory.condCDF ρ a) x
-- ProbabilityTheory.setLIntegral_condCDF
theorem setLIntegral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) {s : Set α}
    (hs : MeasurableSet s) :
    ∫⁻ a in s, ENNReal.ofReal (condCDF ρ a x) ∂ρ.fst = ρ (s ×ˢ Iic x) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (x : ℝ) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (a : α) in s, ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) x) ∂ρ.fst = ρ (s ×ˢ Set.Iic x)
-- ProbabilityTheory.lintegral_condCDF
theorem lintegral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) :
    ∫⁻ a, ENNReal.ofReal (condCDF ρ a x) ∂ρ.fst = ρ (univ ×ˢ Iic x) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (x : ℝ)
Goal: ∫⁻ (a : α), ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) x) ∂ρ.fst = ρ (Set.univ ×ˢ Set.Iic x)
-- ProbabilityTheory.integrable_condCDF
theorem integrable_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) :
    Integrable (fun a ↦ condCDF ρ a x) ρ.fst 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (x : ℝ)
Goal: MeasureTheory.Integrable (fun a => ↑(ProbabilityTheory.condCDF ρ a) x) ρ.fst
-- ProbabilityTheory.setIntegral_condCDF
theorem setIntegral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) {s : Set α}
    (hs : MeasurableSet s) : ∫ a in s, condCDF ρ a x ∂ρ.fst = (ρ (s ×ˢ Iic x)).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (x : ℝ) {s : Set α} (hs : MeasurableSet s)
Goal: ∫ (a : α) in s, ↑(ProbabilityTheory.condCDF ρ a) x ∂ρ.fst = (ρ (s ×ˢ Set.Iic x)).toReal
-- ProbabilityTheory.integral_condCDF
theorem integral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) :
    ∫ a, condCDF ρ a x ∂ρ.fst = (ρ (univ ×ˢ Iic x)).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) [MeasureTheory.IsFiniteMeasure ρ] (x : ℝ)
Goal: ∫ (a : α), ↑(ProbabilityTheory.condCDF ρ a) x ∂ρ.fst = (ρ (Set.univ ×ˢ Set.Iic x)).toReal
-- ProbabilityTheory.measure_condCDF_Iic
theorem measure_condCDF_Iic (ρ : Measure (α × ℝ)) (a : α) (x : ℝ) :
    (condCDF ρ a).measure (Iic x) = ENNReal.ofReal (condCDF ρ a x) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α) (x : ℝ)
Goal: (ProbabilityTheory.condCDF ρ a).measure (Set.Iic x) = ENNReal.ofReal (↑(ProbabilityTheory.condCDF ρ a) x)
-- ProbabilityTheory.measure_condCDF_univ
theorem measure_condCDF_univ (ρ : Measure (α × ℝ)) (a : α) : (condCDF ρ a).measure univ = 1 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ)) (a : α)
Goal: (ProbabilityTheory.condCDF ρ a).measure Set.univ = 1
-- ProbabilityTheory.measurable_measure_condCDF
theorem measurable_measure_condCDF (ρ : Measure (α × ℝ)) :
    Measurable fun a => (condCDF ρ a).measure 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (ρ : MeasureTheory.Measure (α × ℝ))
Goal: Measurable fun a => (ProbabilityTheory.condCDF ρ a).measure
-- MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id
theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id [TopologicalSpace F]
    (hm : m ≤ mΩ) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x : Ω × Ω => f x.2)
      (@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) 
Arguments: {Ω : Type u_1} {F : Type u_2} {m mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → F} [TopologicalSpace F] (hm : m ≤ mΩ) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (id ω, id ω)) μ)
-- MeasureTheory.Integrable.comp_snd_map_prod_id
theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_id [NormedAddCommGroup F] (hm : m ≤ mΩ)
    (hf : Integrable f μ) : Integrable (fun x : Ω × Ω => f x.2)
      (@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) 
Arguments: {Ω : Type u_1} {F : Type u_2} {m mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → F} [NormedAddCommGroup F] (hm : m ≤ mΩ) (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (id ω, id ω)) μ)
-- ProbabilityTheory.condexpKernel_def
noncomputable irreducible_def condexpKernel (μ : Measure Ω) [IsFiniteMeasure μ]
    (m : MeasurableSpace Ω) : @Kernel Ω Ω m mΩ 
Arguments: {Ω : Type u_3} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (m : MeasurableSpace Ω)
Goal: ProbabilityTheory.condexpKernel μ m = (ProbabilityTheory.condDistrib id id μ).comap id ⋯
-- ProbabilityTheory.condexpKernel_apply_eq_condDistrib
lemma condexpKernel_apply_eq_condDistrib {ω : Ω} :
    condexpKernel μ m ω = @condDistrib Ω Ω Ω mΩ _ _ mΩ (m ⊓ mΩ) id id μ _ (id ω) 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {ω : Ω}
Goal: (ProbabilityTheory.condexpKernel μ m) ω = (ProbabilityTheory.condDistrib id id μ) (id ω)
-- ProbabilityTheory.measurable_condexpKernel
theorem measurable_condexpKernel {s : Set Ω} (hs : MeasurableSet s) :
    Measurable[m] fun ω => condexpKernel μ m ω s 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s)
Goal: Measurable fun ω => ((ProbabilityTheory.condexpKernel μ m) ω) s
-- ProbabilityTheory.stronglyMeasurable_condexpKernel
theorem stronglyMeasurable_condexpKernel {s : Set Ω} (hs : MeasurableSet s) :
    StronglyMeasurable[m] fun ω => condexpKernel μ m ω s 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s)
Goal: MeasureTheory.StronglyMeasurable fun ω => ((ProbabilityTheory.condexpKernel μ m) ω) s
-- MeasureTheory.AEStronglyMeasurable.integral_condexpKernel
theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_condexpKernel [NormedSpace ℝ F]
    (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun ω => ∫ y, f y ∂condexpKernel μ m ω) μ 
Arguments: {Ω : Type u_1} {F : Type u_2} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [NormedAddCommGroup F] {f : Ω → F} [NormedSpace ℝ F] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω) μ
-- ProbabilityTheory.aestronglyMeasurable'_integral_condexpKernel
theorem aestronglyMeasurable'_integral_condexpKernel [NormedSpace ℝ F]
    (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable' m (fun ω => ∫ y, f y ∂condexpKernel μ m ω) μ 
Arguments: {Ω : Type u_1} {F : Type u_2} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [NormedAddCommGroup F] {f : Ω → F} [NormedSpace ℝ F] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable' m (fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω) μ
-- MeasureTheory.Integrable.condexpKernel_ae
theorem _root_.MeasureTheory.Integrable.condexpKernel_ae (hf_int : Integrable f μ) :
    ∀ᵐ ω ∂μ, Integrable f (condexpKernel μ m ω) 
Arguments: {Ω : Type u_1} {F : Type u_2} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [NormedAddCommGroup F] {f : Ω → F} (hf_int : MeasureTheory.Integrable f μ)
Goal: ∀ᵐ (ω : Ω) ∂μ, MeasureTheory.Integrable f ((ProbabilityTheory.condexpKernel μ m) ω)
-- MeasureTheory.Integrable.integral_norm_condexpKernel
theorem _root_.MeasureTheory.Integrable.integral_norm_condexpKernel (hf_int : Integrable f μ) :
    Integrable (fun ω => ∫ y, ‖f y‖ ∂condexpKernel μ m ω) μ 
Arguments: {Ω : Type u_1} {F : Type u_2} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [NormedAddCommGroup F] {f : Ω → F} (hf_int : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun ω => ∫ (y : Ω), ‖f y‖ ∂(ProbabilityTheory.condexpKernel μ m) ω) μ
-- MeasureTheory.Integrable.norm_integral_condexpKernel
theorem _root_.MeasureTheory.Integrable.norm_integral_condexpKernel [NormedSpace ℝ F]
    (hf_int : Integrable f μ) :
    Integrable (fun ω => ‖∫ y, f y ∂condexpKernel μ m ω‖) μ 
Arguments: {Ω : Type u_1} {F : Type u_2} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [NormedAddCommGroup F] {f : Ω → F} [NormedSpace ℝ F] (hf_int : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun ω => ‖∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω‖) μ
-- MeasureTheory.Integrable.integral_condexpKernel
theorem _root_.MeasureTheory.Integrable.integral_condexpKernel [NormedSpace ℝ F]
    (hf_int : Integrable f μ) :
    Integrable (fun ω => ∫ y, f y ∂condexpKernel μ m ω) μ 
Arguments: {Ω : Type u_1} {F : Type u_2} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [NormedAddCommGroup F] {f : Ω → F} [NormedSpace ℝ F] (hf_int : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.Integrable (fun ω => ∫ (y : Ω), f y ∂(ProbabilityTheory.condexpKernel μ m) ω) μ
-- ProbabilityTheory.integrable_toReal_condexpKernel
theorem integrable_toReal_condexpKernel {s : Set Ω} (hs : MeasurableSet s) :
    Integrable (fun ω => (condexpKernel μ m ω s).toReal) μ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s)
Goal: MeasureTheory.Integrable (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) μ
-- ProbabilityTheory.condexpKernel_ae_eq_condexp'
lemma condexpKernel_ae_eq_condexp' [IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s) :
    (fun ω ↦ (condexpKernel μ m ω s).toReal) =ᵐ[μ] μ⟦s | m ⊓ mΩ⟧ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s)
Goal: (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω => 1|m ⊓ mΩ]
-- ProbabilityTheory.condexpKernel_ae_eq_condexp
lemma condexpKernel_ae_eq_condexp [IsFiniteMeasure μ]
    (hm : m ≤ mΩ) {s : Set Ω} (hs : MeasurableSet s) :
    (fun ω ↦ (condexpKernel μ m ω s).toReal) =ᵐ[μ] μ⟦s | m⟧ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hm : m ≤ mΩ) {s : Set Ω} (hs : MeasurableSet s)
Goal: (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) =ᵐ[μ] μ[s.indicator fun ω => 1|m]
-- ProbabilityTheory.condexpKernel_ae_eq_trim_condexp
lemma condexpKernel_ae_eq_trim_condexp [IsFiniteMeasure μ]
    (hm : m ≤ mΩ) {s : Set Ω} (hs : MeasurableSet s) :
    (fun ω ↦ (condexpKernel μ m ω s).toReal) =ᵐ[μ.trim hm] μ⟦s | m⟧ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hm : m ≤ mΩ) {s : Set Ω} (hs : MeasurableSet s)
Goal: (fun ω => (((ProbabilityTheory.condexpKernel μ m) ω) s).toReal) =ᵐ[μ.trim hm] μ[s.indicator fun ω => 1|m]