File: Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean

full_name: ProbabilityTheory.IsRatStieltjesPoint
code: /-- `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. -/
structure IsRatStieltjesPoint (f : α → ℚ → ℝ) (a : α) : Prop where
  mono : Monotone (f a)
  tendsto_atTop_one : Tendsto (f a) atTop (𝓝 1)
  tendsto_atBot_zero : Tendsto (f a) atBot (𝓝 0)
  iInf_rat_gt_eq : ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t
goal: Prop

full_name: ProbabilityTheory.IsRatStieltjesPoint.iInf_rat_gt_eq
code: /-- `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. -/
structure IsRatStieltjesPoint (f : α → ℚ → ℝ) (a : α) : Prop where
  mono : Monotone (f a)
  tendsto_atTop_one : Tendsto (f a) atTop (𝓝 1)
  tendsto_atBot_zero : Tendsto (f a) atBot (𝓝 0)
  iInf_rat_gt_eq : ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t
goal: ⨅ r, f a ↑r = f a t

full_name: ProbabilityTheory.IsRatStieltjesPoint.tendsto_atBot_zero
code: /-- `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. -/
structure IsRatStieltjesPoint (f : α → ℚ → ℝ) (a : α) : Prop where
  mono : Monotone (f a)
  tendsto_atTop_one : Tendsto (f a) atTop (𝓝 1)
  tendsto_atBot_zero : Tendsto (f a) atBot (𝓝 0)
  iInf_rat_gt_eq : ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t
goal: Filter.Tendsto (f a) Filter.atBot (𝓝 0)

full_name: ProbabilityTheory.IsRatStieltjesPoint.tendsto_atTop_one
code: /-- `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. -/
structure IsRatStieltjesPoint (f : α → ℚ → ℝ) (a : α) : Prop where
  mono : Monotone (f a)
  tendsto_atTop_one : Tendsto (f a) atTop (𝓝 1)
  tendsto_atBot_zero : Tendsto (f a) atBot (𝓝 0)
  iInf_rat_gt_eq : ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t
goal: Filter.Tendsto (f a) Filter.atTop (𝓝 1)

full_name: ProbabilityTheory.IsRatStieltjesPoint.mono
code: /-- `a : α` is a Stieltjes point for `f : α → ℚ → ℝ` if `f a` is monotone with limit 0 at -∞
and 1 at +∞ and satisfies a continuity property. -/
structure IsRatStieltjesPoint (f : α → ℚ → ℝ) (a : α) : Prop where
  mono : Monotone (f a)
  tendsto_atTop_one : Tendsto (f a) atTop (𝓝 1)
  tendsto_atBot_zero : Tendsto (f a) atBot (𝓝 0)
  iInf_rat_gt_eq : ∀ t : ℚ, ⨅ r : Ioi t, f a r = f a t
goal: Monotone (f a)

full_name: ProbabilityTheory.isRatStieltjesPoint_unit_prod_iff
code: lemma isRatStieltjesPoint_unit_prod_iff (f : α → ℚ → ℝ) (a : α) :
    IsRatStieltjesPoint (fun p : Unit × α ↦ f p.2) ((), a)
      ↔ IsRatStieltjesPoint f a 
goal: ProbabilityTheory.IsRatStieltjesPoint (fun p => f p.2) ((), a) ↔ ProbabilityTheory.IsRatStieltjesPoint f a

full_name: ProbabilityTheory.measurableSet_isRatStieltjesPoint
code: lemma measurableSet_isRatStieltjesPoint (hf : Measurable f) :
    MeasurableSet {a | IsRatStieltjesPoint f a} 
goal: MeasurableSet {a | ProbabilityTheory.IsRatStieltjesPoint f a}

full_name: ProbabilityTheory.IsRatStieltjesPoint.ite
code: lemma IsRatStieltjesPoint.ite {f g : α → ℚ → ℝ} {a : α} (p : α → Prop) [DecidablePred p]
    (hf : p a → IsRatStieltjesPoint f a) (hg : ¬ p a → IsRatStieltjesPoint g a) :
    IsRatStieltjesPoint (fun a ↦ if p a then f a else g a) a 
goal: ProbabilityTheory.IsRatStieltjesPoint (fun a => if p a then f a else g a) a

full_name: ProbabilityTheory.IsMeasurableRatCDF.measurable
code: /-- A function `f : α → ℚ → ℝ` is a (kernel) rational cumulative distribution function if it is
measurable in the first argument and if `f a` satisfies a list of properties for all `a : α`:
monotonicity between 0 at -∞ and 1 at +∞ and a form of continuity.

A function with these properties can be extended to a measurable function `α → StieltjesFunction`.
See `ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction`.
-/
structure IsMeasurableRatCDF (f : α → ℚ → ℝ) : Prop where
  isRatStieltjesPoint : ∀ a, IsRatStieltjesPoint f a
  measurable : Measurable f
goal: Measurable f

full_name: ProbabilityTheory.IsMeasurableRatCDF.isRatStieltjesPoint
code: /-- A function `f : α → ℚ → ℝ` is a (kernel) rational cumulative distribution function if it is
measurable in the first argument and if `f a` satisfies a list of properties for all `a : α`:
monotonicity between 0 at -∞ and 1 at +∞ and a form of continuity.

A function with these properties can be extended to a measurable function `α → StieltjesFunction`.
See `ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction`.
-/
structure IsMeasurableRatCDF (f : α → ℚ → ℝ) : Prop where
  isRatStieltjesPoint : ∀ a, IsRatStieltjesPoint f a
  measurable : Measurable f
goal: ProbabilityTheory.IsRatStieltjesPoint f a

full_name: ProbabilityTheory.IsMeasurableRatCDF
code: /-- A function `f : α → ℚ → ℝ` is a (kernel) rational cumulative distribution function if it is
measurable in the first argument and if `f a` satisfies a list of properties for all `a : α`:
monotonicity between 0 at -∞ and 1 at +∞ and a form of continuity.

A function with these properties can be extended to a measurable function `α → StieltjesFunction`.
See `ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction`.
-/
structure IsMeasurableRatCDF (f : α → ℚ → ℝ) : Prop where
  isRatStieltjesPoint : ∀ a, IsRatStieltjesPoint f a
  measurable : Measurable f
goal: Prop

full_name: ProbabilityTheory.IsMeasurableRatCDF.nonneg
code: lemma IsMeasurableRatCDF.nonneg {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) (q : ℚ) :
    0 ≤ f a q 
goal: 0 ≤ f a q

full_name: ProbabilityTheory.IsMeasurableRatCDF.le_one
code: lemma IsMeasurableRatCDF.le_one {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) (q : ℚ) :
    f a q ≤ 1 
goal: f a q ≤ 1

full_name: ProbabilityTheory.IsMeasurableRatCDF.tendsto_atTop_one
code: lemma IsMeasurableRatCDF.tendsto_atTop_one {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) :
    Tendsto (f a) atTop (𝓝 1) 
goal: Filter.Tendsto (f a) Filter.atTop (𝓝 1)

full_name: ProbabilityTheory.IsMeasurableRatCDF.tendsto_atBot_zero
code: lemma IsMeasurableRatCDF.tendsto_atBot_zero {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α) :
    Tendsto (f a) atBot (𝓝 0) 
goal: Filter.Tendsto (f a) Filter.atBot (𝓝 0)

full_name: ProbabilityTheory.IsMeasurableRatCDF.iInf_rat_gt_eq
code: lemma IsMeasurableRatCDF.iInf_rat_gt_eq {f : α → ℚ → ℝ} (hf : IsMeasurableRatCDF f) (a : α)
    (q : ℚ) :
    ⨅ r : Ioi q, f a r = f a q 
goal: ⨅ r, f a ↑r = f a q

full_name: ProbabilityTheory.defaultRatCDF
code: /-- A function with the property `IsMeasurableRatCDF`.
Used in a piecewise construction to convert a function which only satisfies the properties
defining `IsMeasurableRatCDF` on some set into a true `IsMeasurableRatCDF`. -/
def defaultRatCDF (q : ℚ) 
goal: ℝ

full_name: ProbabilityTheory.monotone_defaultRatCDF
code: lemma monotone_defaultRatCDF : Monotone defaultRatCDF 
goal: Monotone ProbabilityTheory.defaultRatCDF

full_name: ProbabilityTheory.defaultRatCDF_nonneg
code: lemma defaultRatCDF_nonneg (q : ℚ) : 0 ≤ defaultRatCDF q 
goal: 0 ≤ ProbabilityTheory.defaultRatCDF q

full_name: ProbabilityTheory.defaultRatCDF_le_one
code: lemma defaultRatCDF_le_one (q : ℚ) : defaultRatCDF q ≤ 1 
goal: ProbabilityTheory.defaultRatCDF q ≤ 1

full_name: ProbabilityTheory.tendsto_defaultRatCDF_atTop
code: lemma tendsto_defaultRatCDF_atTop : Tendsto defaultRatCDF atTop (𝓝 1) 
goal: Filter.Tendsto ProbabilityTheory.defaultRatCDF Filter.atTop (𝓝 1)

full_name: ProbabilityTheory.tendsto_defaultRatCDF_atBot
code: lemma tendsto_defaultRatCDF_atBot : Tendsto defaultRatCDF atBot (𝓝 0) 
goal: Filter.Tendsto ProbabilityTheory.defaultRatCDF Filter.atBot (𝓝 0)

full_name: ProbabilityTheory.iInf_rat_gt_defaultRatCDF
code: lemma iInf_rat_gt_defaultRatCDF (t : ℚ) :
    ⨅ r : Ioi t, defaultRatCDF r = defaultRatCDF t 
goal: ⨅ r, ProbabilityTheory.defaultRatCDF ↑r = ProbabilityTheory.defaultRatCDF t

full_name: ProbabilityTheory.isRatStieltjesPoint_defaultRatCDF
code: lemma isRatStieltjesPoint_defaultRatCDF (a : α) :
    IsRatStieltjesPoint (fun (_ : α) ↦ defaultRatCDF) a 
goal: ProbabilityTheory.IsRatStieltjesPoint (fun x => ProbabilityTheory.defaultRatCDF) a

full_name: ProbabilityTheory.IsMeasurableRatCDF_defaultRatCDF
code: lemma IsMeasurableRatCDF_defaultRatCDF (α : Type*) [MeasurableSpace α] :
    IsMeasurableRatCDF (fun (_ : α) (q : ℚ) ↦ defaultRatCDF q) 
goal: ProbabilityTheory.IsMeasurableRatCDF fun x q => ProbabilityTheory.defaultRatCDF q

full_name: ProbabilityTheory.toRatCDF
code: /-- Turn a function `f : α → ℚ → ℝ` into another with the property `IsRatStieltjesPoint f a`
everywhere. At `a` that does not satisfy that property, `f a` is replaced by an arbitrary suitable
function.
Mainly useful when `f` satisfies the property `IsRatStieltjesPoint f a` almost everywhere with
respect to some measure. -/
noncomputable
def toRatCDF (f : α → ℚ → ℝ) : α → ℚ → ℝ 
goal: α → ℚ → ℝ

full_name: ProbabilityTheory.toRatCDF_of_isRatStieltjesPoint
code: lemma toRatCDF_of_isRatStieltjesPoint {a : α} (h : IsRatStieltjesPoint f a) (q : ℚ) :
    toRatCDF f a q = f a q 
goal: ProbabilityTheory.toRatCDF f a q = f a q

full_name: ProbabilityTheory.measurable_toRatCDF
code: lemma measurable_toRatCDF (hf : Measurable f) : Measurable (toRatCDF f) 
goal: Measurable (ProbabilityTheory.toRatCDF f)

full_name: ProbabilityTheory.isMeasurableRatCDF_toRatCDF
code: lemma isMeasurableRatCDF_toRatCDF (hf : Measurable f) :
    IsMeasurableRatCDF (toRatCDF f) 
goal: ProbabilityTheory.IsMeasurableRatCDF (ProbabilityTheory.toRatCDF f)

full_name: ProbabilityTheory.toRatCDF_unit_prod
code: lemma toRatCDF_unit_prod (a : α) :
    toRatCDF (fun (p : Unit × α) ↦ f p.2) ((), a) = toRatCDF f a 
goal: ProbabilityTheory.toRatCDF (fun p => f p.2) ((), a) = ProbabilityTheory.toRatCDF f a

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux
code: /-- Auxiliary definition for `IsMeasurableRatCDF.stieltjesFunction`: turn `f : α → ℚ → ℝ` into
a function `α → ℝ → ℝ` by assigning to `f a x` the infimum of `f a q` over `q : ℚ` with `x < q`. -/
noncomputable irreducible_def IsMeasurableRatCDF.stieltjesFunctionAux (f : α → ℚ → ℝ) :
    α → ℝ → ℝ 
goal: α → ℝ → ℝ

full_name: ProbabilityTheory.definition._@.Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes._hyg.2681
code: /-- Auxiliary definition for `IsMeasurableRatCDF.stieltjesFunction`: turn `f : α → ℚ → ℝ` into
a function `α → ℝ → ℝ` by assigning to `f a x` the infimum of `f a q` over `q : ℚ` with `x < q`. -/
noncomputable irreducible_def IsMeasurableRatCDF.stieltjesFunctionAux (f : α → ℚ → ℝ) :
    α → ℝ → ℝ 
goal: α → ℝ → ℝ

full_name: ProbabilityTheory.wrapped._@.Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes._hyg.2681
code: /-- Auxiliary definition for `IsMeasurableRatCDF.stieltjesFunction`: turn `f : α → ℚ → ℝ` into
a function `α → ℝ → ℝ` by assigning to `f a x` the infimum of `f a q` over `q : ℚ` with `x < q`. -/
noncomputable irreducible_def IsMeasurableRatCDF.stieltjesFunctionAux (f : α → ℚ → ℝ) :
    α → ℝ → ℝ 
goal: Subtype (Eq @ProbabilityTheory.definition✝)

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def
code: /-- Auxiliary definition for `IsMeasurableRatCDF.stieltjesFunction`: turn `f : α → ℚ → ℝ` into
a function `α → ℝ → ℝ` by assigning to `f a x` the infimum of `f a q` over `q : ℚ` with `x < q`. -/
noncomputable irreducible_def IsMeasurableRatCDF.stieltjesFunctionAux (f : α → ℚ → ℝ) :
    α → ℝ → ℝ 
goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a x = ⨅ q, f a ↑q

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def'
code: lemma IsMeasurableRatCDF.stieltjesFunctionAux_def' (f : α → ℚ → ℝ) (a : α) :
    IsMeasurableRatCDF.stieltjesFunctionAux f a
      = fun (t : ℝ) ↦ ⨅ r : { r' : ℚ // t < r' }, f a r 
goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a = fun t => ⨅ r, f a ↑r

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod
code: lemma IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod {f : α → ℚ → ℝ} (a : α) :
    IsMeasurableRatCDF.stieltjesFunctionAux (fun (p : Unit × α) ↦ f p.2) ((), a)
      = IsMeasurableRatCDF.stieltjesFunctionAux f a 
goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux (fun p => f p.2) ((), a) =
  ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_eq
code: lemma IsMeasurableRatCDF.stieltjesFunctionAux_eq (a : α) (r : ℚ) :
    IsMeasurableRatCDF.stieltjesFunctionAux f a r = f a r 
goal: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a ↑r = f a r

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_nonneg
code: lemma IsMeasurableRatCDF.stieltjesFunctionAux_nonneg (a : α) (r : ℝ) :
    0 ≤ IsMeasurableRatCDF.stieltjesFunctionAux f a r 
goal: 0 ≤ ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a r

full_name: ProbabilityTheory.IsMeasurableRatCDF.monotone_stieltjesFunctionAux
code: lemma IsMeasurableRatCDF.monotone_stieltjesFunctionAux (a : α) :
    Monotone (IsMeasurableRatCDF.stieltjesFunctionAux f a) 
goal: Monotone (ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a)

full_name: ProbabilityTheory.IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici
code: lemma IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici (a : α) (x : ℝ) :
    ContinuousWithinAt (IsMeasurableRatCDF.stieltjesFunctionAux f a) (Ici x) x 
goal: ContinuousWithinAt (ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux f a) (Set.Ici x) x

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction
code: /-- Extend a function `f : α → ℚ → ℝ` with property `IsMeasurableRatCDF` from `ℚ` to `ℝ`,
to a function `α → StieltjesFunction`. -/
noncomputable def IsMeasurableRatCDF.stieltjesFunction (a : α) : StieltjesFunction 
goal: StieltjesFunction

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_eq
code: lemma IsMeasurableRatCDF.stieltjesFunction_eq (a : α) (r : ℚ) : hf.stieltjesFunction a r = f a r 
goal: ↑(hf.stieltjesFunction a) ↑r = f a r

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_nonneg
code: lemma IsMeasurableRatCDF.stieltjesFunction_nonneg (a : α) (r : ℝ) : 0 ≤ hf.stieltjesFunction a r 
goal: 0 ≤ ↑(hf.stieltjesFunction a) r

full_name: ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_le_one
code: lemma IsMeasurableRatCDF.stieltjesFunction_le_one (a : α) (x : ℝ) :
    hf.stieltjesFunction a x ≤ 1 
goal: ↑(hf.stieltjesFunction a) x ≤ 1

full_name: ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot
code: lemma IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot (a : α) :
    Tendsto (hf.stieltjesFunction a) atBot (𝓝 0) 
goal: Filter.Tendsto (↑(hf.stieltjesFunction a)) Filter.atBot (𝓝 0)

full_name: ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop
code: lemma IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop (a : α) :
    Tendsto (hf.stieltjesFunction a) atTop (𝓝 1) 
goal: Filter.Tendsto (↑(hf.stieltjesFunction a)) Filter.atTop (𝓝 1)

full_name: ProbabilityTheory.IsMeasurableRatCDF.measurable_stieltjesFunction
code: lemma IsMeasurableRatCDF.measurable_stieltjesFunction (x : ℝ) :
    Measurable fun a ↦ hf.stieltjesFunction a x 
goal: Measurable fun a => ↑(hf.stieltjesFunction a) x

full_name: ProbabilityTheory.IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction
code: lemma IsMeasurableRatCDF.stronglyMeasurable_stieltjesFunction (x : ℝ) :
    StronglyMeasurable fun a ↦ hf.stieltjesFunction a x 
goal: MeasureTheory.StronglyMeasurable fun a => ↑(hf.stieltjesFunction a) x

full_name: ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_Iic
code: lemma IsMeasurableRatCDF.measure_stieltjesFunction_Iic (a : α) (x : ℝ) :
    (hf.stieltjesFunction a).measure (Iic x) = ENNReal.ofReal (hf.stieltjesFunction a x) 
goal: (hf.stieltjesFunction a).measure (Set.Iic x) = ENNReal.ofReal (↑(hf.stieltjesFunction a) x)

full_name: ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_univ
code: lemma IsMeasurableRatCDF.measure_stieltjesFunction_univ (a : α) :
    (hf.stieltjesFunction a).measure univ = 1 
goal: (hf.stieltjesFunction a).measure Set.univ = 1

full_name: ProbabilityTheory.IsMeasurableRatCDF.instIsProbabilityMeasure_stieltjesFunction
code: instance IsMeasurableRatCDF.instIsProbabilityMeasure_stieltjesFunction (a : α) :
    IsProbabilityMeasure (hf.stieltjesFunction a).measure 
goal: MeasureTheory.IsProbabilityMeasure (hf.stieltjesFunction a).measure

full_name: ProbabilityTheory.IsMeasurableRatCDF.measurable_measure_stieltjesFunction
code: lemma IsMeasurableRatCDF.measurable_measure_stieltjesFunction :
    Measurable fun a ↦ (hf.stieltjesFunction a).measure 
goal: Measurable fun a => (hf.stieltjesFunction a).measure

full_name: ProbabilityTheory.stieltjesOfMeasurableRat
code: /-- Turn a measurable function `f : α → ℚ → ℝ` into a measurable function `α → StieltjesFunction`.
Composition of `toRatCDF` and `IsMeasurableRatCDF.stieltjesFunction`. -/
noncomputable
def stieltjesOfMeasurableRat (f : α → ℚ → ℝ) (hf : Measurable f) : α → StieltjesFunction 
goal: α → StieltjesFunction

full_name: ProbabilityTheory.stieltjesOfMeasurableRat_eq
code: lemma stieltjesOfMeasurableRat_eq (hf : Measurable f) (a : α) (r : ℚ) :
    stieltjesOfMeasurableRat f hf a r = toRatCDF f a r 
goal: ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) ↑r = ProbabilityTheory.toRatCDF f a r

full_name: ProbabilityTheory.stieltjesOfMeasurableRat_unit_prod
code: lemma stieltjesOfMeasurableRat_unit_prod (hf : Measurable f) (a : α) :
    stieltjesOfMeasurableRat (fun (p : Unit × α) ↦ f p.2) (hf.comp measurable_snd) ((), a)
      = stieltjesOfMeasurableRat f hf a 
goal: ProbabilityTheory.stieltjesOfMeasurableRat (fun p => f p.2) ⋯ ((), a) =
  ProbabilityTheory.stieltjesOfMeasurableRat f hf a

full_name: ProbabilityTheory.stieltjesOfMeasurableRat_nonneg
code: lemma stieltjesOfMeasurableRat_nonneg (hf : Measurable f) (a : α) (r : ℝ) :
    0 ≤ stieltjesOfMeasurableRat f hf a r 
goal: 0 ≤ ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) r

full_name: ProbabilityTheory.stieltjesOfMeasurableRat_le_one
code: lemma stieltjesOfMeasurableRat_le_one (hf : Measurable f) (a : α) (x : ℝ) :
    stieltjesOfMeasurableRat f hf a x ≤ 1 
goal: ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x ≤ 1

full_name: ProbabilityTheory.tendsto_stieltjesOfMeasurableRat_atBot
code: lemma tendsto_stieltjesOfMeasurableRat_atBot (hf : Measurable f) (a : α) :
    Tendsto (stieltjesOfMeasurableRat f hf a) atBot (𝓝 0) 
goal: Filter.Tendsto (↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a)) Filter.atBot (𝓝 0)

full_name: ProbabilityTheory.tendsto_stieltjesOfMeasurableRat_atTop
code: lemma tendsto_stieltjesOfMeasurableRat_atTop (hf : Measurable f) (a : α) :
    Tendsto (stieltjesOfMeasurableRat f hf a) atTop (𝓝 1) 
goal: Filter.Tendsto (↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a)) Filter.atTop (𝓝 1)

full_name: ProbabilityTheory.measurable_stieltjesOfMeasurableRat
code: lemma measurable_stieltjesOfMeasurableRat (hf : Measurable f) (x : ℝ) :
    Measurable fun a ↦ stieltjesOfMeasurableRat f hf a x 
goal: Measurable fun a => ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x

full_name: ProbabilityTheory.stronglyMeasurable_stieltjesOfMeasurableRat
code: lemma stronglyMeasurable_stieltjesOfMeasurableRat (hf : Measurable f) (x : ℝ) :
    StronglyMeasurable fun a ↦ stieltjesOfMeasurableRat f hf a x 
goal: MeasureTheory.StronglyMeasurable fun a => ↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x

full_name: ProbabilityTheory.measure_stieltjesOfMeasurableRat_Iic
code: lemma measure_stieltjesOfMeasurableRat_Iic (hf : Measurable f) (a : α) (x : ℝ) :
    (stieltjesOfMeasurableRat f hf a).measure (Iic x)
      = ENNReal.ofReal (stieltjesOfMeasurableRat f hf a x) 
goal: (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure (Set.Iic x) =
  ENNReal.ofReal (↑(ProbabilityTheory.stieltjesOfMeasurableRat f hf a) x)

full_name: ProbabilityTheory.measure_stieltjesOfMeasurableRat_univ
code: lemma measure_stieltjesOfMeasurableRat_univ (hf : Measurable f) (a : α) :
    (stieltjesOfMeasurableRat f hf a).measure univ = 1 
goal: (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure Set.univ = 1

full_name: ProbabilityTheory.instIsProbabilityMeasure_stieltjesOfMeasurableRat
code: instance instIsProbabilityMeasure_stieltjesOfMeasurableRat
    (hf : Measurable f) (a : α) :
    IsProbabilityMeasure (stieltjesOfMeasurableRat f hf a).measure 
goal: MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure

full_name: ProbabilityTheory.measurable_measure_stieltjesOfMeasurableRat
code: lemma measurable_measure_stieltjesOfMeasurableRat (hf : Measurable f) :
    Measurable fun a ↦ (stieltjesOfMeasurableRat f hf a).measure 
goal: Measurable fun a => (ProbabilityTheory.stieltjesOfMeasurableRat f hf a).measure