File: Mathlib/MeasureTheory/Measure/Stieltjes.lean

full_name: StieltjesFunction.mono'
code: /-- Bundled monotone right-continuous real functions, used to construct Stieltjes measures. -/
structure StieltjesFunction where
  toFun : ℝ → ℝ
  mono' : Monotone toFun
  right_continuous' : ∀ x, ContinuousWithinAt toFun (Ici x) x
goal: Monotone ↑self

full_name: StieltjesFunction.toFun
code: /-- Bundled monotone right-continuous real functions, used to construct Stieltjes measures. -/
structure StieltjesFunction where
  toFun : ℝ → ℝ
  mono' : Monotone toFun
  right_continuous' : ∀ x, ContinuousWithinAt toFun (Ici x) x
goal: ℝ → ℝ

full_name: StieltjesFunction.right_continuous'
code: /-- Bundled monotone right-continuous real functions, used to construct Stieltjes measures. -/
structure StieltjesFunction where
  toFun : ℝ → ℝ
  mono' : Monotone toFun
  right_continuous' : ∀ x, ContinuousWithinAt toFun (Ici x) x
goal: ContinuousWithinAt (↑self) (Set.Ici x) x

full_name: StieltjesFunction
code: /-- Bundled monotone right-continuous real functions, used to construct Stieltjes measures. -/
structure StieltjesFunction where
  toFun : ℝ → ℝ
  mono' : Monotone toFun
  right_continuous' : ∀ x, ContinuousWithinAt toFun (Ici x) x
goal: Type

full_name: StieltjesFunction.instCoeFun
code: instance instCoeFun : CoeFun StieltjesFunction fun _ => ℝ → ℝ 
goal: CoeFun StieltjesFunction fun x => ℝ → ℝ

full_name: StieltjesFunction.ext
code: @[ext] lemma ext {f g : StieltjesFunction} (h : ∀ x, f x = g x) : f = g 
goal: f = g

full_name: StieltjesFunction.mono
code: theorem mono : Monotone f 
goal: Monotone ↑f

full_name: StieltjesFunction.right_continuous
code: theorem right_continuous (x : ℝ) : ContinuousWithinAt f (Ici x) x 
goal: ContinuousWithinAt (↑f) (Set.Ici x) x

full_name: StieltjesFunction.rightLim_eq
code: theorem rightLim_eq (f : StieltjesFunction) (x : ℝ) : Function.rightLim f x = f x 
goal: Function.rightLim (↑f) x = ↑f x

full_name: StieltjesFunction.iInf_Ioi_eq
code: theorem iInf_Ioi_eq (f : StieltjesFunction) (x : ℝ) : ⨅ r : Ioi x, f r = f x 
goal: ⨅ r, ↑f ↑r = ↑f x

full_name: StieltjesFunction.iInf_rat_gt_eq
code: theorem iInf_rat_gt_eq (f : StieltjesFunction) (x : ℝ) :
    ⨅ r : { r' : ℚ // x < r' }, f r = f x 
goal: ⨅ r, ↑f ↑↑r = ↑f x

full_name: StieltjesFunction.id_apply
code: /-- The identity of `ℝ` as a Stieltjes function, used to construct Lebesgue measure. -/
@[simps]
protected def id : StieltjesFunction 
goal: ↑StieltjesFunction.id a = id a

full_name: StieltjesFunction.id
code: /-- The identity of `ℝ` as a Stieltjes function, used to construct Lebesgue measure. -/
@[simps]
protected def id : StieltjesFunction 
goal: StieltjesFunction

full_name: StieltjesFunction.id_leftLim
code: @[simp]
theorem id_leftLim (x : ℝ) : leftLim StieltjesFunction.id x = x 
goal: Function.leftLim (↑StieltjesFunction.id) x = x

full_name: StieltjesFunction.instInhabited
code: instance instInhabited : Inhabited StieltjesFunction 
goal: Inhabited StieltjesFunction

full_name: StieltjesFunction.const
code: /-- Constant functions are Stieltjes function. -/
protected def const (c : ℝ) : StieltjesFunction 
goal: StieltjesFunction

full_name: StieltjesFunction.const_apply
code: @[simp] lemma const_apply (c x : ℝ) : (StieltjesFunction.const c) x = c 
goal: ↑(StieltjesFunction.const c) x = c

full_name: StieltjesFunction.add
code: /-- The sum of two Stieltjes functions is a Stieltjes function. -/
protected def add (f g : StieltjesFunction) : StieltjesFunction 
goal: StieltjesFunction

full_name: StieltjesFunction.instAddZeroClass
code: instance : AddZeroClass StieltjesFunction 
goal: AddZeroClass StieltjesFunction

full_name: StieltjesFunction.instAddCommMonoid
code: instance : AddCommMonoid StieltjesFunction 
goal: AddCommMonoid StieltjesFunction

full_name: StieltjesFunction.instModuleNNReal
code: instance : Module ℝ≥0 StieltjesFunction 
goal: Module ℝ≥0 StieltjesFunction

full_name: StieltjesFunction.add_apply
code: @[simp] lemma add_apply (f g : StieltjesFunction) (x : ℝ) : (f + g) x = f x + g x 
goal: ↑(f + g) x = ↑f x + ↑g x

full_name: Monotone.stieltjesFunction
code: /-- If a function `f : ℝ → ℝ` is monotone, then the function mapping `x` to the right limit of `f`
at `x` is a Stieltjes function, i.e., it is monotone and right-continuous. -/
noncomputable def _root_.Monotone.stieltjesFunction {f : ℝ → ℝ} (hf : Monotone f) :
    StieltjesFunction 
goal: StieltjesFunction

full_name: Monotone.stieltjesFunction_eq
code: theorem _root_.Monotone.stieltjesFunction_eq {f : ℝ → ℝ} (hf : Monotone f) (x : ℝ) :
    hf.stieltjesFunction x = rightLim f x 
goal: ↑hf.stieltjesFunction x = Function.rightLim f x

full_name: StieltjesFunction.countable_leftLim_ne
code: theorem countable_leftLim_ne (f : StieltjesFunction) : Set.Countable { x | leftLim f x ≠ f x } 
goal: {x | Function.leftLim (↑f) x ≠ ↑f x}.Countable

full_name: StieltjesFunction.length
code: /-- Length of an interval. This is the largest monotone function which correctly measures all
intervals. -/
def length (s : Set ℝ) : ℝ≥0∞ 
goal: ℝ≥0∞

full_name: StieltjesFunction.length_empty
code: @[simp]
theorem length_empty : f.length ∅ = 0 
goal: f.length ∅ = 0

full_name: StieltjesFunction.length_Ioc
code: @[simp]
theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) 
goal: f.length (Set.Ioc a b) = ENNReal.ofReal (↑f b - ↑f a)

full_name: StieltjesFunction.length_mono
code: theorem length_mono {s₁ s₂ : Set ℝ} (h : s₁ ⊆ s₂) : f.length s₁ ≤ f.length s₂ 
goal: f.length s₁ ≤ f.length s₂

full_name: StieltjesFunction.outer
code: /-- The Stieltjes outer measure associated to a Stieltjes function. -/
protected def outer : OuterMeasure ℝ 
goal: MeasureTheory.OuterMeasure ℝ

full_name: StieltjesFunction.outer_le_length
code: theorem outer_le_length (s : Set ℝ) : f.outer s ≤ f.length s 
goal: f.outer s ≤ f.length s

full_name: StieltjesFunction.length_subadditive_Icc_Ioo
code: /-- If a compact interval `[a, b]` is covered by a union of open interval `(c i, d i)`, then
`f b - f a ≤ ∑ f (d i) - f (c i)`. This is an auxiliary technical statement to prove the same
statement for half-open intervals, the point of the current statement being that one can use
compactness to reduce it to a finite sum, and argue by induction on the size of the covering set. -/
theorem length_subadditive_Icc_Ioo {a b : ℝ} {c d : ℕ → ℝ} (ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i)) :
    ofReal (f b - f a) ≤ ∑' i, ofReal (f (d i) - f (c i)) 
goal: ENNReal.ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ENNReal.ofReal (↑f (d i) - ↑f (c i))

full_name: StieltjesFunction.outer_Ioc
code: @[simp]
theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) 
goal: f.outer (Set.Ioc a b) = ENNReal.ofReal (↑f b - ↑f a)

full_name: StieltjesFunction.measurableSet_Ioi
code: theorem measurableSet_Ioi {c : ℝ} : MeasurableSet[f.outer.caratheodory] (Ioi c) 
goal: MeasurableSet (Set.Ioi c)

full_name: StieltjesFunction.outer_trim
code: theorem outer_trim : f.outer.trim = f.outer 
goal: f.outer.trim = f.outer

full_name: StieltjesFunction.borel_le_measurable
code: theorem borel_le_measurable : borel ℝ ≤ f.outer.caratheodory 
goal: borel ℝ ≤ f.outer.caratheodory

full_name: StieltjesFunction.measure
code: /-- The measure associated to a Stieltjes function, giving mass `f b - f a` to the
interval `(a, b]`. -/
protected irreducible_def measure : Measure ℝ 
goal: MeasureTheory.Measure ℝ

full_name: StieltjesFunction.wrapped._@.Mathlib.MeasureTheory.Measure.Stieltjes._hyg.3843
code: /-- The measure associated to a Stieltjes function, giving mass `f b - f a` to the
interval `(a, b]`. -/
protected irreducible_def measure : Measure ℝ 
goal: Subtype (Eq StieltjesFunction.definition✝)

full_name: StieltjesFunction.definition._@.Mathlib.MeasureTheory.Measure.Stieltjes._hyg.3843
code: /-- The measure associated to a Stieltjes function, giving mass `f b - f a` to the
interval `(a, b]`. -/
protected irreducible_def measure : Measure ℝ 
goal: MeasureTheory.Measure ℝ

full_name: StieltjesFunction.measure_def
code: /-- The measure associated to a Stieltjes function, giving mass `f b - f a` to the
interval `(a, b]`. -/
protected irreducible_def measure : Measure ℝ 
goal: f.measure = { toOuterMeasure := f.outer, m_iUnion := ⋯, trim_le := ⋯ }

full_name: StieltjesFunction.measure_Ioc
code: @[simp]
theorem measure_Ioc (a b : ℝ) : f.measure (Ioc a b) = ofReal (f b - f a) 
goal: f.measure (Set.Ioc a b) = ENNReal.ofReal (↑f b - ↑f a)

full_name: StieltjesFunction.measure_singleton
code: @[simp]
theorem measure_singleton (a : ℝ) : f.measure {a} = ofReal (f a - leftLim f a) 
goal: f.measure {a} = ENNReal.ofReal (↑f a - Function.leftLim (↑f) a)

full_name: StieltjesFunction.measure_Icc
code: @[simp]
theorem measure_Icc (a b : ℝ) : f.measure (Icc a b) = ofReal (f b - leftLim f a) 
goal: f.measure (Set.Icc a b) = ENNReal.ofReal (↑f b - Function.leftLim (↑f) a)

full_name: StieltjesFunction.measure_Ioo
code: @[simp]
theorem measure_Ioo {a b : ℝ} : f.measure (Ioo a b) = ofReal (leftLim f b - f a) 
goal: f.measure (Set.Ioo a b) = ENNReal.ofReal (Function.leftLim (↑f) b - ↑f a)

full_name: StieltjesFunction.measure_Ico
code: @[simp]
theorem measure_Ico (a b : ℝ) : f.measure (Ico a b) = ofReal (leftLim f b - leftLim f a) 
goal: f.measure (Set.Ico a b) = ENNReal.ofReal (Function.leftLim (↑f) b - Function.leftLim (↑f) a)

full_name: StieltjesFunction.measure_Iic
code: theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) :
    f.measure (Iic x) = ofReal (f x - l) 
goal: f.measure (Set.Iic x) = ENNReal.ofReal (↑f x - l)

full_name: StieltjesFunction.measure_Ici
code: theorem measure_Ici {l : ℝ} (hf : Tendsto f atTop (𝓝 l)) (x : ℝ) :
    f.measure (Ici x) = ofReal (l - leftLim f x) 
goal: f.measure (Set.Ici x) = ENNReal.ofReal (l - Function.leftLim (↑f) x)

full_name: StieltjesFunction.measure_univ
code: theorem measure_univ {l u : ℝ} (hfl : Tendsto f atBot (𝓝 l)) (hfu : Tendsto f atTop (𝓝 u)) :
    f.measure univ = ofReal (u - l) 
goal: f.measure Set.univ = ENNReal.ofReal (u - l)

full_name: StieltjesFunction.isFiniteMeasure
code: lemma isFiniteMeasure {l u : ℝ} (hfl : Tendsto f atBot (𝓝 l)) (hfu : Tendsto f atTop (𝓝 u)) :
    IsFiniteMeasure f.measure 
goal: MeasureTheory.IsFiniteMeasure f.measure

full_name: StieltjesFunction.isProbabilityMeasure
code: lemma isProbabilityMeasure (hf_bot : Tendsto f atBot (𝓝 0)) (hf_top : Tendsto f atTop (𝓝 1)) :
    IsProbabilityMeasure f.measure 
goal: MeasureTheory.IsProbabilityMeasure f.measure

full_name: StieltjesFunction.instIsLocallyFiniteMeasure
code: instance instIsLocallyFiniteMeasure : IsLocallyFiniteMeasure f.measure 
goal: MeasureTheory.IsLocallyFiniteMeasure f.measure

full_name: StieltjesFunction.eq_of_measure_of_tendsto_atBot
code: lemma eq_of_measure_of_tendsto_atBot (g : StieltjesFunction) {l : ℝ}
    (hfg : f.measure = g.measure) (hfl : Tendsto f atBot (𝓝 l)) (hgl : Tendsto g atBot (𝓝 l)) :
    f = g 
goal: f = g

full_name: StieltjesFunction.eq_of_measure_of_eq
code: lemma eq_of_measure_of_eq (g : StieltjesFunction) {y : ℝ}
    (hfg : f.measure = g.measure) (hy : f y = g y) :
    f = g 
goal: f = g

full_name: StieltjesFunction.measure_const
code: @[simp]
lemma measure_const (c : ℝ) : (StieltjesFunction.const c).measure = 0 
goal: (StieltjesFunction.const c).measure = 0

full_name: StieltjesFunction.measure_add
code: @[simp]
lemma measure_add (f g : StieltjesFunction) : (f + g).measure = f.measure + g.measure 
goal: (f + g).measure = f.measure + g.measure

full_name: StieltjesFunction.measure_smul
code: @[simp]
lemma measure_smul (c : ℝ≥0) (f : StieltjesFunction) : (c • f).measure = c • f.measure 
goal: (c • f).measure = c • f.measure