File: Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean

full_name: MeasureTheory._aux_Mathlib_MeasureTheory_Integral_VitaliCaratheodory___unexpand_MeasureTheory_SimpleFunc_1
code: local infixr:25 " →ₛ " => SimpleFunc
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Integral_VitaliCaratheodory___macroRules_MeasureTheory_term_→ₛ__1»
code: local infixr:25 " →ₛ " => SimpleFunc
goal: Lean.Macro

full_name: MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge
code: /-- Given a simple function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
`lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ≥0) {ε : ℝ≥0∞}
    (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧
      (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
goal: ∃ g, (∀ (x : α), ↑f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑(↑f x) ∂μ + ε

full_name: MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
code: /-- Given a measurable function `f` with values in `ℝ≥0`, there exists a lower semicontinuous
function `g ≥ f` with integral arbitrarily close to that of `f`. Formulation in terms of
`lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf : Measurable f) {ε : ℝ≥0∞}
    (εpos : ε ≠ 0) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
goal: ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ + ε

full_name: MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge
code: /-- Given a measurable function `f` with values in `ℝ≥0` in a sigma-finite space, there exists a
lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
Formulation in terms of `lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α → ℝ≥0)
    (fmeas : Measurable f) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
goal: ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

full_name: MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable
code: /-- Given an almost everywhere measurable function `f` with values in `ℝ≥0` in a sigma-finite space,
there exists a lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
Formulation in terms of `lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite μ] (f : α → ℝ≥0)
    (fmeas : AEMeasurable f μ) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε 
goal: ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

full_name: MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal
code: /-- Given an integrable function `f` with values in `ℝ≥0` in a sigma-finite space, there exists a
lower semicontinuous function `g > f` with integral arbitrarily close to that of `f`.
Formulation in terms of `integral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : α → ℝ≥0)
    (fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → ℝ≥0∞,
      (∀ x, (f x : ℝ≥0∞) < g x) ∧
      LowerSemicontinuous g ∧
      (∀ᵐ x ∂μ, g x < ⊤) ∧
      Integrable (fun x => (g x).toReal) μ ∧ (∫ x, (g x).toReal ∂μ) < (∫ x, ↑(f x) ∂μ) + ε 
goal: ∃ g,
  (∀ (x : α), ↑(f x) < g x) ∧
    LowerSemicontinuous g ∧
      (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧
        MeasureTheory.Integrable (fun x => (g x).toReal) μ ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), ↑(f x) ∂μ + ε

full_name: MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le
code: /-- Given a simple function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
`lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ≥0)
    (int_f : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
      (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε 
goal: ∃ g, (∀ (x : α), g x ≤ ↑f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(↑f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε

full_name: MeasureTheory.exists_upperSemicontinuous_le_lintegral_le
code: /-- Given an integrable function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
`lintegral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f : (∫⁻ x, f x ∂μ) ≠ ∞)
    {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
    ∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
      (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε 
goal: ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε

full_name: MeasureTheory.exists_upperSemicontinuous_le_integral_le
code: /-- Given an integrable function `f` with values in `ℝ≥0`, there exists an upper semicontinuous
function `g ≤ f` with integral arbitrarily close to that of `f`. Formulation in terms of
`integral`.
Auxiliary lemma for Vitali-Carathéodory theorem `exists_lt_lower_semicontinuous_integral_lt`. -/
theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
    (fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → ℝ≥0,
      (∀ x, g x ≤ f x) ∧
      UpperSemicontinuous g ∧
      Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, ↑(g x) ∂μ 
goal: ∃ g,
  (∀ (x : α), g x ≤ f x) ∧
    UpperSemicontinuous g ∧
      MeasureTheory.Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ε ≤ ∫ (x : α), ↑(g x) ∂μ

full_name: MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt
code: /-- **Vitali-Carathéodory Theorem**: given an integrable real function `f`, there exists an
integrable function `g > f` which is lower semicontinuous, with integral arbitrarily close
to that of `f`. This function has to be `EReal`-valued in general. -/
theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
    {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → EReal,
      (∀ x, (f x : EReal) < g x) ∧
      LowerSemicontinuous g ∧
      Integrable (fun x => EReal.toReal (g x)) μ ∧
      (∀ᵐ x ∂μ, g x < ⊤) ∧ (∫ x, EReal.toReal (g x) ∂μ) < (∫ x, f x ∂μ) + ε 
goal: ∃ g,
  (∀ (x : α), ↑(f x) < g x) ∧
    LowerSemicontinuous g ∧
      MeasureTheory.Integrable (fun x => (g x).toReal) μ ∧
        (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε

full_name: MeasureTheory.exists_upperSemicontinuous_lt_integral_gt
code: /-- **Vitali-Carathéodory Theorem**: given an integrable real function `f`, there exists an
integrable function `g < f` which is upper semicontinuous, with integral arbitrarily close to that
of `f`. This function has to be `EReal`-valued in general. -/
theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
    {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → EReal,
      (∀ x, (g x : EReal) < f x) ∧
      UpperSemicontinuous g ∧
      Integrable (fun x => EReal.toReal (g x)) μ ∧
      (∀ᵐ x ∂μ, ⊥ < g x) ∧ (∫ x, f x ∂μ) < (∫ x, EReal.toReal (g x) ∂μ) + ε 
goal: ∃ g,
  (∀ (x : α), g x < ↑(f x)) ∧
    UpperSemicontinuous g ∧
      MeasureTheory.Integrable (fun x => (g x).toReal) μ ∧
        (∀ᵐ (x : α) ∂μ, ⊥ < g x) ∧ ∫ (x : α), f x ∂μ < ∫ (x : α), (g x).toReal ∂μ + ε