l3lab/ntp-mathlib huggingface dataset:
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by 

Proof state:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ ≠ ⊤

Next tactic:
simpa
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
⊢ ∃ g,
    (∀ (x : α), ↑(f x) < g x) ∧
      LowerSemicontinuous g ∧
        (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧
          (Integrable fun x => ENNReal.toReal (g x)) ∧ ∫ (x : α), ENNReal.toReal (g x) ∂μ < ∫ (x : α), ↑(f x) ∂μ + ↑ε

Next tactic:
have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
⊢ ∃ g,
    (∀ (x : α), ↑(f x) < g x) ∧
      LowerSemicontinuous g ∧
        (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧
          (Integrable fun x => ENNReal.toReal (g x)) ∧ ∫ (x : α), ENNReal.toReal (g x) ∂μ < ∫ (x : α), ↑(f x) ∂μ + ↑ε

Next tactic:
have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    filter_upwards [g_lt_top] with _ hx
    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    

Proof state:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
⊢ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ = ∫⁻ (a : α), g a ∂μ

Next tactic:
apply lintegral_congr_ae
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    

Proof state:
case h
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
⊢ (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =ᶠ[ae μ] fun a => g a

Next tactic:
filter_upwards [g_lt_top] with _ hx
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    filter_upwards [g_lt_top] with _ hx
    

Proof state:
case h
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
a✝ : α
hx : g a✝ < ⊤
⊢ ENNReal.ofReal (ENNReal.toReal (g a✝)) = g a✝

Next tactic:
simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    filter_upwards [g_lt_top] with _ hx
    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
  

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
Ig : ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ ∃ g,
    (∀ (x : α), ↑(f x) < g x) ∧
      LowerSemicontinuous g ∧
        (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧
          (Integrable fun x => ENNReal.toReal (g x)) ∧ ∫ (x : α), ENNReal.toReal (g x) ∂μ < ∫ (x : α), ↑(f x) ∂μ + ↑ε

Next tactic:
refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    filter_upwards [g_lt_top] with _ hx
    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
  refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
Ig : ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ Integrable fun x => ENNReal.toReal (g x)

Next tactic:
refine' ⟨gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    filter_upwards [g_lt_top] with _ hx
    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
  refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
  · refine' ⟨gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
    

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
Ig : ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ HasFiniteIntegral fun x => ENNReal.toReal (g x)

Next tactic:
simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
/-- 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 := by
  have fmeas : AEMeasurable f μ := by
    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable
    simp only [Real.toNNReal_coe]
  lift ε to ℝ≥0 using εpos.le
  obtain ⟨δ, δpos, hδε⟩ : ∃ δ : ℝ≥0, 0 < δ ∧ δ < ε; exact exists_between εpos
  have int_f_ne_top : (∫⁻ a : α, f a ∂μ) ≠ ∞ :=
    (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne
  rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable μ f fmeas
      (ENNReal.coe_ne_zero.2 δpos.ne') with
    ⟨g, f_lt_g, gcont, gint⟩
  have gint_ne : (∫⁻ x : α, g x ∂μ) ≠ ∞ := ne_top_of_le_ne_top (by simpa) gint
  have g_lt_top : ∀ᵐ x : α ∂μ, g x < ∞ := ae_lt_top gcont.measurable gint_ne
  have Ig : (∫⁻ a : α, ENNReal.ofReal (g a).toReal ∂μ) = ∫⁻ a : α, g a ∂μ := by
    apply lintegral_congr_ae
    filter_upwards [g_lt_top] with _ hx
    simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]
  refine' ⟨g, f_lt_g, gcont, g_lt_top, _, _⟩
  · refine' ⟨gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
    simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
    

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : WeaklyRegular μ
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable fun x => ↑(f x)
fmeas : AEMeasurable f
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
Ig : ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ ∫⁻ (a : α), ENNReal.ofReal (ENNReal.toReal (g a)) ∂μ < ⊤

Next tactic:
convert gint_ne.lt_top using 1