l3lab/ntp-mathlib huggingface dataset:
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  

Proof state:
case 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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, gint⟩
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, 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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
refine' ⟨g, fun x => _, gcont, _⟩
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, gint⟩
  refine' ⟨g, fun x => _, gcont, _⟩
  · 

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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
x : α
⊢ ↑(f x) < g x

Next tactic:
calc
      (f x : ℝ≥0∞) < f' x := by
        simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)
      _ ≤ g x := le_g x
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, gint⟩
  refine' ⟨g, fun x => _, gcont, _⟩
  · calc
      (f x : ℝ≥0∞) < f' x := 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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
x : α
⊢ ↑(f x) < f' x

Next tactic:
simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, gint⟩
  refine' ⟨g, fun x => _, gcont, _⟩
  · calc
      (f x : ℝ≥0∞) < f' x := by
        simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)
      _ ≤ g x := le_g x
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_2
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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
calc
      (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
      _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
        rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
      _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, gint⟩
  refine' ⟨g, fun x => _, gcont, _⟩
  · calc
      (f x : ℝ≥0∞) < f' x := by
        simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)
      _ ≤ g x := le_g x
  · calc
      (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
      _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := 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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) + ↑(w x) ∂μ + ε / 2 = ∫⁻ (x : α), ↑(f x) ∂μ + ∫⁻ (x : α), ↑(w x) ∂μ + ε / 2

Next tactic:
rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩
  let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal
      this with
    ⟨g, le_g, gcont, gint⟩
  refine' ⟨g, fun x => _, gcont, _⟩
  · calc
      (f x : ℝ≥0∞) < f' x := by
        simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)
      _ ≤ g x := le_g x
  · calc
      (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
      _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
        rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)
      _ = (∫⁻ x : α, f x ∂μ) + ε := 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
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x => ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2 + ε / 2 = ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
rw [add_assoc, ENNReal.add_halves]
/-- 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 := 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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  

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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩
  

Proof state:
case 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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩
  rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
  

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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : ↑↑μ s = 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
rcases exists_le_lowerSemicontinuous_lintegral_ge μ (s.indicator fun _x => ∞)
      (measurable_const.indicator smeas) this with
    ⟨g1, le_g1, g1_cont, g1_int⟩
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩
  rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ (s.indicator fun _x => ∞)
      (measurable_const.indicator smeas) this with
    ⟨g1, le_g1, g1_cont, g1_int⟩
  

Proof state:
case intro.intro.intro.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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : ↑↑μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), Set.indicator s (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), Set.indicator s (fun _x => ⊤) x ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Next tactic:
refine' ⟨fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _⟩
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩
  rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ (s.indicator fun _x => ∞)
      (measurable_const.indicator smeas) this with
    ⟨g1, le_g1, g1_cont, g1_int⟩
  refine' ⟨fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _⟩
  · 

Proof state:
case intro.intro.intro.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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : ↑↑μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), Set.indicator s (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), Set.indicator s (fun _x => ⊤) x ∂μ + ε / 2
x : α
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

Next tactic:
by_cases h : x ∈ s
/-- 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 := by
  have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
  rcases exists_lt_lowerSemicontinuous_lintegral_ge μ (fmeas.mk f) fmeas.measurable_mk this with
    ⟨g0, f_lt_g0, g0_cont, g0_int⟩
  rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with ⟨s, hs, smeas, μs⟩
  rcases exists_le_lowerSemicontinuous_lintegral_ge μ (s.indicator fun _x => ∞)
      (measurable_const.indicator smeas) this with
    ⟨g1, le_g1, g1_cont, g1_int⟩
  refine' ⟨fun x => g0 x + g1 x, fun x => _, g0_cont.add g1_cont, _⟩
  · by_cases h : x ∈ s
    · 

Proof state:
case pos
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
fmeas : AEMeasurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x => f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : ↑↑μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), Set.indicator s (fun _x => ⊤) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), Set.indicator s (fun _x => ⊤) x ∂μ + ε / 2
x : α
h : x ∈ s
⊢ ↑(f x) < (fun x => g0 x + g1 x) x

Next tactic:
have := le_g1 x