/-- **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 := by
  let δ : ℝ≥0 := ⟨ε / 2, (half_pos εpos).le⟩
  have δpos : 0 < δ := half_pos εpos
  let fp : α → ℝ≥0 := fun x => Real.toNNReal (f x)
  have int_fp : Integrable (fun x => (fp x : ℝ)) μ := hf.real_toNNReal
  rcases exists_lt_lowerSemicontinuous_integral_gt_nnreal fp int_fp δpos with
    ⟨gp, fp_lt_gp, gpcont, gp_lt_top, gp_integrable, gpint⟩
  let fm : α → ℝ≥0 := fun x => Real.toNNReal (-f x)
  have int_fm : Integrable (fun x => (fm x : ℝ)) μ := hf.neg.real_toNNReal
  rcases exists_upperSemicontinuous_le_integral_le fm int_fm δpos with
    ⟨gm, gm_le_fm, gmcont, gm_integrable, gmint⟩
  let g : α → EReal := fun x => (gp x : EReal) - gm x
  have ae_g : ∀ᵐ x ∂μ, (g x).toReal = (gp x : EReal).toReal - (gm x : EReal).toReal := by
    filter_upwards [gp_lt_top] with _ hx
    rw [EReal.toReal_sub] <;> simp [hx.ne]
  refine' ⟨g, ?lt, ?lsc, ?int, ?aelt, ?intlt⟩
  case int =>
    show Integrable (fun x => EReal.toReal (g x)) μ
    rw [integrable_congr ae_g]
    convert gp_integrable.sub gm_integrable
    simp
  

Proof state:
case lt
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
hf : Integrable f
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := { val := ε / 2, property := (_ : 0 ≤ ε / 2) }
δpos : 0 < δ
fp : α → ℝ≥0 := fun x => Real.toNNReal (f x)
int_fp : Integrable fun x => ↑(fp x)
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ⊤
gp_integrable : Integrable fun x => ENNReal.toReal (gp x)
gpint : ∫ (x : α), ENNReal.toReal (gp x) ∂μ < ∫ (x : α), ↑(fp x) ∂μ + (fun a => ↑a) δ
fm : α → ℝ≥0 := fun x => Real.toNNReal (-f x)
int_fm : Integrable fun x => ↑(fm x)
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable fun x => ↑(gm x)
gmint : ∫ (x : α), ↑(fm x) ∂μ - (fun a => ↑a) δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x => ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, EReal.toReal (g x) = EReal.toReal ↑(gp x) - EReal.toReal ↑↑(gm x)
⊢ ∀ (x : α), ↑(f x) < g x
case lsc
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
hf : Integrable f
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := { val := ε / 2, property := (_ : 0 ≤ ε / 2) }
δpos : 0 < δ
fp : α → ℝ≥0 := fun x => Real.toNNReal (f x)
int_fp : Integrable fun x => ↑(fp x)
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ⊤
gp_integrable : Integrable fun x => ENNReal.toReal (gp x)
gpint : ∫ (x : α), ENNReal.toReal (gp x) ∂μ < ∫ (x : α), ↑(fp x) ∂μ + (fun a => ↑a) δ
fm : α → ℝ≥0 := fun x => Real.toNNReal (-f x)
int_fm : Integrable fun x => ↑(fm x)
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable fun x => ↑(gm x)
gmint : ∫ (x : α), ↑(fm x) ∂μ - (fun a => ↑a) δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x => ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, EReal.toReal (g x) = EReal.toReal ↑(gp x) - EReal.toReal ↑↑(gm x)
⊢ LowerSemicontinuous g
case aelt
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
hf : Integrable f
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := { val := ε / 2, property := (_ : 0 ≤ ε / 2) }
δpos : 0 < δ
fp : α → ℝ≥0 := fun x => Real.toNNReal (f x)
int_fp : Integrable fun x => ↑(fp x)
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ⊤
gp_integrable : Integrable fun x => ENNReal.toReal (gp x)
gpint : ∫ (x : α), ENNReal.toReal (gp x) ∂μ < ∫ (x : α), ↑(fp x) ∂μ + (fun a => ↑a) δ
fm : α → ℝ≥0 := fun x => Real.toNNReal (-f x)
int_fm : Integrable fun x => ↑(fm x)
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable fun x => ↑(gm x)
gmint : ∫ (x : α), ↑(fm x) ∂μ - (fun a => ↑a) δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x => ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, EReal.toReal (g x) = EReal.toReal ↑(gp x) - EReal.toReal ↑↑(gm x)
⊢ ∀ᵐ (x : α) ∂μ, g x < ⊤
case intlt
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
hf : Integrable f
ε : ℝ
εpos : 0 < ε
δ : ℝ≥0 := { val := ε / 2, property := (_ : 0 ≤ ε / 2) }
δpos : 0 < δ
fp : α → ℝ≥0 := fun x => Real.toNNReal (f x)
int_fp : Integrable fun x => ↑(fp x)
gp : α → ℝ≥0∞
fp_lt_gp : ∀ (x : α), ↑(fp x) < gp x
gpcont : LowerSemicontinuous gp
gp_lt_top : ∀ᵐ (x : α) ∂μ, gp x < ⊤
gp_integrable : Integrable fun x => ENNReal.toReal (gp x)
gpint : ∫ (x : α), ENNReal.toReal (gp x) ∂μ < ∫ (x : α), ↑(fp x) ∂μ + (fun a => ↑a) δ
fm : α → ℝ≥0 := fun x => Real.toNNReal (-f x)
int_fm : Integrable fun x => ↑(fm x)
gm : α → ℝ≥0
gm_le_fm : ∀ (x : α), gm x ≤ fm x
gmcont : UpperSemicontinuous gm
gm_integrable : Integrable fun x => ↑(gm x)
gmint : ∫ (x : α), ↑(fm x) ∂μ - (fun a => ↑a) δ ≤ ∫ (x : α), ↑(gm x) ∂μ
g : α → EReal := fun x => ↑(gp x) - ↑↑(gm x)
ae_g : ∀ᵐ (x : α) ∂μ, EReal.toReal (g x) = EReal.toReal ↑(gp x) - EReal.toReal ↑↑(gm x)
⊢ ∫ (x : α), EReal.toReal (g x) ∂μ < ∫ (x : α), f x ∂μ + ε

Next tactic:
case intlt =>
    show (∫ x : α, (g x).toReal ∂μ) < (∫ x : α, f x ∂μ) + ε;
    exact
      calc
        (∫ x : α, (g x).toReal ∂μ) = ∫ x : α, EReal.toReal (gp x) - EReal.toReal (gm x) ∂μ :=
          integral_congr_ae ae_g
        _ = (∫ x : α, EReal.toReal (gp x) ∂μ) - ∫ x : α, ↑(gm x) ∂μ := by
          simp only [EReal.toReal_coe_ennreal, ENNReal.coe_toReal]
          exact integral_sub gp_integrable gm_integrable
        _ < (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ∫ x : α, ↑(gm x) ∂μ := by
          apply sub_lt_sub_right
          convert gpint
          simp only [EReal.toReal_coe_ennreal]
        _ ≤ (∫ x : α, ↑(fp x) ∂μ) + ↑δ - ((∫ x : α, ↑(fm x) ∂μ) - δ) := (sub_le_sub_left gmint _)
        _ = (∫ x : α, f x ∂μ) + 2 * δ := by
          simp_rw [integral_eq_integral_pos_part_sub_integral_neg_part hf]; ring
        _ = (∫ x : α, f x ∂μ) + ε := by congr 1; field_simp [mul_comm]