-- Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
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

Proof state:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∫⁻ (a : α), ∑' (i : ℕ), ↑((SimpleFunc.eapproxDiff f i) a) ∂μ ≤ ∫⁻ (x : α), f x ∂μ

Tactic:
simp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl]

Result:
no goals

Proof state:
case intro.intro.refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
x : α
⊢ f x ≤ (fun x => ∑' (n : ℕ), ↑(g n x)) x

Tactic:
rw [← SimpleFunc.tsum_eapproxDiff f hf]

Result:
case intro.intro.refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
x : α
⊢ ∑' (n : ℕ), ↑((SimpleFunc.eapproxDiff f n) x) ≤ (fun x => ∑' (n : ℕ), ↑(g n x)) x

Proof state:
case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
this :
  ∀ (n : ℕ),
    ∃ g,
      (∀ (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g x) ∧
        LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ + ε

Tactic:
choose g f_le_g gcont hg using this

Result:
case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ + ε

Proof state:
case intro.intro.refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ LowerSemicontinuous fun x => ∑' (n : ℕ), ↑(g n x)

Tactic:
refine lowerSemicontinuous_tsum fun n => ?_

Result:
case intro.intro.refine_2
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
n : ℕ
⊢ LowerSemicontinuous fun x => ↑(g n x)

Proof state:
case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
⊢ ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ + ε

Tactic:
have :
  ∀ n,
    ∃ g : α → ℝ≥0,
      (∀ x, SimpleFunc.eapproxDiff f n x ≤ g x) ∧
        LowerSemicontinuous g ∧
          (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n :=
  fun n =>
  SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge μ (SimpleFunc.eapproxDiff f n)
    (δpos n).ne'

Result:
case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
this :
  ∀ (n : ℕ),
    ∃ g,
      (∀ (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g x) ∧
        LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∃ g, (∀ (x : α), f x ≤ g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ + ε

Proof state:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∑' (n : ℕ), ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ ≤ ∫⁻ (x : α), f x ∂μ

Tactic:
rw [← lintegral_tsum]

Result:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∫⁻ (a : α), ∑' (i : ℕ), ↑((SimpleFunc.eapproxDiff f i) a) ∂μ ≤ ∫⁻ (x : α), f x ∂μ

α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∀ (i : ℕ), AEMeasurable (fun x => ↑((SimpleFunc.eapproxDiff f i) x)) μ