Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
⊢ ∑ x ∈ Finset.range n, dist (∫ (v : ℝ) in c..(I x).1, f v) (∫ (v : ℝ) in c..(I x).2, f v) ≤
    ∑ i ∈ Finset.range (n, I).1, ∫ (x : ℝ) in uIoc ((n, I).2 i).1 ((n, I).2 i).2, ‖f x‖ ∂volume.restrict (uIoc a b)

pp: f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
⊢ ∀ i ∈ Finset.range (n, I).1,
    IntegrableOn (fun x ↦ ‖f x‖) (uIoc ((n, I).2 i).1 ((n, I).2 i).2) (volume.restrict (uIoc a b))

Tactic:
]

Goal state after:
f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
⊢ ∑ x ∈ Finset.range n, dist (∫ (v : ℝ) in c..(I x).1, f v) (∫ (v : ℝ) in c..(I x).2, f v) ≤
    ∑ i ∈ Finset.range (n, I).1, ∫ (x : ℝ) in uIoc ((n, I).2 i).1 ((n, I).2 i).2, ‖f x‖ ∂volume.restrict (uIoc a b)

f : ℝ → ℝ
a b c : ℝ
h : IntervalIntegrable f volume a b
hc : c ∈ uIcc a b
s : ℕ × (ℕ → ℝ × ℝ) → Set ℝ := fun E ↦ ⋃ i ∈ Finset.range E.1, uIoc (E.2 i).1 (E.2 i).2
this✝¹ :
  Tendsto (fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this✝ :
  Tendsto (ENNReal.toReal ∘ fun i ↦ ∫⁻ (x : ℝ) in s i, ‖f x‖ₑ ∂volume.restrict (uIoc a b))
    (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)
this : ∀ᶠ (E : ℕ × (ℕ → ℝ × ℝ)) in totalLengthFilter ⊓ 𝓟 (disjWithin a b), E ∈ disjWithin a b
n : ℕ
I : ℕ → ℝ × ℝ
hnI : (n, I) ∈ disjWithin a b
hnI1 : ∀ i ∈ Finset.range (n, I).1, ((n, I).2 i).1 ∈ uIcc a b ∧ ((n, I).2 i).2 ∈ uIcc a b
hnI2 : (↑(Finset.range (n, I).1)).PairwiseDisjoint fun i ↦ uIoc ((n, I).2 i).1 ((n, I).2 i).2
⊢ ∀ i ∈ Finset.range (n, I).1,
    IntegrableOn (fun x ↦ ‖f x‖) (uIoc ((n, I).2 i).1 ((n, I).2 i).2) (volume.restrict (uIoc a b))