theorem extracted_formal_statement_295.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : Fin n → Ω) (i : ι)
  (h_original_goal :
    |∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ| ≤ ↑n * (b + b)) :
  ∀ᵐ (x : Fin n → Ω) ∂Measure.pi fun x => μ, |(|∑ k, (f i (X (ω k)) - f i (X (x k)))|)| ≤ ↑n * (b + b) := sorry

State before:
case refine_2.h.hf
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω
i : ι
⊢ |∫ (ω' : Fin n → Ω), |∑ k, (f i (X (ω k)) - f i (X (ω' k)))| ∂Measure.pi fun x => μ| ≤ ↑n * (b + b)

Tactic: apply abs_expectation_le_of_abs_le_const

State after:
[case refine_2.h.hf.h
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 X : Ω → Z
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω
 i : ι
 ⊢ ∀ᵐ (x : Fin n → Ω) ∂Measure.pi fun x => μ, |(|∑ k, (f i (X (ω k)) - f i (X (x k)))|)| ≤ ↑n * (b + b)]

Relevant lemmas:
abs_expectation_le_of_abs_le_const : ∀ {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_4 : IsProbabilityMeasure μ] {f : Ω → ℝ} {C : ℝ}, (∀ᵐ (x : Ω) ∂μ, |f x| ≤ C) → |∫ (x : Ω), f x ∂μ| ≤ C