theorem extracted_formal_statement_278.{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] [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] {X : Ω → Z} (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ}
  (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b) (ω : (Fin n → Ω) × (Fin n → Ω))
  (h_original_goal : ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ∑ σ, ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|)
  (i : Signs n) :
  i ∈ Finset.univ → ⨆ i_1, |∑ k, ↑↑(i k) * f i_1 (X (ω.1 k))| ≤ ⨆ i_1, ∑ k, |↑↑(i k) * f i_1 (X (ω.1 k))| := sorry

State before:
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
inst✝¹ : Nonempty ι
inst✝ : Countable ι
X : Ω → Z
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : (Fin n → Ω) × (Fin n → Ω)
⊢ ∑ σ, ⨆ i, |∑ k, ↑↑(σ k) * f i (X (ω.1 k))| ≤ ∑ σ, ⨆ i, ∑ k, |↑↑(σ k) * f i (X (ω.1 k))|

Tactic: apply Finset.sum_le_sum

State after:
[case 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
 inst✝¹ : Nonempty ι
 inst✝ : Countable ι
 X : Ω → Z
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : (Fin n → Ω) × (Fin n → Ω)
 ⊢ ∀ i ∈ Finset.univ, ⨆ i_1, |∑ k, ↑↑(i k) * f i_1 (X (ω.1 k))| ≤ ⨆ i_1, ∑ k, |↑↑(i k) * f i_1 (X (ω.1 k))|]

Relevant lemmas:
Finset.sum_le_sum : ∀ {ι : Type u_1} {N : Type u_5} [inst : OrderedAddCommMonoid N] {f g : ι → N} {s : Finset ι},
  (∀ i ∈ s, f i ≤ g i) → ∑ i ∈ s, f i ≤ ∑ i ∈ s, g i