theorem extracted_formal_statement_127.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] [inst_2 : Countable ι]
  [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ} (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {n : ℕ}
  {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    AEStronglyMeasurable
      (fun ω =>
        2⁻¹ ^ (n + 1) *
          ∑ σ,
            ⨆ I,
              ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
                (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I))
      ((μ.prod μ).prod (Measure.pi fun x => μ.prod μ))) :
  Measurable fun ω =>
    2⁻¹ ^ (n + 1) *
      ∑ σ,
        ⨆ I,
          ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
            (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I) := sorry

State before:
case left
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
n : ℕ
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ AEStronglyMeasurable
    (fun ω =>
      2⁻¹ ^ (n + 1) *
        ∑ σ,
          ⨆ I,
            ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
              (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I))
    ((μ.prod μ).prod (Measure.pi fun x => μ.prod μ))

Tactic: apply Measurable.aestronglyMeasurable

State after:
[case left.hf
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 n : ℕ
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ Measurable fun ω =>
     2⁻¹ ^ (n + 1) *
       ∑ σ,
         ⨆ I,
           ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) +
             (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I)]

Relevant lemmas:
Measurable.aestronglyMeasurable : ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m m₀ : MeasurableSpace α}
  {μ : Measure α} {f : α → β} [inst_4 : MeasurableSpace β] [inst_5 : TopologicalSpace.PseudoMetrizableSpace β]
  [inst_6 : SecondCountableTopology β] [inst_7 : OpensMeasurableSpace β], Measurable f → AEStronglyMeasurable f μ