theorem extracted_formal_statement_94.{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] {n : ℕ}
  [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) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C)
  (h_original_goal :
    AEStronglyMeasurable (fun a => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + c I)
      (Measure.pi fun x => μ.prod μ)) :
  Measurable fun a => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).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
n : ℕ
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
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
⊢ AEStronglyMeasurable (fun a => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).2)) + c I)
    (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
 n : ℕ
 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
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 ⊢ Measurable fun a => 2⁻¹ ^ n * ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (a i).1) - f I (X (a i).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