∀ {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {γ : Type u_8} {γ' : Type u_9} {_mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'}
  {_mγ : MeasurableSpace γ} {_mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'},
  ProbabilityTheory.IndepFun f g μ →
    AEMeasurable f μ →
      AEMeasurable g μ →
        AEMeasurable φ (MeasureTheory.Measure.map f μ) →
          AEMeasurable ψ (MeasureTheory.Measure.map g μ) → ProbabilityTheory.IndepFun (φ ∘ f) (ψ ∘ g) μ