∀ {ι : Type u_1} {κ : Type u_2} {X : ι → Type u_5} {Y : κ → Type u_6} {mX : (i : ι) → MeasurableSpace (X i)}
  [inst : (i : ι) → TopologicalSpace (X i)] [∀ (i : ι), BorelSpace (X i)] [∀ (i : ι), HasOuterApproxClosed (X i)]
  {mY : (j : κ) → MeasurableSpace (Y j)} [inst_3 : (j : κ) → TopologicalSpace (Y j)] [∀ (j : κ), BorelSpace (Y j)]
  [∀ (j : κ), HasOuterApproxClosed (Y j)] [inst_6 : Fintype ι] [inst_7 : Fintype κ]
  {μ ν : MeasureTheory.Measure (((i : ι) → X i) × ((j : κ) → Y j))} [MeasureTheory.IsFiniteMeasure μ]
  [MeasureTheory.IsFiniteMeasure ν],
  (∀ (f : (i : ι) → BoundedContinuousFunction (X i) ℝ) (g : (j : κ) → BoundedContinuousFunction (Y j) ℝ),
      ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂μ =
        ∫ (p : ((i : ι) → X i) × ((j : κ) → Y j)), (∏ i, (f i) (p.1 i)) * ∏ j, (g j) (p.2 j) ∂ν) →
    μ = ν