$ such that the following statements hold:\
(i) For $ i=1,2 $, the law on $ G^{-\N} \times G^{-\N} $ of $ (\eta^i,\xi) $ under $ \bQ $ coincides with that of $ (\eta,\xi) $ under $ \bP^i $ where $ \xi $ is defined in .\
(ii) For any $ k \in -\N $, $$\xi_k \mbox{ is independent of } \F^{\eta^1,\eta^2}_{k-1} \mbox{ under $ \bQ $} .$$

Let $ \bP^1_{\xi}(\cdot) $ and $ \bP^2_{\xi}(\cdot) $ denote the regular conditional probability given $ \F^{\xi}=\sigma(\xi_k: \, k \in -\N) $ such that $$\bP^i(\eta \in A , \xi \in B) = \int_B \bP^i_{\xi}(A) \mu(d\xi)$$ for arbitrary measurable sets $ A $ and $ B $ of $ G^{-\N} $, $ i=1,2 $. Then the desired probability measure $ \bQ $ is obtained as $$\bQ(d\eta^1 \, d\eta^2 \, d\xi) = \bP^1_{\xi}(d\eta^1) \, \bP^2_{\xi}(d\eta^2) \, \mu(d\xi) .$$ We can prove the claims (i) and (ii) in the same way as in the proof of Yamada–Watanabe’s theorem, so we omit the proofs of Lemma \[coupling\].

Let $ \bQ $ be the probability measure given in Lemma \[coupling\]. As far as the end of this paragraph, we omit to write “$ \bQ $-a.s." By Lemma \[coupling\] 