be fixed. We denote by $ H_\mu(\bP^o) $ the isotropic subgroup at $ \bP^o $: $$H_\mu(\bP^o) = \{ g \in G : \, T_g(\bP^o) = \bP^o \} .$$ It is easy to see that $ H_\mu(\bP^o) $ is a closed subgroup of $ G $, and hence the quotient set $ G/H_\mu(\bP^o) $ is a compact Hausdorff set. Thus Theorem \[Gal1\] implies the following corollary, which reveals the homogeneous space structure of the set $ \ext $.

The action of $ (T_g : \, g \in G) $ restricted on $ \ext $ induces a homeomorphism: $$G/H_\mu(\bP^o) \stackrel{\sim}{\longrightarrow} \ext. 
\label{homeo}$$

The key to the proof of Theorem \[Gal1\] is the following lemma, which we follow the proof of Yamada–Watanabe’s theorem [@YW Proposition 1]. See also [@IW pp. 163–166].

We consider a product space $ G^{-\N} \times G^{-\N} \times G^{-\N} $ with its coordinate written as $ (\eta^1,\eta^2,\xi) $. Define $$\F^{\eta^1,\eta^2}_k 
= \sigma {\left(  \eta^1_j , \eta^2_j : \, j \le k  \right)} , 
\quad k \in -\N .$$

\[coupling\] Let $ \bP^1 $ and $ \bP^2 $ be two solutions in $ \sP_\mu $. Then there exists a probability measure $ \bQ $ on $ G^{-\N} \times G^{-\N} \times G^{-\N} 