$ G $ is compact, we have the normalized Haar measure $ \nu $: That is, there exists a unique positive Borel measure $ \nu $ on $ G $ such that $ \nu(G)=1 $ and such that $$\int_G \phi(g h) \nu(dh) = 
\int_G \phi(h g) \nu(dh) = 
\int_G \phi(h) \nu(dh)$$ for any $ g \in G $ and any bounded measurable function $ \phi $ on $ G $.

A random variable $ U $ is said to be [*uniformly distributed on $ G $*]{} if the law of $ U $ is equal to the normalized Haar measure $ \nu $.

Now we restate Theorem \[Intro exist\].

\[YorTh\] For an arbitrary noise law $ \mu = {\left( \mu_k \right)} $, there exists a unique element $ \bP^*_{\mu} \in \sP_\mu $ whose marginal distributions are uniform: $$\lambda_k = \nu , 
\quad k \in -\N. 
\label{YorTh marginal}$$ Moreover, under $ \bP^*_{\mu} $, $$\mbox{each $ \eta_k $ is independent of the noise $ \F^{\xi}=\sigma(\xi_k : \, k \in -\N) $} . 
\label{YorTh indep}$$

In particular, the solution $ \bP^*_{\mu} $ is [*non-strong*]{} unless $ G $ is trivial: i.e. $ G = \{ {e}\} $.

Theorem \[YorTh\] immediately implies the following.

