, \ \ \int_{\NN_\psi[r]}  \abs{ \psi }^p \, d\mu  \leq \kappa \, r^{{\gamma}/2}.$$ In particular $\psi \in L^p(\mu)$.

[ [<span style="font-variant:small-caps;">Proof:</span> ]{}]{}Let $\psi \in \UU$ and $\NN:= \NN_\psi$. We may assume that $0 \leq e^\psi \leq 1$ by adding some constant to $\psi$. Let $r < 1/2$ and $\QQ_j:= \NN[r/2^j] \setminus \NN[r/2^{j+1}]$. Since $e^\psi \geq (r/2^{j+1})^\rho$ on $\QQ_j$, we obtain: $$\int_{\NN[r]}  \abs {\psi }^p \, d\mu  = \sum_{j \geq 0}
\int_{\QQ_j}  \abs{ \log e^\psi} ^p \, d\mu \leq \sum_{j \geq 0}
\left \vert \, \rho \log \left( {r \over 2^{j+1}} \right) \, \right  \vert ^p \cdot \mu (\QQ_j).$$ The inequalities $\mu (\QQ_j) \leq c (r/2^j)^{\gamma}$ and $\abs{\log {r \over 2^{j+1}}} =  (j+1) \log 2 + \log {1 \over r}
 \leq   (j+2)   \log {1 \over r}$ yield: $$\int_{\NN[r]}  \abs{ \psi }^p \, d\mu \leq  \left[ c \, \rho^p \,  \sum_{j \geq
  0} {(j+2)^p \over 2^{{\gamma}j}} \right]  \left( \log {1 \over r}
\right)^p   r^{\gamma}= M_{\rho,{\gamma}} \cdot \left(  \log {1 \over r} \right)^p  r^{{\gamma}/2} \cdot  r^{{\gamma}/2} .$$ The lemma follows with $\kappa:=   M_{\rho,{\gamma}} \cdot \sup_{ 0 < r < 1/2 } \left( \log {1 \over r} \right)^p r^{{\gamma}/2}$.

The Bernoulli space $(\Sigma,s,\nu)$ {#consaaa}
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We endow $\AA:= \{ 1, \ldots, d_t \}$ with 