Z_n&\subset W_n. \label{eq:second_inclusion}
  \end{aligned}$$ Let $\phi\in W_n\cap Z_n$. Extend $\phi$ by zero to a function $\bar\phi:G\to k^s$. Let $y\in kG^r$. Then we can decompose $y$ as $y=y_0+y_1$ with ${\operatorname{supp}}(y_0)\subset F_n$ and ${\operatorname{supp}}(y_1)\subset
  G\backslash F_n$. By definition of the radius $R$ it is clear that ${\operatorname{supp}}(f(y_1))\subset G\backslash F_n\cup\partial_RF_n$. Because of $\phi\in Z_n$ we have $\langle f(y_1),\bar\phi\rangle=0$. The fact that $\phi\in W_n$ implies that $$\langle f(y_0), \bar\phi\rangle=\langle {\operatorname{pr}}_n\circ f\circ
  j_n(y_0), \phi\rangle=0.$$ So we obtain that $\langle
  f(y),\bar\phi\rangle=0$, meaning that $\phi\in V_n$. The proof of  is similar.

The following theorem is due to Weiss. Its proof can be found in [@Deninger-Schmidt(2007) Proposition 5.5].

\[the:Weiss\_Theorem\] Let $G$ be a countable amenable group. Let $G_n\subset G$, $n\ge 1$, be a sequence of normal subgroups of finite index with $\bigcap_{n\ge 1}G_n = \{1\}$. Then there exists, for every $R\ge
  1$ and every $\epsilon > 0$, an integer $M = M(R,\epsilon)\ge 1$ such that for $n \ge M$ there is a fundamental domain $Q_n\subset G$ of the coset space $G/G_n$ such that $$\frac{\bigl|\partial_R Q_n\bigr|}{\bigl|Q_n\bigr|} < \epsilon.$$

Now we are ready to prove Theorem \[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\]

According to Theorem \[the:Weiss\_Theorem\] let $(Q_n)_{n \ge 0}$ be a Følner sequence of $G$ such that $Q_n\subset G$ is a fundamental domain for 