Elek actually defines $\dim_G^{{\mathcal A}}(V)$ using Følner exhaustions, i.e. increasing Følner sequences $(F_{n \in \mathbb{N}})$ with $\bigcup_{n\in{{\mathbb N}}} F_n =G$. This makes no difference since the existence of the limit of $(\dim_k \bigl(V\vert_{F_n}\bigr)/|F_n|)_{n\in{{\mathbb N}}}$ for arbitrary Følner sequences (and thus its independence of the choice) follows from [@Lindenstrauss-Weiss(2000)]\*[Theorem 6.1]{}.

Let $M$ be a finitely generated left $kG$-module. The $k$-dual $M^\ast=\hom_k(M,k)$ (where $M$ and $k$ are viewed as left $k$-modules, and $(\phi a)m = \phi(am)$ for $\phi \in M^*$, $a \in k$ and $m\in M$) carries the natural right $G$-action $(\phi g)(m)=\phi(g m)$. The dual of the free left $kG$-module $kG^n$ is canonically isomorphic to ${\operatorname{map}}(G, k^n)$. Any left $kG$-surjection $f\colon kG^n\twoheadrightarrow M$ induces a right $kG$-injection $f^\ast\colon M^\ast\to{\operatorname{map}}(G, k^n)$ such that ${\operatorname{im}}(f^\ast)$ is a $G$-invariant $k$-subspace.

\[def:elek\_dimension\_function\] Let $M$ be a finitely generated left $kG$-module. Its *dimension in the sense of Elek* is defined by choosing a left $kG$-surjection $f\colon kG^n\twoheadrightarrow M$ and setting $$\label{eq:elek_dimension_and_average_dimension}
    \dim^{{\operatorname{Elek}}}_{kG}(M) =  \dim_G^{{\mathcal A}}\bigl({\operatorname{im}}(f^\ast)\bigr).$$

\[the:Eleks-dimension\] Let $G$ be a finitely generated amenable group. The definition  of $\dim^{{\operatorname{Elek}}}_{kG}(M)$ is independent of the choice of the surjection $f$, and $\dim^{{\operatorname{Elek}}}_{kG}$ has the following properties:

1.  $\dim^{{\operatorname{Elek}}}_{kG}(kG) = 1$;

2.  For any short exact sequence of finitely generated 