Q1. Are there "canonical" ways to construct Følner sequences for locally compact amenable groups? Possibly via representations or perhaps characters ?

To elaborate, let $G$ be a l.c. amenable group with the left Haar measure $\lambda$. We say the sequence $F_n$ of compact non-zero measure subsets with $G=\cup A_n$ is a Følner sequence if for any compact set $K$ and $\delta>0$ $$\lambda(KF_n\Delta F_n)\leq \delta \lambda(F_n),$$ for large enough $n$. People sometimes assume extra restrictions such as symmetry of$F_n$ and that each $F_n$ contains of the identity. But let us not worry for about that for now.

Q2. Are there ergodic theorems similar to Lindensrauss' for amenable groups with conditions on Følner sequences rather than Shulman's (that is $\lambda(\cup_{k<n} F^{-1}_kF_n)\leq C\lambda(F_n)$ for all $n$?)

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    $\begingroup$ Remark: if $G$ is not $\sigma$-compact then there is no such sequence: indeed, let $U$ be an open $\sigma$-compact subgroup containing $\bigcup F_n$. Then for all $g\notin U$, $gF_n\cap F_n$ is empty. In this generality, we need a net of Følner subsets rather than a sequence. $\endgroup$ – YCor Jan 17 '17 at 8:43
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    $\begingroup$ Second remark: the assumption that $F_n$ is mild since it can be achieved up to replace $F_n$ with a right translate. $\endgroup$ – YCor Jan 17 '17 at 8:44
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    $\begingroup$ Third remark: the assumption that $F_n$ is symmetric is not usual at all. Actually, there is a net of symmetric Følner subsets in $G$ if and only $G$ is amenable and unimodular. $\endgroup$ – YCor Jan 17 '17 at 8:46
  • $\begingroup$ Very interesting. I was not aware of the statement in your third remark and actually showed a little proposition stating that a certain amenable L.C group (non-unimodular) has no compact symmetric subset! Now Im trying to cook up a net(or sequence) satisfying Folner condition, a task that Greenleaf-Emerson couldn't accomplish. $\endgroup$ – BigM Jan 17 '17 at 14:27
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    $\begingroup$ Yes, this is called "compactly generated". In restriction to discrete groups it's called "finitely generated". It's quite a trivial exercise to construct countable groups that are not finitely generated (amenable if you like, e.g. abelian). An example is a locally finite countable group, that is a group $G$ that is an increasing union of finite subgroups $F_n$. In this case $(F_n)$ is Følner. $\endgroup$ – YCor Jan 17 '17 at 20:21

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