Theo Buehler
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 Feb 20 awarded Good Answer Dec 18 awarded Good Answer Dec 14 awarded Guru Nov 25 awarded Yearling Feb 13 awarded Enlightened Jan 17 awarded Enlightened Jan 17 awarded Nice Answer Nov 25 awarded Yearling Aug 6 awarded Notable Question Nov 25 awarded Yearling Sep 9 awarded Good Question Jun 25 awarded Excavator Jun 25 awarded Pundit Mar 29 awarded Good Answer Jan 9 comment reference for “X compact <=> C_b(X) separable” (X metric space) I don't have a reference, but I'd suggest this argument: if $X$ is not compact, there is an infinite closed discrete subset $D$ of $X$. For every $A \subset D$ choose a continuous function $f_A \colon X \to [0,1]$ such that $f_A|_A = 1$ and $f_A|_{D \setminus A} = 0$ (by Urysohn). This gives an uncountable family of continuous bounded functions such that $\|f_A - f_B\| = 1$ whenever $A \neq B$. Alternatively, embed $\ell^\infty$ using a similar trick. If $X$ is compact then $C(X)$ is separable by Stone-Weierstrass. Jan 9 comment Why are abelian groups amenable? In the definition of $K$ you should replace $\lVert \Phi\rVert \leq 1$ by $\Phi(\chi_G) = 1$ (which together with $\Phi(F) \geq 0$ implies $\lVert \Phi \rVert = 1$). The way you define it, $0 \in K$ is always a fixed-point, whether $G$ is amenable or not. Jan 9 revised Why are abelian groups amenable? fixed LaTeX Jan 9 comment Why are abelian groups amenable? (and the $F$ and $F_n$ must have positive measure, of course). Jan 9 comment Why are abelian groups amenable? I'm not sure what exactly the deleted comments you responded to said, but it is not true that Følner limits you to discrete groups. A locally compact group $G$ with Haar measure $\mu$ is amenable if and only if for every compact $K$ and every $\varepsilon \gt 0$ there is a compact set $F$ such that $\mu(F \mathbin{\Delta} kF) \leq \varepsilon \mu(F)$ for all $k \in K$. If $G$ happens to be $\sigma$-compact then you can arrange $F$ to be from an exhaustive sequence $F_n$. Jan 4 comment About the definition of Borel and Radon measures Thanks! Meanwhile, I remembered an old construction due to Oxtoby dx.doi.org/10.1090/S0002-9947-1946-0018188-5 which produces a non-trivial (invariant and inner regular) Borel measure on every separable completely metrizable group. If the group is not locally compact then every open set has infinite measure, but there are always sets of finite measure, thus Borel regularity fails. This construction can be adapted to give a non-Borel regular Borel measure even on $\mathbb{R}$, by working on the set of irrationals and using that they are homeomorphic to $\Bbb{Z^N}$.