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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 $\Bbb{Z^N}$.
Jan
1
comment The Practical Impact of Set-Theoretic Axioms on Measure Theory
A standard application of Martin's axiom en.wikipedia.org/wiki/Martin%27s_axiom is the existence of Banach limits satisfying some measurability conditions (medial means in the sense of Mokobodzki). See my answer to math.stackexchange.com/q/54554 for some details and references and part 4 of that answer for a basic sample application that might illustrate their power.
Jan
1
comment About the definition of Borel and Radon measures
What would be a good example of a non-Borel regular Borel measure on a second countable metric space? All the standard procedures for constructing measures I'm aware of seem to yield measures satisfying this condition. The examples I was able to produce either live on large (i.e. non-separable) spaces or fail to measure all Borel sets.
Jan
1
revised Math Zeitgeist 2012
spelling of Agol's last name
Dec
30
comment Continuum Hypothesis
The point of "naturality" is exactly what I meant by emphasizing "algebraic". Thanks a lot for all the additional pointers and all these interesting links in your answer.
Dec
30
comment What do I need to understand Wiles' proof of Fermat's Last Theorem?
I believe this would be the relevant meta thread. tea.mathoverflow.net/discussion/1327/closing-questions-as-spam
Dec
30
comment Continuum Hypothesis
Hi Andres. Some minor points: 1) The question asked especially about algebraic equivalents of CH. Do you happen to know one beyond the one mentioned by Mariano in another thread? 2) I think the original article by Erdős is very readable projecteuclid.org/euclid.mmj/1028999028 and crystal clear. 3) I would add Gödel's Monthly article jstor.org/stable/2304666 to recommended reading. 4) Since you brought up MA: As a non-set theorist who happened to need to understand it I can recommend Fremlin's book books.google.com/books?id=tXVrPwAACAAJ Happy New Year!
Dec
28
comment Old books still used
Maybe Dunford-Schwartz (1957) could be added here?
Dec
27
comment Automorphisms of higher-genus Riemann surfaces act nontrivially on homology (Reference Request)
Hurwitz's paper is freely available from the Göttinger Digitalisierungszentrum here: resolver.sub.uni-goettingen.de/purl?GDZPPN002253941