Theo Buehler
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Registered User
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Mar 29 |
awarded | ● Good Answer |
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Jan 9 |
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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. |
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Jan 9 |
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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. |
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Jan 9 |
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Why are abelian groups amenable? fixed LaTeX |
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Jan 9 |
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Why are abelian groups amenable? (and the $F$ and $F_n$ must have positive measure, of course). |
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Jan 9 |
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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 $F_n$. |
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Jan 4 |
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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 $\Bbb{Z^N}$. |
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Jan 1 |
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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. |
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Jan 1 |
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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. |
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Jan 1 |
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Math Zeitgeist 2012 spelling of Agol's last name |
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Dec 30 |
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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. |
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Dec 30 |
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What do I need to understand Wiles' proof of Fermat’s Last Theorem? I believe this would be the relevant meta thread. meta.mathoverflow.net/discussion/1327/… |
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Dec 30 |
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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! |
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Dec 28 |
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Old books still used Maybe Dunford-Schwartz (1957) could be added here? |
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Dec 27 |
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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/… |
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Dec 27 |
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Why is a ring called a “ring”? Hi quid. It seems to me that the sources I give in the answer to where does the term “integral domain” come from? are somewhat relevant to your answer (bridging the gap between Hilbert and Moderne Algebra): math.stackexchange.com/q/45945 The history of the concept of rings (as opposed to the etymology) was also discussed in this thread: math.stackexchange.com/q/362 |
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Dec 27 |
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Why is a ring called a “ring”? Several lengthy discussion of parts of this question can be found on math.SE, e.g. in this thread: math.stackexchange.com/q/61497 and the threads mentioned there. |
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Dec 26 |
awarded | ● Good Answer |
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Dec 26 |
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Freyd-Mitchell’s embedding theorem @Axel: thanks again, I finally found the time for doing it. |
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Dec 26 |
revised |
Freyd-Mitchell’s embedding theorem fixed a typo noted by a reader, used markdown instead of abusing MathJax |
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Dec 26 |
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Lebesgue differentiation theorem beyond Euclidean spaces Fixed typo in a name, included links to the publishers websites dedicated to the recommended books. |
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Dec 26 |
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When is a space of measures a measurable space? Since you seem to be excited about the approach emphasized by Dmitri Pavlov, I'd like to point out to you that a lot of what DP is talking about in his answers is discussed in volume 3 of Fremlin's measure theory: essex.ac.uk/maths/people/fremlin/mt.htm (mostly in the notes and exercises). Instead of working with von Neumann algebras, Fremlin emphasizes Boolean and measure algebras and Riesz spaces where the translation to/from classical measure theory, probability theory and ergodic theory seems to become clearer, at least to someone with a more classical analytic slant like me. |
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Dec 26 |
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On similar concepts in mathematics whose similarity is a non-trivial fact. You rang...? I'm sorry to disappoint, I don't have all that much to say on this matter right now. We apparently noticed this analogy around the same time, and I agree that it is not totally trivial to make it precise. I will contact you in the first few weeks of January with some more specifics. Meanwhile, happy holidays and a happy new year to you! |
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Dec 26 |
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Topologizing the category of measure spaces @Ronnie Brown: It is possible to develop Mackey's theory of virtual groups along lines closely parallel to the approach to orbifolds via localizations of suitable categories of Lie groupoids. First steps in this direction were taken in the seventies mainly by Arlan Ramsay and more recently by Anantharaman-Delaroche and Renault in their book on amenable groupoids. None of the available references are very "algebraic" or "categorical", though. |
