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 Feb9 comment noisy 1-second GPS data I voted to close as off-topic with the intention of migrating the question to stats.SE, but this did not happen, I don't know why. Feb9 comment Math and Logic of Infinite Chess I cast the third vote to undelete this post since the contest's results are published (linked in the comments to the question), so there seems to be no harm in making this answer available to everyone. Feb3 comment Non-isomorphic atomless Boolean algebras I suppose it's a matter of background and culture. I've seen reduced measure algebra used for the same thing (I think Givant/Halmos use that). However, it doesn't make much sense to me to use the term measure algebra as a synonym of $\sigma$-algebra and from this point of view $\Sigma/\mathcal{N}$ is pretty much the only possible interpretation. It seems that for analysts $\Sigma/\mathcal{N}$ is the interesting thing to consider, not $\Sigma$ itself. Feb3 comment Spectrum of the sum of two commuting matrices @Theorem: Section 11 in the current version is called Nullstellensätze and in any case the pdf is fully searchable... Feb3 comment Proof of inequality involving surds There were a few edits that seem to have changed/corrected mathematical content in the OP. I don't think it's a good idea to do that and reviewers should look more closely... Feb3 comment Non-isomorphic atomless Boolean algebras For me the measure algebra of a measure space is the algebra of measurable sets modulo the ideal $\mathcal{N}$ of null sets. Thus, $\mathfrak{A} = \mathfrak{L}/\mathcal{N} \cong \mathfrak{B} / (\mathfrak{B} \cap \mathcal{N})$ has cardinality $\mathfrak{c}$. Otherwise ccc wouldn't tell $\mathfrak{A}$ and $\mathfrak{B}$ apart, would it? :) Feb3 comment Is Topology an important class to take before Functional Analysis? Thanks for the feedback, that's very nice to hear and I really appreciate it. Good luck with your further studies! Dec26 comment Is Hom$(G,-)$ left exact if morphisms are required to be continuous? Hi Ben, I just stumbled over this thread: in view of the accepted answer, maybe you're interested in having a look at my expository paper on exact categories where I give a hands-on approach to Quillen's theory. Best wishes and a happy New Year, Theo Dec20 comment Inner regularity property of Radon measures in metric spaces You understand Michael's answer correctly. You can find proofs in many places, for example Theorem 3.2 in Parthasarathy's Probability Measures on Metric Spaces or Theorem 17.11 of Kechris's Classical Descriptive Set Theory. Dec20 comment Inner regularity property of Radon measures in metric spaces I see. Here's the idea: The example of a non-tight Radon measure on an uncountable disjoint union of $\mathbb{R}$ I linked you to can be modified to use the Cantor set with the "coin-flipping measure" instead of $\mathbb{R}$. Thus the task is essentially reduced to find a closed subspace of a metric space which is homeomorphic to an uncountable disjoint union of Cantor sets. This can be done using this construction and a locally finite disjoint family of open sets (and some effort). I'm not so sure if the actual construction is all that enlightening. Dec19 comment Inner regularity property of Radon measures in metric spaces I didn't work it out in full detail, but I think that one can use the example I alluded to above to show that on every non-separable complete metric space without isolated points there is an outer measure as in Evans-Gariepy, but that the associated measure (from Carathéodory) will not be tight while satisfying all your conditions. On the other hand, one can always construct a tight version of it which will fail to be outer regular. Dec19 comment Weak-to-weak continuous operator which is not norm-continuous @Mike: that's my favorite example, too :-) It may be worth adding that the summation functional is adjoint to the limit functional on $c$ (convergent sequences). That is, if you view $\ell^1$ as the dual space of $c$ via the pairing $\langle x,y \rangle_{\ell^1,c} = \left(\sum x_n\right) \lim y_n + \sum x_n y_n$ then $T$ is adjoint to $S(y) = (\lim y_n) \cdot e_1$. Dec19 comment Inner regularity property of Radon measures in metric spaces What kind of spaces are you interested in? Do you care about large spaces (not $\sigma$-compact)? Would you mind adding completeness? Have you considered the example of Lebesgue measure times counting measure on the reals times the discrete reals (it has two incarnations: one is inner regular but not outer regular and the other is outer regular but not inner regular)? Dec19 comment Why for a compact metric probability space, any Borel subset can be approximated by compact set? @Jochen: The passage on Wikipedia is correct in that they are talking about inner regularity with respect to closed sets as opposed to inner regularity with respect to compact sets. The former holds for every semi-finite Borel measure on a metric space while the latter needs additional assumptions such as the ones you note. Dec16 comment Abstract characterization of Borel $\sigma$-algebras If you are willing to enter the realm of measure algebras then there's no need of invoking Maharam's classification theorem. The much more elementary Stone representation theorem gives you an honest topological measure space whose measure algebra is (canonically isomorphic to) the one you started with. Since Jon is an operator algebraist it might be worth pointing out that the resulting space is the same as the Gelfand spectrum of $L^\infty(\mu)$. In this context variants of this result also go by the name of "Mackey's point realization theorem". Dec10 comment Does the open mapping theorem imply the Baire category theorem? @MattN. Thank you very much, but I agree with Asaf's assessment. A bounty would not be of use in this case. I doubt the answer is known at all, that's why I didn't take the question to MO either. Nov29 comment construction of a linear functional in $\mathcal{C}([0,1])$ Try again: $$f_\varepsilon(x) = \begin{cases} 1 & 0 \leq x \leq 1/2 - \varepsilon \\ \text{linear} & \text{on }1/2 - \varepsilon \lt x \lt 1/2+\varepsilon \\ -1 & 1/2 + \varepsilon \leq x \leq 1.\end{cases}$$ You should get $\varphi(f_\varepsilon) = 1 - \varepsilon$ while $\lVert f_\varepsilon \rVert = 1$, now let $\varepsilon \to 0$. Yes, no reflexivity needed: see here and here. Nov29 comment Is a convex function defined on a convex open subset of $\mathbb R^n$ continuous? @Jeff: See e.g. Theorem 3.3.1 in these notes Nov29 comment A strange ring category Sorry, I'm very late, but anyway: product of comm. rings = cartesian product, coproduct of comm. rings = tensor product. You were probably thinking of modules which is an additive category and finite coproducts and products are indeed equal. Nov29 comment construction of a linear functional in $\mathcal{C}([0,1])$ triangle inequality + $\lvert \int f \rvert \leq \int \rvert f\lvert$.