Fabrizio Polo
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 Sep 24 awarded Autobiographer Jul 7 awarded Popular Question Mar 30 awarded Nice Question Mar 21 awarded Yearling May 20 comment Why is Lebesgue integration taught using positive and negative parts of functions? Completion of what? I understand how the extended reals can be seen as an order completion of ${\mathbb R}.$ But what does metric completion have to do with vector valued integrals? What are we metrically completing? ${\mathbb R}$ is already complete. The connection between order and monotone convergence makes sense. But the examples of the "metric" approach seem off topic. Are you saying convergence in measure and completeness of $L_p$ are more natural if you avoid the extended reals? I don't get it. May 18 answered Why is Lebesgue integration taught using positive and negative parts of functions? May 18 answered a.e. convergence of the powers of an operator built from rotations May 10 comment How can one characterise compactness-by-experiment? This was already covered in his question when he mentioned homotopy theory. May 10 comment Examples of common false beliefs in mathematics Remember being confused by this too. It became much clearer when I formally was taught about short exact sequences. Then you can see exactly the obstruction to such a decomposition. May 2 comment Two commuting mappings in the disk A related tangent: applying this other Kakutani Fixed-point theorem (en.wikipedia.org/wiki/Kakutani_fixed_point_theorem) to the multi-valued function $g^{-1} \circ f$ looks like it solves the problem... unfortunately the sets $f^{-1}( g(x))$ aren't convex (which is a pretty significant assumption!) Apr 14 comment 2, 3, and 4 (a possible fixed point result ?) @Ady: Since your question is whether or not $FP(2,3,4)$ holds, I'm confused. You seem to be claiming that BGK resolves your question. Apr 6 comment 2, 3, and 4 (a possible fixed point result ?) For fun, I started considering variants of this question but made no real progress on them either. Let ${\bf FP}(p,q,r)$ be the same statement with $2,3,4$ replaced by $p,q,r$. The BGK fixed point theorem gives us ${\bf FP}(p,p,p)$ for $1 < p < \infty.$ For what other values of $p,q,r$ can you prove or disprove this statement? Do you know of a counterexample to $FP(1,\infty, \infty)$? Mar 30 revised Topologizing free abelian groups added 26 characters in body Mar 30 revised Topologizing free abelian groups fixed the topology Mar 30 awarded Commentator Mar 30 comment Topologizing free abelian groups I've edited the uniqueness part to make it more convincing. Mar 30 revised Topologizing free abelian groups clarified uniqueness Mar 30 comment Topologizing free abelian groups Chris Schomer-Preis' comment made me think a little more. I was under the impression that my construction worked at least when $S$ was something nice like compact metric. If it goes wrong with a stranger space $S$ (as suggested by Dominguez below) then it would be really interesting to see some examples. Mar 30 revised Topologizing free abelian groups Added existence portion of argument Mar 30 answered Topologizing free abelian groups